Hecke Actions on Loops and Periods of Iterated Shimura Integrals

TL;DR

This paper explores how classical Hecke correspondences act on conjugacy classes of the modular group and its profinite completion, inducing dual actions on rings of class functions related to iterated integrals of modular forms, with implications for mixed Hodge structures and Galois actions.

Contribution

It introduces a new framework for Hecke actions on conjugacy classes and their induced dual actions on rings of class functions associated with iterated modular integrals, revealing non-commutative algebraic structures.

Findings

Hecke correspondences act on conjugacy classes of SL_2(Z) and its profinite completion.

Induces a dual action on rings of class functions with mixed Hodge structures.

Hecke operators preserve Hodge structures and commute with Galois actions, but generate a non-commutative algebra.

Abstract

In this paper we show that the classical Hecke correspondences T_N, N>0, act on the free abelian groups generated by the conjugacy classes of the modular group SL_2(Z) and the conjugacy classes of its profinite completion. We show that this action induces a dual action on the ring of class functions of a certain relative unipotent completion of the modular group. This ring contains all iterated integrals of modular forms that are constant on conjugacy classes. It possesses a natural mixed Hodge structure and, after tensoring with Q_ell$, a natural action of the absolute Galois group. Each Hecke operator preserves this mixed Hodge structure and commutes with the action of the absolute Galois group. Unlike in the classical case, the algebra generated by these Hecke operators is not commutative. The appendix by Pham Tiep is not included. It can be found at arXiv:2303.02807.

Equations

$$0\to {\mathrm{Sym}}^{r_1}{V}_{f_1}\otimes \cdots \otimes {\mathrm{Sym}}^{r_m}{V}_{f_m}(d)\to E\to \mathbb{Q}(0)\to 0$$

$$\mathbf{\lambda }({\mathrm{SL}}_2(\mathbb{Z}))\to \mathbf{\lambda }({\mathrm{PSL}}_2(\mathbb{Z}))$$

$${\pi }_{\ast }:\mathbb{Z}\lambda (Y)\to \mathbb{Z}\lambda (X)\text{ and }{\pi }^{\ast }:\mathbb{Z}\lambda (X)\to \mathbb{Z}\lambda (Y),$$

$${\pi }_{\ast }:\mathbb{Z}\mathbf{\lambda }({\Gamma }^{\prime })\to \mathbb{Z}\mathbf{\lambda }(\Gamma )\text{ and }{\pi }^{\ast }:\mathbb{Z}\mathbf{\lambda }(\Gamma )\to \mathbb{Z}\mathbf{\lambda }({\Gamma }^{\prime })$$

$${\pi }_{\ast }{\pi }^{\ast }(\gamma )=\sum _j{\gamma }^{m_j}\gamma \in \mathbf{\lambda }(\Gamma ),$$

$$T_p:\mathbb{Z}\mathbf{\lambda }({\mathrm{SL}}_2(\mathbb{Z}))\to \mathbb{Z}\mathbf{\lambda }({\mathrm{SL}}_2(\mathbb{Z}))$$

$$\mathbb{Z}\mathbf{\lambda }({\mathrm{SL}}_2(\mathbb{Z}))\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\pi }^{\mathrm{op}}:\gamma \mapsto \left(\begin{array}{cc}{p}^{-1}& 0\\ 0& 1\end{array}\right){\gamma }^{-T}\left(\begin{array}{cc}p& 0\\ 0& 1\end{array}\right)$$

$${\mathbf{e}}_p:\mathbb{Z}\mathbf{\lambda }({\mathrm{SL}}_2(\mathbb{Z}))\to \mathbb{Z}\mathbf{\lambda }({\mathrm{SL}}_2(\mathbb{Z}))$$

$$\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$$

$${\mathbf{e}}_p({\sigma }_o^n)=\{\begin{array}{ll}p{\sigma }_o^n& p|n,\\ {\sigma }_o^{np}& p\nmid n.\end{array}$$

$$m_2(x)=x(x+1)(x-2)\text{ and }{m}_p(x)=x(x^2-1)(x-p)\text{ when p is odd}.$$

$$T_p(\alpha )=\sum _{\text{finite}}\text{classes of GL2(Q)-conjugates of powers of α},$$

$$T_p({\sigma }_o^n)={\sigma }_o^{np}+\{\begin{array}{ll}p{\sigma }_o^{n/p}& p|n,\\ {\sigma }_o^n& p\nmid n.\end{array}$$

$$T_{p^n}\circ T_p=T_{p^{n+1}}+T_{p^{n-1}}\circ {\mathbf{e}}_p.$$

$$T_{p^n}\circ T_p=T_{p^{n+1}}+p{T}_{p^{n-1}}$$

$$T_{p^2}=T_p^2-{\mathbf{e}}_p$$

$${\hat{\mathbb{T}}}_p:=\mathbb{Z}⟨T_p,{\mathbf{e}}_p⟩/(m_p({\mathbf{e}}_p))$$

$$T_{p^{n+1}}=T_{p^n}{T}_p-T_{p^{n-1}}{\mathbf{e}}_p.$$

$$\hat{\mathbb{T}}=\text{sideset}{}^{\prime }{⨂}_{p\text{ prime}}{\hat{\mathbb{T}}}_p:=\underset{N>0}{\underrightarrow{\lim }}\underset{p\le N}{⨂}{\hat{\mathbb{T}}}_p.$$

$$\mathbb{T}=\hat{\mathbb{T}}/({\mathbf{e}}_p-p:p\text{ prime})\cong \mathbb{Z}[T_p:p\text{ prime}].$$

$$T_N:\mathbb{Z}\mathbf{\lambda }({\mathrm{GL}}_n(\mathbb{Z}))\to \mathbb{Z}\mathbf{\lambda }({\mathrm{GL}}_n(\mathbb{Z}))\text{ and }\mathbb{Z}\mathbf{\lambda }({\mathrm{GL}}_n(\hat{\mathbb{Z}}))\to \mathbb{Z}\mathbf{\lambda }({\mathrm{GL}}_n(\hat{\mathbb{Z}})).$$

$$1\to U\to G\to R\to 1$$

$${\mathrm{SL}}_2\times {\mathrm{SL}}_2(\hat{\mathbb{Z}})$$

$$H^1(\mathfrak{u})\otimes \overline{\mathbb{Q}}=\underset{N}{\underrightarrow{\lim }}\underset{\chi ,m}{⨁}{H}^1(\Gamma (N),S^{m}H{)}_{\chi }\otimes S^{m}H$$

$${\mathrm{Sym}}^{r_1}{V}_{f_1}\otimes \cdots \otimes {\mathrm{Sym}}^{r_m}{V}_{f_m}(d),$$

$${\mathrm{Ext}}_{\mathsf{MHS}}^1(\mathbb{Q},{\mathrm{Sym}}^{r_1}{V}_{f_1}\otimes \cdots \otimes {\mathrm{Sym}}^{r_m}{V}_{f_m}(d)).$$

$${\check{T}}_N:\mathcal{C}\ell (\mathcal{G})\to \mathcal{C}\ell (\mathcal{G})$$

$$⟨\alpha ,{\check{T}}_{N}F⟩=⟨T_N\alpha ,F⟩\text{ and }⟨\alpha ,{\check{\mathbf{e}}}_{p}F⟩=⟨{\mathbf{e}}_p\alpha ,F⟩$$

$${\check{T}}_p\circ {\check{T}}_{p^n}={\check{T}}_{p^{n+1}}+{\check{\mathbf{e}}}_p\circ {\check{T}}_{p^{n-1}}.$$

Full text

In the appendix, Pham Tiep proves that for all primes $p\geq 5$, every irreducible character of ${\mathrm{SL}}_{2}({\mathbb{Z}}/p^{n})/(\pm\operatorname{id})$ appears in its conjugation action on the complex group algebra of ${\mathrm{SL}}_{2}({\mathbb{Z}}/p^{n})$, a result needed in the body of the paper.

ORCID: 0000-0002-7009-6971

with an appendix by pham huu tiep

1. 1 Introduction

2. 2 Preliminaries

3. I Unramified Correspondences

4. 3 Quick Review of Covering Space Theory

5. 4 Pushforward and Pullback

6. 5 Unramified Correspondences act on Conjugacy Classes

7. 6 The Profinite Case

8. 7 The Dual Action on Class Functions

9. II The Hecke Action on Conjugacy Classes of ${\mathrm{SL}}_{2}({\mathbb{Z}})$

10. 8 Conjugacy classes in ${\mathrm{SL}}_{2}({\mathbb{Z}})$

11. 9 The Hecke Action on ${\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}}))$

12. 10 The Action of $T_{p}$ on ${\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}}))$

13. 11 The Hecke Action in Higher Level, Higher Rank, etc

14. 12 The Hecke Action on ${\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}}))$ and its characters

15. III Relative Unipotent Completions of Modular Groups

16. 13 The Modular Curve as an Orbifold and as a Stack

17. 14 Fundamental Groups and the Galois Action on ${\mathrm{SL}}_{2}({\mathbb{Z}})^{\wedge}$

18. 15 Local systems and Connections on the Modular Curve

19. 16 Relative Unipotent Completion in the Abstract

20. 17 Relative Unipotent Completion of the Modular Group

21. 18 Variants

22. IV Class Functions from Relative Completions

23. 19 Class Functions

24. 20 Pushforward of Class Functions

25. 21 Hecke Action on ${\mathscr{C}\!\ell}({\mathcal{G}})$

26. 22 Constructing Class Functions

27. 23 Examples of Conjugation Invariant Iterated Integrals

28. 24 Filtrations

29. A The Conjugation Representation of ${\mathrm{SL}}_{2}({\mathbb{Z}}/p^{n})$

1. Introduction

In this paper we show that the action of the Hecke operators on invariants of modular curves, such as their cohomology groups, lifts to some mildly non-abelian invariants. More precisely, we show that the classical Hecke operators $T_{N}$ ($N\in{\mathbb{N}}_{+}$) act on the free abelian groups generated by the conjugacy classes of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ and the conjugacy classes of its profinite completion. We show that this action induces a dual action on those iterated integrals of modular forms (iterated Shimura integrals in Manin’s terminology [24]) — and their non-holomorphic generalizations — that are constant on conjugacy classes. These form a ring of class functions on ${\mathrm{SL}}_{2}({\mathbb{Z}})$ which possesses a natural mixed Hodge structure. Each Hecke operator preserves this mixed Hodge structure and commutes with the action of the absolute Galois group on the $\ell$-adic analogue of this ring. Unlike in the classical case, the algebra generated by these Hecke operators is not commutative.

The overall goal of the project is to use this Hecke action to understand periods of iterated Shimura integrals and the extensions (Galois and Hodge) of the form

[TABLE]

that occur in subquotients of the coordinate ring of relative unipotent completions of modular groups. These are expected to be motivic. The coordinate rings of such relative completions contain all iterated Shimura integrals. It is known by the work of Francis Brown on multiple modular values [4, Ex. 17.6] that all of the extensions of ${\mathbb{Q}}$ by $V_{f}(d)$ predicted by the conjectures of Beilinson [2, Conj. 3.4a] do occur in the coordinate ring of this completion of ${\mathrm{SL}}_{2}({\mathbb{Z}})$. This is implied by the fact that the periods of twice iterated integrals of Eisenstein series can contain periods of cusp forms, which was proved by Brown in [4]. It is hoped that almost all of the of the extensions (1) predicted by Beilinson’s conjectures occur in such coordinate rings.111Brown has shown that not all can occur — see [4, §17].

The construction of the Hecke action on conjugacy classes is elementary and natural. Denote the set of conjugacy classes of a discrete (or profinite) group ${\Gamma}$ by $\boldsymbol{\lambda}({\Gamma})$ and by ${\Bbbk}\boldsymbol{\lambda}({\Gamma})$ the free ${\Bbbk}$ module generated by it, where ${\Bbbk}$ is a commutative ring. The central objects of this paper are ${\Bbbk}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}}))$ and, dually, the functions on it that arise from conjugation invariant iterated integrals of modular forms.

Before proceeding, it is worth recalling the relation between conjugacy classes in ${\mathrm{SL}}_{2}({\mathbb{Z}})$ and closed geodesics on the modular curve, which we regard as an orbifold, or more accurately, a stack. The map

[TABLE]

is 2-to-1 with fibers $\pm\gamma$. Apart from the two conjugacy classes of elements of order 4 of ${\mathrm{SL}}_{2}({\mathbb{Z}})$, the two preimages of $\mu\in\boldsymbol{\lambda}({\mathrm{PSL}}_{2}({\mathbb{Z}}))$ are distinguished by the signs of their traces. The conjugacy classes of non-torsion elements of ${\mathrm{PSL}}_{2}({\mathbb{Z}})$ correspond to powers of oriented closed geodesics in the modular curve and powers of the horocycle. So a non torsion conjugacy class of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ corresponds to either a (not necessarily prime) closed geodesic or horocycle, together with the sign of its trace.

Theorem 1**.**

The classical Hecke operators $T_{N}$, $N\in{\mathbb{N}}$, act on ${\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}}))$. The operators $T_{N}$ and $T_{M}$ commute when $M$ and $N$ are relatively prime. These actions of the $T_{N}$ descend to ${\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{PSL}}_{2}({\mathbb{Z}}))$.

The basic observation behind the existence of this Hecke action is that if $\pi:Y\to X$ is a finite unramified cover of topological spaces, then there are pushforward and pullback maps

[TABLE]

where, for a topological space $X$, $\lambda(X)$ denotes the set of free homotopy classes of maps from the circle to $X$.222If $X$ is path connected, then $\lambda(X)=\boldsymbol{\lambda}(\pi_{1}(X,x))$. The pushforward map is simply composition with $\pi$; the pullback map takes a loop in $X$ to the sum of closed loops in $Y$ that cover its preimage under $\pi$. The precise definition can be found in Section 4.1.333This notion also occurs independently in [30], where it is called a transfer map. In particular, this gives a definition of pushforward and pullback maps

[TABLE]

associated to the inclusion ${\Gamma}^{\prime}\hookrightarrow{\Gamma}$ of a finite index subgroup by taking $X$ and $Y$ to be appropriate models of their classifying spaces $B{\Gamma}$ and $B{\Gamma}^{\prime}$. One finds that

[TABLE]

where the $m_{j}$ are positive integers that depend on $\gamma$ and whose sum is the degree of $\pi$, which is the index $|{\Gamma}:{\Gamma}^{\prime}|$.

When $p$ is a prime number, the Hecke operator

[TABLE]

is the map

[TABLE]

induced by the inclusions $\pi:{\Gamma}_{0}(p)\hookrightarrow{\mathrm{SL}}_{2}({\mathbb{Z}})$ and $\pi^{\mathrm{op}}:{\Gamma}_{0}(p)\hookrightarrow{\mathrm{SL}}_{2}({\mathbb{Z}})$, where

[TABLE]

Here $({\phantom{x}})^{-T}$ denotes inverse transpose and ${\Gamma}_{0}(N)$ is the subgroup of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ whose elements are upper triangular mod $N$.

Since $\pi_{\ast}\circ\pi^{\ast}$ is not simply multiplication by $\deg\pi$, the classical relation that expresses $T_{p^{n}}$ as a polynomial in $T_{p}$ no longer holds and has to be modified. To this end, for each prime number $p$, define

[TABLE]

to be the map $\pi_{\ast}\pi^{\ast}-\operatorname{id}$ associated to the inclusion ${\Gamma}_{0}(p)\hookrightarrow{\mathrm{SL}}_{2}({\mathbb{Z}})$. For example, if $\sigma_{o}$ is the class of

[TABLE]

then

[TABLE]

In particular, $\mathbf{e}_{p}$ is not, in general, multiplication by $p$. The operator $\mathbf{e}_{p}$ satisfies the polynomial relation $m_{p}(\mathbf{e}_{p})=0$, where

[TABLE]

It commutes with $\mathbf{e}_{q}$ for all primes $q$. (See Section 10.3.)

The general shape of the formula for the action of $T_{p}$ on a conjugacy class $\alpha$ is

[TABLE]

where each term of the sum lies in ${\mathrm{SL}}_{2}({\mathbb{Z}})$. For example,

[TABLE]

Theorem 2**.**

The actions of the Hecke operators $T_{p^{n}}$ on ${\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}}))$ satisfy

[TABLE]

This relation reduces to the familiar relation444See, for example, Prop. 10 in Chapter VII of [34]. For loops, $R_{p}=1$.

[TABLE]

that holds for invariants for which $\mathbf{e}_{p}=p$, such as cohomology and spaces of modular forms.

The formulas (2) and (4) for the action of $T_{p}$ and $\mathbf{e}_{p}$ on $\sigma_{o}$ imply that $T_{p}$ and $\mathbf{e}_{p}$ do not commute. (See Example 10.9.) This and the relation

[TABLE]

imply that $T_{p}$ does not commute with $T_{p^{2}}$, unlike in the classical case.

In view of this, it is natural to consider, for each prime number $p$, the quotient

[TABLE]

of the free associative algebra generated by symbols $T_{p}$ and $\mathbf{e}_{p}$, by the two-sided ideal generated by $m_{p}(\mathbf{e}_{p})$, where $m_{p}(x)$ is the polynomial (3). When $n>1$, one can define elements $T_{p^{n}}$ of ${\widehat{{\mathbb{T}}}}_{p}$ inductively by

[TABLE]

One can then define ${\widehat{{\mathbb{T}}}}$ to be the restricted tensor product

[TABLE]

This is a non-commutative generalization of the classical Hecke algebra ${\mathbb{T}}$ that acts on ${\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}}))$. The classical Hecke algebra is obtained from it by adding the relations $\mathbf{e}_{p}=p$:

[TABLE]

This suggests the purely topological question:

Question 1.1*.*

Is the Hecke action ${\widehat{{\mathbb{T}}}}\to\operatorname{End}{\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}}))$ is injective? Or are there universal relations that hold between the $T_{p}$ and $\mathbf{e}_{p}$?

For each $m\geq 0$ one also has the Adams operator $\psi^{m}:{\mathbb{Z}}\boldsymbol{\lambda}(\Gamma)\to{\mathbb{Z}}\boldsymbol{\lambda}(\Gamma)$, which takes the class of $\gamma\in{\Gamma}$ to the class of $\gamma^{m}$.

For each appropriate choice of a base point of the modular curve, there is a natural action of the absolute Galois group on the profinite completion ${\mathrm{SL}}_{2}({\mathbb{Z}})^{\wedge}$ of ${\mathrm{SL}}_{2}({\mathbb{Z}})$. (See Section 14.) It induces a Galois action on ${\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}})^{\wedge})$ which commutes with the Adams operators. This action does not depend on the choice of a base point. The group theoretic description of the action of ${\widehat{{\mathbb{T}}}}$ on ${\mathbb{Z}}\lambda({\mathrm{SL}}_{2}({\mathbb{Z}}))$ implies that ${\widehat{{\mathbb{T}}}}$ also acts on ${\mathbb{Z}}\lambda({\mathrm{SL}}_{2}({\mathbb{Z}})^{\wedge})$.

Theorem 3**.**

There is an action of ${\widehat{{\mathbb{T}}}}$ on ${\mathbb{Z}}({\mathrm{SL}}_{2}({\mathbb{Z}})^{\wedge})$. This action extends the action of ${\widehat{{\mathbb{T}}}}$ on ${\mathbb{Z}}({\mathrm{SL}}_{2}({\mathbb{Z}})^{\wedge})$ in the sense that the canonical map ${\mathbb{Z}}({\mathrm{SL}}_{2}({\mathbb{Z}}))\to{\mathbb{Z}}({\mathrm{SL}}_{2}({\mathbb{Z}})^{\wedge})$ is ${\widehat{{\mathbb{T}}}}$ invariant. The ${\widehat{{\mathbb{T}}}}$ action on ${\mathbb{Z}}({\mathrm{SL}}_{2}({\mathbb{Z}})^{\wedge})$ commutes with the action of the absolute Galois group on it.

Note that it is not necessary to complete the coefficient ring at this stage. If this seems odd, note that the absolute Galois group acts on ${\mathrm{SL}}_{2}({\mathbb{Z}})^{\wedge}$, and thus on its integral group ring ${\mathbb{Z}}[{\mathrm{SL}}_{2}({\mathbb{Z}})^{\wedge}]$.

Similarly one can define Hecke operators for lattices in higher rank groups. For example, one has Hecke operators

[TABLE]

One difference with the rank 1 case is that, when $n\geq 3$, the congruence kernel of ${\mathrm{SL}}_{n}({\mathbb{Z}})$ is trivial by [1], so that the profinite completion of ${\mathrm{SL}}_{n}({\mathbb{Z}})$ is ${\mathrm{SL}}_{n}({\widehat{{\mathbb{Z}}}})$ unlike in the case of ${\mathrm{SL}}_{2}({\mathbb{Z}})$. (See [25].)

1.1. Relative completions of the modular group

Relative unipotent completion (or relative completion for short) of a discrete or profinite group replaces it by an affine group scheme over a field of characteristic zero. In more classical language, relative completion replaces a discrete or profinite group by a proalgebraic group. (Relative completion is reviewed in Section 16. A gentler introduction can be found in [13].)

In characteristic zero, every affine group $G$ is an extension

[TABLE]

of a proreductive group $R$ by a prounipotent group $U$. Every finite group can be regarded as a reductive algebraic group, so every profinite group can be regarded as a proreductive group. The completion ${\mathcal{G}}$ of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ that we consider in this paper is constructed in Section 17. It is a ${\mathbb{Q}}$-group that is an extension of the proreductive ${\mathbb{Q}}$-group

[TABLE]

by a prounipotent group ${\mathcal{U}}$ whose Lie algebra ${\mathfrak{u}}$ is free as a pronilpotent Lie algebra555This is a proalgebraic analogue of a result of Mel’nikov [25] that sates that the kernel of ${\mathrm{SL}}_{2}({\mathbb{Z}})^{\wedge}\to{\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$ is a countably generated free profinite group. and has abelianization $H_{1}({\mathfrak{u}})$ which, after tensoring with ${\overline{{\mathbb{Q}}}}$, is canonically dual (as an ${\mathrm{SL}}_{2}\times{\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$ module) to

[TABLE]

Here ${\Gamma}(N)$ denotes the full level $N$ subgroup of ${\mathrm{SL}}({\mathbb{Z}})$, $H$ denotes the fundamental representation of ${\mathrm{SL}}_{2}$, $S^{m}H$ its $m$th symmetric power, and $\chi$ an irreducible character of ${\mathrm{SL}}_{2}({\mathbb{Z}}/N)$. The subscript $\chi$ on the cohomology group, signifies the $\chi$ isotypical summand. There is a canonical Zariski dense homomorphism ${\mathrm{SL}}_{2}({\mathbb{Z}})\to{\mathcal{G}}({\mathbb{Q}})$ whose composition with the projection to ${\mathrm{SL}}_{2}({\mathbb{Q}})\times{\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$ is the diagonal inclusion.

The coordinate ring of ${\mathcal{G}}$ has Betti, de Rham and $\ell$-adic étale incarnations, which we denote by ${\mathcal{G}}^{B}_{\mathbb{Q}}$, ${\mathcal{G}}^{\mathrm{DR}}_{\mathbb{Q}}$, ${\mathcal{G}}^{\mathrm{\acute{e}t}}_{{\mathbb{Q}}_{\ell}}$. The first two are ${\mathbb{Q}}$-groups and the third is defined over ${{\mathbb{Q}}_{\ell}}$. These are related by natural comparison isomorphisms. (See Section 17.) Their coordinate rings are expected to be direct limits of realizations of Voevodsky motives, but that it not known at the present time. For this reason we work with “compatible realizations”.

The coordinate rings of each these incarnations of ${\mathcal{G}}$ carries a natural weight filtration. Each $\ell$-adic étale incarnation has a natural action of the absolute Galois group. Elements of the coordinate ring of ${\mathcal{G}}^{\mathrm{DR}}$ include all of Manin’s iterated Shimura integrals [24] of all levels.

Theorem 4**.**

The coordinate ring ${\mathcal{O}}({\mathcal{G}})$ of this relative completion of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ carries a natural ind mixed Hodge structure with non-negative weights. Each of its $\ell$-adic incarnations ${\mathcal{O}}({\mathcal{G}}^{{\mathrm{\acute{e}t}}_{\ell}}_{{\mathbb{Q}}_{\ell}})$ has a natural action of the absolute Galois group $\operatorname{Gal}({\overline{{\mathbb{Q}}}}/{\mathbb{Q}})$. The product, coproduct and antipode respect the Hodge and Galois structures.

Here we have suppressed the role of the base point, which sometimes plays an important role. Our default choice is the tangential base point $\partial/\partial q$, which corresponds to the first order smoothing of the nodal cubic given by the Tate curve. Precise statements can be found in Section 14.

1.2. Class functions and the dual Hecke action

The ring ${\mathscr{C}\!\ell}({\mathcal{G}})$ of class functions on ${\mathcal{G}}$ is, by definition, the subspace of ${\mathcal{O}}({\mathcal{G}})$ that consists of those functions that are invariant under conjugation and thus constant on conjugacy classes of elements of ${\mathrm{SL}}_{2}({\mathbb{Z}})$. It has Betti, de Rham and $\ell$-adic manifestations. As we show in Section 22, it is very large. In particular, it contains the subring of conjugation invariant iterated Shimura integrals, which is not finitely generated. The Hodge and Galois structures on ${\mathcal{O}}({\mathcal{G}})$ described in Theorem 4 restrict to ${\mathscr{C}\!\ell}({\mathcal{G}})$.

Proposition 5**.**

The ring ${\mathscr{C}\!\ell}({\mathcal{G}})$ of class functions on ${\mathcal{G}}$ carries a natural mixed Hodge structure and each of its $\ell$-adic incarnations ${\mathscr{C}\!\ell}({\mathcal{G}}^{{\mathrm{\acute{e}t}}_{\ell}}_{{\mathbb{Q}}_{\ell}})$ has a natural action of $\operatorname{Gal}({\overline{{\mathbb{Q}}}}/{\mathbb{Q}})$. Neither of these structures depends on the choice of a base point. The Adams operators $\psi^{m}$, $m\in{\mathbb{N}}$ are morphisms of MHS and commute with the Galois action.

The weight graded quotients of ${\mathscr{C}\!\ell}({\mathcal{G}})$ are sums of “motives” of the form

[TABLE]

where $V_{f}$ is the “motive” of a Hecke eigenform $f$. Consequently, the tannakian subcategory of (say) the category of mixed Hodge structures generated by ${\mathscr{C}\!\ell}({\mathcal{G}})$ determines many elements of

[TABLE]

The hope is that the extensions that occur conform to Beilinson’s conjectures, subject to Brown’s constraint [4, §17].

Our main result asserts that the Hecke operators act on ${\mathscr{C}\!\ell}({\mathcal{G}})$.

Theorem 6**.**

Each Hecke correspondence $T_{N}$ induces a (dual) Hecke operator

[TABLE]

which is a morphism of mixed Hodge structures and, after tensoring with ${{\mathbb{Q}}_{\ell}}$, is Galois equivariant. The dual operators satisfy

[TABLE]

for all $N\geq 1$, all prime numbers $p$ and all $F\in{\mathscr{C}\!\ell}({\mathcal{G}})$. The operators ${\check{T}}_{N}$ and ${\check{T}}_{M}$ commute when $N$ and $M$ are relatively prime. For all primes $p$ we have the relation

[TABLE]

dual to (5).

Define the dual Hecke algebra ${\widehat{{\mathbb{T}}}}^{\mathrm{op}}$ to be the opposite ring of ${\widehat{{\mathbb{T}}}}$. It is generated by the ${\check{T}}_{p}$ and ${\check{\mathbf{e}}}_{p}$. The previous result says that ${\widehat{{\mathbb{T}}}}^{\mathrm{op}}$ acts on ${\mathscr{C}\!\ell}({\mathcal{G}})$. It also implies that ${\widehat{{\mathbb{T}}}}^{\mathrm{op}}$ acts on the ring of motivic periods (in the sense of Brown [4, §15]) of ${\mathscr{C}\!\ell}({\mathcal{G}})$. These are, by definition, formal linear combinations of symbols $[{\mathscr{C}\!\ell}({\mathcal{G}});\alpha,F]$, where $\alpha\in\lambda({\mathrm{SL}}_{2}({\mathbb{Z}}))$ and $F\in{\mathscr{C}\!\ell}({\mathcal{G}}^{\mathrm{DR}}_{\mathbb{Q}})$, that are subject to some basic relations. Such periods can be thought of as unevaluated iterated integrals. The dual Hecke operator ${\check{T}}_{N}$ acts on them by the formula

[TABLE]

One hope is that the Hecke action on the motivic periods of ${\mathscr{C}\!\ell}({\mathcal{G}})$ — which include all multiple zeta values by Saad’s thesis [32] — will help illuminate their structure and their relation to periods of modular forms.

The most basic elements of ${\mathscr{C}\!\ell}({\mathcal{G}})$ that do not come from class functions on ${\mathrm{SL}}_{2}\times{\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$ are of the form

[TABLE]

where $f$ is a modular form of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ of weight $2n$ and level 1, ${\omega}_{f}$ is the corresponding $S^{2n-2}H$ valued 1-form on the modular curve, $\varphi:S^{2n-2}H\to{\mathcal{O}}({\mathrm{SL}}_{2})$ is an ${\mathrm{SL}}_{2}$ invariant map, where ${\mathrm{SL}}_{2}$ acts on its coordinate ring by conjugation, and ${\omega}_{f}(\varphi):=\varphi\circ{\omega}_{f}$. One computes the value of this integral by first integrating ${\omega}_{f}(\varphi)$ over the loop $\alpha$ to obtain an element of ${\mathcal{O}}({\mathrm{SL}}_{2})$ and then evaluating the result on $\alpha$.

More generally, the cyclic iterated integrals

[TABLE]

of such forms are also in ${\mathscr{C}\!\ell}({\mathcal{G}})$, where $C_{r}$ denotes the cyclic group generated by $(1,2,\dots,r)$. These constructions are discussed in detail in Section 22.3.

Computing the action of the Hecke operators on the motivic periods appears to be a deep and difficult problem. However, we do work out one class of examples in Section 23.6, where we compute the action of $T_{p}$ on certain periods of the class functions $F_{f,\varphi}$ defined above in (8). We show that if $\alpha\in{\mathrm{SL}}_{2}({\mathbb{Z}})$ acts on ${\mathbb{P}}^{1}({{\mathbb{F}}_{p}})$ with one orbit, then

[TABLE]

where $\psi^{p+1}(\varphi)$ is the action of the $(p+1)$st Adams operator on $\varphi$. In particular, if $f$ is a Hecke eigenform with rational Fourier coefficients, then $[{\mathscr{C}\!\ell}({\mathcal{G}});\alpha,F_{f,\varphi}]$ will be an “eigenperiod” of $T_{p}$ with “eigenvalue” a rational multiple of $\psi^{p+1}$.

1.3. Mumford–Tate groups

It is important to know that the mixed Hodge structure on ${\mathscr{C}\!\ell}({\mathcal{G}})$ is as rich as possible — that it generates an interesting subcategory of the tannakian category of mixed Hodge structures. The richness of a mixed Hodge structure is measured by its Mumford–Tate group.

Recall that the Mumford–Tate group ${\mathrm{MT}}_{V}$ of a ${\mathbb{Q}}$ mixed Hodge structure $V$ is the image of the homomorphism

[TABLE]

where ${\mathsf{MHS}}_{\mathbb{Q}}$ is the category of graded polarizable ${\mathbb{Q}}$ mixed Hodge structures, ${\omega}$ is the fiber functor that takes a MHS to its underlying ${\mathbb{Q}}$ vector space, and $\pi_{1}({\mathsf{MHS}}_{\mathbb{Q}},{\omega})$ is its tannakian fundamental group.

One goal of Brown’s program [4] of mixed modular motives is to understand the Mumford–Tate group of ${\mathcal{O}}({\mathcal{G}})$. Proposition 5 and Theorem 6 imply that the action of the Adams and Hecke operators bound the Mumford–Tate group of ${\mathscr{C}\!\ell}({\mathcal{G}})$.

Question 1.2*.*

Is ${\mathrm{MT}}_{{\mathscr{C}\!\ell}({\mathcal{G}})}$ the group of automorphisms of ${\mathscr{C}\!\ell}({\mathcal{G}})$ that commute with all Hecke and Adams operators?

This raises the question of whether the MHS on ${\mathscr{C}\!\ell}({\mathcal{G}})$ is as rich as the MHS on ${\mathcal{O}}({\mathcal{G}})$. In other words, is the natural homomorphism

[TABLE]

is an isomorphism? One might optimistically conjecture that it is based partly on the fact that, when ${\mathcal{V}}$ is the unipotent completion of the fundamental group of a smooth affine curve, the restriction mapping

[TABLE]

is an isomorphism when restricted to prounipotent radicals (i.e., to $W_{-1}{\mathrm{MT}}$). This follows from [15, Cor. 3] and [21, Thm. 4.3].

1.4. Overview

The paper is in four Parts. The first two are elementary. The action of “unramified correspondences” on conjugacy classes of fundamental groups is constructed in Part 1 in both the discrete and profinite settings. The action of the Hecke operators on ${\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}}))$ is constructed in Part 2 and the basic relations between them are established. In Section 10 we give an explicit formula for the action of $T_{p}$ on the elliptic and parabolic conjugacy classes of ${\mathrm{SL}}_{2}({\mathbb{Z}})$, compute the minimal polynomial of $\mathbf{e}_{p}$ and also describe the action of $T_{p}$ on hyperbolic classes. This part concludes with a discussion of the Hecke action on the class functions of ${\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$.

The main task of Part 3 is to construct the relative unipotent completion of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ needed in later sections. In order to accommodate the Hecke action, it is necessary to use a version of relative completion that is larger than the ones used to date in, for example, [4, 13]. Its various incarnations — Betti, de Rham and $\ell$-adic étale — and the comparison isomorphisms between them are constructed in the final section of this part, Section 17. We construct the natural action of the absolute Galois group on each of its $\ell$-adic incarnations as well as the canonical mixed Hodge structure on its coordinate ring. This part begins with a detailed discussion of the modular curve and local systems and connections over it needed in later sections. For example, we construct the action of the absolute Galois group on the profinite completion of ${\mathrm{SL}}_{2}({\mathbb{Z}})^{\wedge}$ in Section 14. Section 16 is a terse review of of relative completion in which we make minor improvements to results in the existing literature.

In Part 4, we show that the Hecke operators act on the class functions ${\mathscr{C}\!\ell}({\mathcal{G}})$ of the relative completion ${\mathcal{G}}$ of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ constructed in Part 3 and show that the mixed Hodge and Galois structures on ${\mathcal{O}}({\mathcal{G}})$ induce a mixed Hodge structure on ${\mathscr{C}\!\ell}({\mathcal{G}})$, as well as a Galois action on each of its $\ell$-adic realizations. We show that each Hecke operator preserves these structures.

An algebraic description of ${\mathscr{C}\!\ell}({\mathcal{G}})$ is given in Section 22 as well as techniques for constructing non-trivial elements of it. These techniques are applied in Section 23 to construct explicit elements of ${\mathscr{C}\!\ell}({\mathcal{G}})$ from modular forms. This section concludes with a computation of the Hecke action on the class functions associated to Hecke eigen forms.

The ring ${\mathscr{C}\!\ell}({\mathcal{G}})$ has several natural filtrations, including the weight filtration, the length filtration, the filtration by level, and a filtration coming from the representation theory of ${\mathrm{SL}}_{2}$, which we call the modular filtration. These are described in Section 24 along with their finiteness properties and their behaviour under the Hecke operators.

Acknowledgments: This work originated during a sabbatical visit to the Institute for Advanced Study in 2014–15, where I had many inspiring discussions with Francis Brown on periods of iterated integrals of modular forms and his “multiple modular value” program. I am especially grateful to him for his interest in this work and the many stimulating discussions we had at IAS and elsewhere. I would like to thank both IAS and Duke University for support during that sabbatical during which this work was born.

Special thanks goes to Pham Tiep who enthusiastically worked on, and almost completely resolved, the problem of determining the characters of ${\mathrm{SL}}_{2}({\mathbb{Z}}/N)$ that appear in its “adjoint representation”. His result appears as Theorem 23.5 and is proved in the Appendix. It is needed to ensure that there are large families of class functions on ${\mathrm{SL}}_{2}({\mathbb{Z}})$ that come from modular forms of higher level. I am also grateful to Florian Naef who long ago contributed a statement equivalent to the $m=1$ case of Lemma 22.5. This helped point me in the right direction when I was trying to understand the class functions on the relative completion of ${\mathrm{SL}}_{2}({\mathbb{Z}})$. Finally, I am grateful to an anonymous reviewer of an early version for pointing out that the operators $\mathbf{e}_{p}$ do indeed satisfy polynomial relations.

2. Preliminaries

2.1. Path multiplication and iterated integrals

We use the topologist’s convention (which is the opposite of the algebraist’s convention) for path multiplication. Two paths $\alpha,\beta:[0,1]\to X$ in a topological space $X$ are composable when $\alpha(1)=\beta(0)$. The product $\alpha\beta$ of two composable paths first traverses $\alpha$ and then $\beta$. With this convention, $\pi_{1}(X,x)$ acts on the left of a pointed universal covering of $X$ and on the right of the fiber $V_{x}$ of a local system ${\mathbb{V}}$ over $X$.

The torsor of paths in a topological space $X$ from $x\in X$ to $y\in X$ will be denoted $\pi(X;x,y)$.

So that the homomorphism from the topological to the étale fundamental group of a variety is an isomorphism, we define the étale fundamental group of a scheme $X$ with respect to a fiber functor $b$ from étale coverings of $X$ to finite sets to the the opposite of the usual definition: $\pi_{1}^{\mathrm{\acute{e}t}}(X,b)$ is the group of right automorphisms of $b$.

2.2. Iterated Integrals

The iterated integral of smooth 1-forms ${\omega}_{1},\dots,{\omega}_{r}$ on a manifold $M$ over a piecewise smooth path $\alpha:[0,1]\to M$ is defined by

[TABLE]

where $f_{j}(t)dt=\alpha^{\ast}{\omega}_{j}$ and $\Delta^{r}$ is the “time ordered” $r$-simplex

[TABLE]

An exposition of the basic properties of iterated integrals can be found in [5] and [10].

2.3. Hodge theory

All mixed Hodge structures will be ${\mathbb{Q}}$ mixed Hodge structures and graded polarizable unless stated otherwise. The category of ${\mathbb{Q}}$ mixed Hodge structures will be denoted by ${\mathsf{MHS}}$. The category of ${\mathbb{R}}$-mixed Hodge structures will be denoted by ${\mathsf{MHS}}_{\mathbb{R}}$.

2.4. Orbifolds and stacks

By an orbifold (resp., an complex analytic orbifold) we mean a stack in the category of topological spaces (resp., complex analytic varieties). In general, the word stack will refer to a Deligne–Mumford stack. We will typically denote the complex analytic orbifold associated to a stack $X$ defined over a subfield of ${\mathbb{C}}$ by $X^{\mathrm{an}}$. The associated orbifold will often be denoted by $X({\mathbb{C}})$.

2.5. Modular Groups

Suppose that $N\geq 1$. We will use the standard notation

[TABLE]

The orbifolds ${\Gamma}_{0}(N){\backslash\negthickspace\backslash}{\mathfrak{h}}$, ${\Gamma}_{1}(N){\backslash\negthickspace\backslash}{\mathfrak{h}}$ and ${\Gamma}(N){\backslash\negthickspace\backslash}{\mathfrak{h}}$ have (orbifold) fundamental groups ${\Gamma}_{0}(N)$, ${\Gamma}_{1}(N)$ and ${\Gamma}(N)$, respectively. They are the complex analytic orbifolds associated to the modular curves $Y_{0}(N)$, $Y_{1}(N)$ and $Y(N)$, which will be regarded as stacks. Their standard compactifications will be denoted $X_{0}(N)$, $X_{1}(N)$ and $X(N)$, respectively.

A convenient model of the homotopy type of the orbifold ${\Gamma}{\backslash\negthickspace\backslash}{\mathfrak{h}}$, where ${\Gamma}$ is a subgroup of ${\mathrm{SL}}_{2}({\mathbb{Z}})$, is ${\Gamma}\backslash\hat{{\mathfrak{h}}}$, where ${\Gamma}$ acts diagonally on

[TABLE]

and $E{\mathrm{SL}}_{2}({\mathbb{Q}})$ is a space on which ${\mathrm{GL}}_{2}^{+}({\mathbb{Q}})$ acts freely and properly discontinuously.

Part I Unramified Correspondences

In this part we define unramified correspondences and show that they act on the free abelian group generated by conjugacy classes in the fundamental group of the source of the correspondence. We also give a group theoretic description of this action. The material in this section is elementary.

3. Quick Review of Covering Space Theory

For clarity we give a quick review of the relevant facts we shall need from covering space theory. This will also serve to fix notation. In all discussions of covering spaces and unramified correspondences, the topological spaces will be assumed to be locally contractible. Thus all such discussions apply to manifolds, complex algebraic varieties and to the geometric realizations of simplicial sets.

Suppose that $X$ is a connected topological space and that $\pi:Y\to X$ is a finite covering of degree $d$. Initially we do not assume that $Y$ is connected. Fix a base point $x_{o}\in X$. For $y\in\pi^{-1}(x_{o})$, we shall identify $\pi_{1}(Y,y)$ with its image under the injection $\pi_{\ast}:\pi_{1}(Y,y)\to\pi_{1}(X,x_{o})$.

There is a natural right action

[TABLE]

of $\pi_{1}(X,x_{o})$ on the fiber over $\pi$ over $x_{o}$. We shall denote it by

[TABLE]

It is characterized by the property that $y\cdot\gamma=y^{\prime}$ if and only the unique lift of a loop in $X$ based at $x_{o}$ that represents $\gamma$ starts at $y\in\pi^{-1}(x_{o})$ ends at $y^{\prime}$. If $y$ is in the fiber over $x_{o}$, then

[TABLE]

and

[TABLE]

Now suppose that $Y$ is connected. Fix a base point $y_{o}$ of $Y$ that lies over $x_{o}$. Then $\pi_{1}(Y,y_{o})$ has index $d$ in $\pi_{1}(X,x_{o})$ and the right $\pi_{1}(X,x_{o})$ action on $\pi^{-1}(x_{o})$ is transitive. Since $\pi_{1}(Y,y_{o})$ stabilizes $y_{o}$, there is a natural isomorphism

[TABLE]

of right $\pi_{1}(X,x_{o})$ sets.

The kernel of the right action $\rho_{Y}:\pi_{1}(X,x_{o})\to\operatorname{Aut}\pi^{-1}(x_{o})$ is the normal subgroup of $\pi_{1}(X,x_{o})$ that corresponds to the Galois closure

[TABLE]

of $\pi$. Set

[TABLE]

The group $G$ (and hence $H$ as well) acts on the left on $Z$. The fiber of ${\tilde{\pi}}$ over $x_{o}$ is a principal left $G$-set. Choose a base point $z_{o}$ of $Z$ that lies over $x_{o}$. This trivializes both of these torsors. So we can identify the fiber of ${\tilde{\pi}}$ over $x_{o}$ with $G$ via the map

[TABLE]

The choice of $z_{o}$ determines an isomorphism of $G$ with the image of $\pi_{1}(X,x_{o})$ in $\operatorname{Aut}\pi^{-1}(x_{o})$. This isomorphism is characterized by the property that

[TABLE]

for all $\gamma\in\pi_{1}(X,x_{o})$. In addition, $\pi_{1}(Y,y_{o})$ is the inverse image of $H$ under $\rho_{Y}$.

The pointed covering $\pi:(Y,y_{o})\to(X,x_{o})$ is naturally identified with the covering $H\backslash(Z,z_{o})\to(X,x_{o})$. This means that we can identify the fiber of $\pi:Y\to X$ over $x_{o}$ with $H\backslash G$. The map induces an isomorphism of right $\pi_{1}(X,x_{o})$ sets

[TABLE]

4. Pushforward and Pullback

As in the introduction, the set of homotopy classes of (unbased) loops $S^{1}\to X$ in a topological space $X$ will be denoted by $\lambda(X)$. The set of conjugacy classes of a group ${\Gamma}$ will be denoted by $\boldsymbol{\lambda}({\Gamma})$. If $X$ is path connected, then $\lambda(X)=\boldsymbol{\lambda}(\pi_{1}(X,x_{o}))$. If ${\Gamma}$ is a discrete group, then $\boldsymbol{\lambda}({\Gamma})=\lambda(B{\Gamma})$, where $B{\Gamma}$ denotes the classifying space of ${\Gamma}$.

Suppose that ${\Bbbk}$ is a commutative ring ${\Bbbk}$. The free ${\Bbbk}$-module generated by $\lambda(X)$ will be denoted by ${\Bbbk}\lambda(X)$ and the free ${\Bbbk}$-module generated by $\boldsymbol{\lambda}({\Gamma})$ will be denoted by ${\Bbbk}\boldsymbol{\lambda}({\Gamma})$. In both cases, the conjugacy class of the identity will be denoted by $1$.

Every continuous map $f:X\to X^{\prime}$ between topological spaces induces a function

[TABLE]

and thus a ${\Bbbk}$-module map ${\Bbbk}\lambda(X)\to{\Bbbk}\lambda(X^{\prime})$. It takes the free homotopy class of the loop $\alpha:S^{1}\to Y$ to the free homotopy class of the loop $f\circ\alpha:S^{1}\to X$.

Likewise, every group homomorphism $\phi:{\Gamma}\to{\Gamma}^{\prime}$ induces a function $\phi_{\ast}:\boldsymbol{\lambda}({\Gamma})\to\boldsymbol{\lambda}({\Gamma}^{\prime})$ and a ${\Bbbk}$-module map ${\Bbbk}\boldsymbol{\lambda}({\Gamma})\to{\Bbbk}\boldsymbol{\lambda}({\Gamma}^{\prime})$. The main task in this section is to show that, when $\pi:Y\to X$ is a finite unramified covering, there is a pullback map

[TABLE]

Applying this to the special case of the covering $B{\Gamma}^{\prime}\to B{\Gamma}$ associated to the inclusion of a finite index subgroup ${\Gamma}^{\prime}$ of a discrete group ${\Gamma}$ implies that there is a pullback map

[TABLE]

We will give an algebraic formula for this map, which will imply that the pullback map is also defined when ${\Gamma}^{\prime}$ is an open subgroup of a profinite group ${\Gamma}$.

4.1. Pullback of loops along unramified coverings

Suppose that $\pi:Y\to X$ is an unramified covering of finite degree. For the time being, we will not assume that either space is connected.

Suppose that $\alpha:S^{1}\to X$ is a loop in $X$. The pullback

[TABLE]

of $\pi$ along $\alpha$ is a covering of $S^{1}$ and thus a disjoint union of oriented circles

[TABLE]

whose components are indexed by a finite set ${\mathscr{S}}_{\alpha}$. Denote the restriction of ${\tilde{\alpha}}$ to the component $s\in{\mathscr{S}}_{\alpha}$ by ${\tilde{\alpha}}_{s}$. The homotopy lifting properties of unramified coverings implies that its class in $\lambda(Y)$ depends only on the class of $\alpha$ in $\lambda(X)$. The pullback of $\alpha$ is defined by

[TABLE]

The following basic property of pullback is an immediate consequence of the definition.

Lemma 4.1**.**

If $\pi_{X}:Y\to X$ and $\pi_{Y}:Z\to Y$ are finite coverings, then the diagram

[TABLE]

commutes.

Denote the connected $r$-fold covering map of the circle by $\pi_{r}:S^{1}\to S^{1}$. Orient the circles so that it is orientation preserving.

Lemma 4.2**.**

The pullback of the covering $\pi_{d}$ along $\pi_{e}$ is a disjoint union of $g:=\gcd(d,e)$ circles. In the pullback diagram

[TABLE]

the top horizontal map has degree $e/g$ on each component and the left hand vertical map has degree $d/g$ on each component.

Proof.

This is an exercise in covering space theory using the elementary fact that $d{\mathbb{Z}}\cap e{\mathbb{Z}}=g{\mathbb{Z}}$. ∎

As an immediate corollary, we obtain a formula for the pullback along $\pi_{d}:S^{1}\to S^{1}$ of a multiple of the positive generator $\sigma$ of $\pi_{1}(S^{1})$. Denote the positive generator of the fundamental group of the top circle by $\mu$.

Corollary 4.3**.**

For all integers $n$, we have

[TABLE]

Consequently, $\pi_{\ast}\pi^{\ast}(\sigma^{n})=\gcd(d,n)\sigma^{nd/\gcd(d,n)}$.

4.2. An algebraic description of the pullback map

When $X$ and $Y$ are connected, the pullback map admits a description in terms of the induced map on fundamental groups. Choose a base point $x_{o}$ of $X$ and $y_{o}$ of $Y$ that lies over $x_{o}$. We will use the notation and conventions established in Section 3.

Suppose that $\alpha\in\pi_{1}(X,x_{o})$. We will abuse notation and also denote its conjugacy class by $\alpha$. The subgroup $\langle\alpha\rangle$ of $\pi_{1}(X,x_{o})$ generated by $\alpha$ acts on $\pi^{-1}(x_{o})$ on the right: $\alpha$ takes $y\in\pi^{-1}(x_{o})$ to $y^{\prime}$ if and only if the unique lift of $\alpha$ to a path in $Y$ that starts at $y$ ends at $y^{\prime}$.

The set ${\mathscr{S}}_{\alpha}$ above is the set of $\langle\alpha\rangle$-orbits $\pi^{-1}(x_{o})/\langle\alpha\rangle$. For each $s\in\pi^{-1}(x_{o})/\langle\alpha\rangle$, choose $\gamma_{s}\in\pi_{1}(X,x_{o})$ such that $y_{o}\cdot\gamma_{s}$ is in the orbit. The cardinality $d_{s}$ of the corresponding orbit is given by

[TABLE]

The $\langle\alpha\rangle$-orbit that contains $y_{o}\cdot\gamma_{s}$ is illustrated in Figure 1. It determines the loop

[TABLE]

where ${\tilde{\alpha}}_{s,j}$ is the unique lift of $\alpha$ to $Y$ that starts at $y_{o}\cdot\gamma_{s}\alpha^{j-1}$. This is the lift of the loop $\gamma_{s}^{-1}\alpha^{d_{s}}\gamma_{s}$ in $X$ to a loop in $Y$ based at $y_{o}\cdot\gamma_{s}$.

With this notation, the formula for the pullback of $\alpha$ is

[TABLE]

This formula can be converted into a group theoretic description of the pullback map. The right action of $\pi_{1}(X,x_{o})$ on $\pi^{-1}(x_{o})$ induces an isomorphism of right $\pi_{1}(X,x_{o})$ sets

[TABLE]

The set of $\langle\alpha\rangle$ orbits is thus the double coset space

[TABLE]

and $\{\gamma_{s}:s\in{\mathscr{S}}_{\alpha}\}$ is a set of double coset representatives. We conclude that $\pi^{\ast}(\alpha)$ is the image of

[TABLE]

in ${\mathbb{Z}}\lambda(Y)$, where $d_{s}$ is defined by (10).

4.3. The pullback map for finite index subgroups

This algebraic description of the pullback map allows us to extend the definition of pullback to the case of the inclusion $j:{\Gamma}^{\prime}\hookrightarrow{\Gamma}$ of a finite index subgroup. There is no restriction on the group ${\Gamma}$; it can be discrete, profinite, algebraic or a Lie group. However, when ${\Gamma}$ is an algebraic or topological group, we will require that ${\Gamma}^{\prime}$ be an open subgroup. We will need the profinite case when discussing Galois equivariance of the Hecke action on conjugacy classes of various profinite completions of ${\mathrm{SL}}_{2}({\mathbb{Z}})$.

We now define the pullback map $j^{\ast}:{\mathbb{Z}}\boldsymbol{\lambda}({\Gamma})\to{\mathbb{Z}}\boldsymbol{\lambda}({\Gamma}^{\prime})$ using the group theoretical description of the pullback (12) given above. To simplify and unify the discussion, we will regard a discrete group ${\Gamma}$ to be a topological group with the discrete topology. In all cases, the set ${\Gamma}^{\prime}\backslash{\Gamma}$ is finite(and discrete as ${\Gamma}^{\prime}$ has finite index and is open in ${\Gamma}$).

Suppose that $\alpha\in{\Gamma}$. Denote by $\langle\alpha\rangle$ the cyclic subgroup of ${\Gamma}$ generated by $\alpha$. Set

[TABLE]

This is a discrete finite space. Choose a representatives $\gamma_{s}\in{\Gamma}$ of each double coset $s$. For each $s$, set

[TABLE]

This is well defined as ${\Gamma}^{\prime}$ has finite index in ${\Gamma}$. Define $j^{\ast}(\alpha)\in{\mathbb{Z}}\boldsymbol{\lambda}({\Gamma}^{\prime})$ to be the image of

[TABLE]

under the quotient map ${\mathbb{Z}}[{\Gamma}^{\prime}]\to{\mathbb{Z}}\boldsymbol{\lambda}({\Gamma}^{\prime})$.

This formula implies that the construction is natural in the following sense.

Proposition 4.4**.**

Suppose that we have a commutative diagram

[TABLE]

of groups where $j$ and $\hat{j}$ are inclusions of finite index open subgroups. If the induced map $\phi:{\Gamma}^{\prime}\backslash{\Gamma}\to\hat{{\Gamma}}^{\prime}\backslash\hat{{\Gamma}}$ is a bijection, then the diagram

[TABLE]

commutes.

One important case for us is where $\phi$ and $\phi^{\prime}$ are a version of profinite completion (such as full profinite completion, (relative) pro-$\ell$ completion, …) of $j:{\Gamma}^{\prime}\hookrightarrow{\Gamma}$ and where $\hat{j}$ is injective.666Completion is not, in general, left exact. See [6].

5. Unramified Correspondences act on Conjugacy Classes

In this section, we define unramified correspondences and show that they act on conjugacy classes. Later we’ll see that Hecke correspondences are unramified correspondences and therefore act on ${\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}}))$.

5.1. Unramified correspondences

By an unramified correspondence between two locally contractible topological spaces $X$ and $Y$, we mean a (not necessarily connected) space $U$ and maps

[TABLE]

where $f$ is a covering projection of finite degree. We will call $X$ the source and $Y$ the target of the correspondence. We will typically denote such a correspondence by a roman letter, such as $F$.

The composition of this correspondence with the unramified correspondence $G$ given by the diagram

[TABLE]

is the unramified correspondence $G\circ F$ given by the diagram

[TABLE]

where $\phi$ and $\kappa$ are the compositions of the two natural projections from $U\times_{Y}V$ to $U$ and $v$ with $f$ and $k$ respectively. The map $\phi$ is a covering map of finite degree as it the composition of $f$ with the pullback of the covering map $h$ along $g$.

Correspondences with the same source and target can be added. The sum of the correspondence (13) with the correspondence

[TABLE]

is the correspondence

[TABLE]

where $\sqcup$ denotes disjoint union. The unramified correspondences from $X$ to $Y$ form an abelian monoid which we denote by $\operatorname{Hom}^{+}(X,Y)$. The multiplication

[TABLE]

is bilinear. Additive inverses can be added formally to $\operatorname{Hom}^{+}(X,Y)$ in the standard way to obtain an abelian group $\operatorname{Hom}(X,Y)$. The multiplication map extends to a bilinear mapping $\operatorname{Hom}(Y,Z)\times\operatorname{Hom}(X,Y)\to\operatorname{Hom}(X,Z)$.

When both the maps in a correspondence $F$ are finite unramified coverings, we can reverse the roles of the source and target to obtain a new unramified correspondence, which we call the adjoint of $F$ and denote by $F^{\vee}$. For example, the adjoint of the correspondence in equation (13) above is

[TABLE]

If both projections in the correspondence $G$ above are also finite unramified coverings, it and $G\circ F$ will have adjoints, and these will satisfy

[TABLE]

5.2. Unramified correspondences act on ${\mathbb{Z}}\lambda(X)$

Is is standard that correspondences act on sheaves, homology, cohomology, etc. For example, the correspondence (13) acts on covariant objects by $g_{\ast}\circ f^{\ast}$ and on contravariant objects by $f_{\ast}\circ g^{\ast}$. Unramified correspondences act on ${\mathbb{Z}}\lambda({\phantom{x}})$ as we can pullback conjugacy classes along finite unramified coverings:

The unramified correspondence (13) induces the map

[TABLE]

Proposition 5.1**.**

The map

[TABLE]

is compatible with multiplication of unramified correspondences.

Proof.

We will reduce this to the case where the spaces are circles. We take our correspondences to be (13) and (14). Using the definition of pushforward and pullback and the linearity of the action of correspondences under disjoint union of coverings, we can (and will) reduce to the case where $X$,$Y$,$Z$, $U$ and $V$ are each just one copy of $S^{1}$. Set

[TABLE]

This will be a disjoint union of circles $W_{j}$. Denote the restriction of $r$ to $W_{j}$ by $r_{j}$. We can further assume that $\alpha\in\lambda(X)$ is the positive generator of $\pi_{1}(X)=\pi_{1}(S^{1})$.

The two correspondences and their composition are summarized in the diagram

[TABLE]

Pick a generator of the fundamental group of each circle. We can choose these such that the degrees of $f$, $h$ and $r$ are positive. Denote the chosen generators of the various circles according to the following table:

[TABLE]

Denote the degrees (with respect to these chosen generators) of the maps in the diagram by

[TABLE]

Then, by Lemma 4.2, the degree of the restriction of $p$ to each component of $W_{j}$ of $W$ is $e/gcd(d,e)$ and the degree of the restriction of $q$ to each $W_{j}$ is $d/gcd(d,e)$. Then

[TABLE]

On the other hand, using Lemma 4.1, we have

[TABLE]

∎

6. The Profinite Case

The analogue of the unramified correspondence (13) in the profinite case, is a diagram

[TABLE]

where ${\Gamma}$ and ${\Gamma}^{\prime}$ are profinite groups, $\{{\Gamma}^{\prime\prime}_{k}\}_{k\in K}$ is a finite set of open subgroups of ${\Gamma}$ with corresponding inclusions $\phi_{k}:{\Gamma}^{\prime\prime}_{k}\hookrightarrow{\Gamma}$, and where $\psi_{k}:{\Gamma}_{k}^{\prime\prime}\to{\Gamma}^{\prime}$, $k\in K$ are continuous group homomorphisms. Such a diagram will be called a group correspondence. It can be regarded as the formal sum of the basic group correspondences

[TABLE]

indexed by $k\in K$. The correspondence (15) induces the map

[TABLE]

where the pullback maps $\phi_{k}^{\ast}$ are defined as in Section 4.3.

As in the discrete case, we can formally define $\operatorname{Hom}({\Gamma},{\Gamma}^{\prime})$, the abelian group of group correspondences from ${\Gamma}$ to ${\Gamma}^{\prime}$. Composition of group correspondences, which is defined in the obvious way, is bilinear.

It is also useful to consider symmetries of correspondences such as (15). To this end, suppose that $G$ is a group that acts continuously on ${\Gamma}$ and ${\Gamma}^{\prime}$ and that permutes the collection $\{{\Gamma}^{\prime\prime}_{k}\}_{k\in K}$ of subgroups. The $G$-action induces an action of $G$ on ${\mathbb{Z}}\boldsymbol{\lambda}({\Gamma})$ and ${\mathbb{Z}}\boldsymbol{\lambda}({\Gamma}^{\prime})$. It also acts on

[TABLE]

Lemma 6.1**.**

With these hypotheses, the homomorphism ${\mathbb{Z}}\boldsymbol{\lambda}({\Gamma}^{\prime})\to{\mathbb{Z}}\boldsymbol{\lambda}({\Gamma}^{\prime})$ induced by the group correspondence (15) is $G$-equivariant.

Proof.

This follows directly from the easily verified facts that the pullback map

[TABLE]

and the pushforward map

[TABLE]

are $G$-equivariant. ∎

6.1. Galois Equivariance

Suppose that $X$ is a geometrically connected scheme over a field $K$. Suppose that ${\overline{x}}$ is a geometric point of $X$ that lies over a $K$-rational point. Denote the absolute Galois group $\operatorname{Gal}({\overline{K}}/K)$ of $K$ by $G_{K}$. It acts continuously on $\pi_{1}^{\mathrm{\acute{e}t}}(X_{/{\overline{K}}},{\overline{x}})$ and therefore on the profinite set $\boldsymbol{\lambda}(\pi_{1}^{\mathrm{\acute{e}t}}(X_{/{\overline{K}}},{\overline{x}}))$. Define $\lambda^{\mathrm{\acute{e}t}}(X)=\boldsymbol{\lambda}(\pi_{1}^{\mathrm{\acute{e}t}}(X_{/{\overline{K}}},{\overline{x}}))$.

Lemma 6.2**.**

The action of $G_{K}$ on $\pi_{1}^{\mathrm{\acute{e}t}}(X_{/{\overline{K}}},{\overline{x}})$ induces an action on $\lambda^{\mathrm{\acute{e}t}}(X)$. This action does not depend on the choice of the base point ${\overline{x}}$. ∎

This Galois action can also be constructed using the canonical outer Galois action $G_{K}\to\operatorname{Out}\pi_{1}^{\mathrm{\acute{e}t}}(X_{/{\overline{K}}},{\overline{x}})$, where ${\overline{x}}$ is any geometric point of $X$.

An unramified correspondence of $K$-schemes is defined to be a correspondence

[TABLE]

of $K$-schemes where $f$ is étale. Applying the construction above to the geometric étale fundamental groups, we see that such a correspondence will induce a map

[TABLE]

Theorem 6.3**.**

The map ${\mathbb{Z}}\lambda^{\mathrm{\acute{e}t}}(X)\to{\mathbb{Z}}\lambda^{\mathrm{\acute{e}t}}(Y)$ induced by the étale correspondence (16) is $G_{K}$ equivariant.

Proof.

We will reduce the proof to Lemma 6.2. We’ll suppose that $Y$ is geometrically connected and that $U$ is irreducible over $K$. The general case is left to the reader. Write

[TABLE]

where each $U_{c}$ is a connected ${\overline{K}}$-scheme. The set $C$ is a transitive finite $G_{K}$ set. Choose a geometric point ${\overline{x}}$ of $X$ and geometric points ${\overline{u}}_{c}$ of each component $U_{c}$, $c\in C$ that lie over ${\overline{x}}$. Choose a geometric point ${\overline{y}}$ of $Y$. For each $c\in C$, choose étale paths $\gamma_{c}$ in $Y$ from ${\overline{y}}$ to $g({\overline{u}}_{c})$ such that $\gamma_{\sigma(c)}=\sigma(\gamma_{c})$ for all $c\in C$. Define

[TABLE]

by $\alpha\mapsto\gamma_{c}\alpha\gamma_{c}^{-1}$ and $\phi_{c}$ to be the homomorphism induced by $f$. The diagram

[TABLE]

satisfies the hypotheses of Lemma 6.2, which implies that the induced map is $G_{K}$ equivariant. ∎

6.2. A comparison theorem

Suppose that $K$ is a subfield of ${\mathbb{C}}$. Fix an embedding ${\overline{K}}\hookrightarrow{\mathbb{C}}$. If $Z$ is a scheme over $K$ and ${\overline{z}}\in Z({\mathbb{C}})$, there is a natural homomorphism

[TABLE]

where we regard the complex points $Z({\mathbb{C}})$ of $Z$ to be a complex analytic variety. This homomorphism becomes an isomorphism after taking the profinite completion of the topological fundamental group of $Z({\mathbb{C}})$. There is therefore a natural comparison map

[TABLE]

Given the unramified correspondence (16), we also has the unramified correspondence

[TABLE]

of complex analytic varieties.

Proposition 6.4**.**

The diagram

[TABLE]

commutes, where the horizontal maps are induced by the correspondences (17) and (16), respectively.

7. The Dual Action on Class Functions

Suppose that ${\Gamma}$ is a topological group (e.g., a discrete or profinite group) and that ${\Bbbk}$ is a commutative topological ring such as ${\mathbb{Z}}$ or a field of any characteristic endowed with the discrete topology. Denote the set of continuous functions ${\Gamma}\to{\Bbbk}$ that are constant on each conjugacy class of ${\Gamma}$ by ${\mathscr{C}\!\ell}_{\Bbbk}({\Gamma})$.777We will extend this definition to affine groups schemes in Section 19.

When ${\Gamma}$ is finite and ${\Bbbk}$ is a field, ${\mathscr{C}\!\ell}_{\Bbbk}({\Gamma})$ is spanned by the characters of representations. This is typically not the case when ${\Gamma}$ is a lattice in an algebraic group, as can be seen by considering the case of a lattice in a non-abelian unipotent ${\mathbb{Q}}$-group $U$. In this case, the non-constant characters are pullbacks of characters of finite quotients of ${\Gamma}$. Class functions on ${\Gamma}$ that are not characters can be obtained by restricting non-constant class functions on $U$ to ${\Gamma}$.888For a general construction of class functions on unipotent groups, see Remark 22.7(i).

Denote the group correspondence (15) by $F$.

Proposition 7.1**.**

For all commutative rings ${\Bbbk}$, the group correspondence $F$ induces a function

[TABLE]

It is defined by

[TABLE]

for all $\gamma\in\boldsymbol{\lambda}({\Gamma})$ and $\chi\in{\mathscr{C}\!\ell}_{\Bbbk}({\Gamma}^{\prime})$.

If a group $G$ acts on the correspondence $F$ as in the setup of Lemma 15, then it acts on ${\mathscr{C}\!\ell}_{\Bbbk}({\Gamma})$ and ${\mathscr{C}\!\ell}_{\Bbbk}({\Gamma}^{\prime})$. Lemma 15 then implies that $\check{F}$ is $G$-equivariant.

We can now apply this in the setup of Section 6.1.

Corollary 7.2**.**

The étale correspondence (16) induces a linear function

[TABLE]

Part II The Hecke Action on Conjugacy Classes of ${\mathrm{SL}}_{2}({\mathbb{Z}})$

In this part we define an action of each Hecke operator $T_{N}$ on ${\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}}))$ and determine the basic relations that hold between them. This part is mainly elementary topology. All modular curves in this part will be regarded as complex analytic orbifolds. In particular, ${\mathcal{M}}_{1,1}$ will denote the complex analytic orbifold ${\mathrm{SL}}_{2}({\mathbb{Z}}){\backslash\negthickspace\backslash}{\mathfrak{h}}$.

8. Conjugacy classes in ${\mathrm{SL}}_{2}({\mathbb{Z}})$

As remarked in the introduction, the map $\lambda({\mathrm{SL}}_{2}({\mathbb{Z}}))\to\lambda({\mathrm{PSL}}_{2}({\mathbb{Z}}))$ is 2 to 1. The fibers of this map are $\{\gamma,-\gamma\}$. Except when $\operatorname{tr}(\gamma)=0$ (in which case $\gamma$ has order 4), these are distinguished by the trace map.

If $X$ is a compact hyperbolic manifold, then $\lambda(X)-\{1\}$ can be identified with the closed, oriented (not necessarily prime) geodesics in $X$. If $X$ is a complete hyperbolic surface with finite topology, then

[TABLE]

The modular curve ${\mathcal{M}}_{1,1}$ is the orbifold quotient ${\mathrm{SL}}_{2}({\mathbb{Z}}){\backslash\negthickspace\backslash}{\mathfrak{h}}$ of the upper half plane ${\mathfrak{h}}$.999We use the convention of [12, §3]. With this convention, the orbifold fundamental group of ${\Gamma}{\backslash\negthickspace\backslash}{\mathfrak{h}}$ is ${\Gamma}$. A closed geodesic on ${\Gamma}{\backslash\negthickspace\backslash}{\mathfrak{h}}$ will be a ${\Gamma}$-invariant geodesic in ${\mathfrak{h}}$ and two distinct points on it that lie in the same ${\Gamma}$-orbit. Since $-\operatorname{id}$ acts trivially on ${\mathfrak{h}}$, we also have the orbifold ${\mathrm{PSL}}_{2}({\mathbb{Z}}){\backslash\negthickspace\backslash}{\mathfrak{h}}$. In both cases the orbifold has has a hyperbolic metric.

Each element of ${\mathrm{SL}}_{2}({\mathbb{C}})$ that is not a scalar matrix has one or two fixed points on ${\mathbb{P}}^{1}({\mathbb{C}})$. These are its projectivized eigenspaces. Except for $\pm\operatorname{id}$, all elements $\gamma$ of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ have either one fixed point (necessarily in ${\mathbb{P}}^{1}({\mathbb{R}})$), or two real fixed points, or a complex conjugate pair of non-real fixed points. These correspond to the three types of non-trivial elements $\gamma$ of ${\mathrm{SL}}_{2}({\mathbb{Z}})-\{\pm\operatorname{id}\}$:

[TABLE]

This decomposition of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ gives a partition

[TABLE]

of $\lambda({\mathrm{SL}}_{2}({\mathbb{Z}}))$ into $\{\pm\operatorname{id}\}$, elliptic, parabolic and hyperbolic classes and gives the decomposition

[TABLE]

9. The Hecke Action on ${\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}}))$

In this section we prove Theorems 1 and 2. For an integer $N\geq 1$, define $\operatorname{Cov}_{N}$ to be the set of isomorphism classes of pairs of lattices

[TABLE]

in ${\mathbb{C}}$, where $\Lambda^{\prime}$ has index $N$ in $\Lambda$. It is an orbifold model of the moduli space of $N$-fold coverings $E^{\prime}\to E$ of elliptic curves. The pair of lattices $(\Lambda,\Lambda^{\prime})$ corresponds to the covering ${\mathbb{C}}/\Lambda^{\prime}\to{\mathbb{C}}/\Lambda$. Note that $\operatorname{Cov}_{N}$ is connected if an only if $N$ is square free.

The Hecke correspondence $T_{N}$ is the unramified correspondence

[TABLE]

where $\pi_{N}$ and $\pi_{N}^{\prime}$ are the two projections $\pi_{N}:(\Lambda,\Lambda^{\prime})\mapsto\Lambda$ and $\pi_{N}^{\prime}:(\Lambda,\Lambda^{\prime})\mapsto\Lambda^{\prime}$. It induces the map $(\pi_{N}^{\prime})_{\ast}\circ\pi_{N}^{\ast}$ on ${\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}}))$ that we shall (by abuse of notation) denote by

[TABLE]

For each prime number $p$, define

[TABLE]

to be $(\pi_{p})_{\ast}\circ\pi_{p}^{\ast}-\operatorname{id}$. The restriction of $\mathbf{e}_{p}$ to the image of ${\mathbb{Z}}\boldsymbol{\lambda}({\Gamma}(p))$ is multiplication by $p$, but in general the action is more complicated and one has

[TABLE]

where $\sum_{j}d_{j}=p+1$ and each $d_{j}$ divides $p(p+1)$. (See Corollary 4.3.)

Theorem 9.1**.**

Each Hecke correspondence $T_{N}$ acts on ${\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}}))$. This action preserves the decomposition (18) and satisfies the identities

[TABLE]

The operators $\mathbf{e}_{p}$ and $\mathbf{e}_{q}$ commute for all pairs of prime numbers $p$ and $q$.

In Example 10.9, we will see that $\mathbf{e}_{p}$ does not commute with $T_{p}$ and that $T_{p^{2}}$ does not commute with $T_{p}$.

The two identities follow from refinements of the standard arguments that one finds in [34, VII,§5]. The first is a direct consequence of the following easily proved fact.

Proposition 9.2**.**

If the positive integers $M$ and $N$ are relatively prime, then the diagram

[TABLE]

of orbifolds is a pullback square. The isomorphism $\operatorname{Cov}_{M}\times_{{\mathcal{M}}_{1,1}}\operatorname{Cov}_{N}\to\operatorname{Cov}_{MN}$ is defined by

[TABLE]

on lattices, or by fibered product

[TABLE]

of coverings. ∎

The rest of this section is devoted to proving the second identity. We begin with some useful background.

9.1. The involution $\iota_{N}$

The orbifold $\operatorname{Cov}_{N}$ has an involution $\iota_{N}$ that is defined by

[TABLE]

It is an involution because $[\Lambda\supset\Lambda^{\prime}]=[N\Lambda\supset N\Lambda^{\prime}]$. In more algebro-geometric language, $\iota_{N}$ takes the isomorphism class of the isogeny $E^{\prime}\to E$ to the class of the dual isogeny $\operatorname{Pic}^{0}E\to\operatorname{Pic}^{0}E^{\prime}$. This description is equivalent to the first by Abel’s Theorem: the Abel–Jacobi map $X\to\operatorname{Pic}^{0}X$ is an isomorphism for all elliptic curves $X$. Apparently $\iota_{N}$ is called the Fricke involution.

The relevance of the involution is that $\pi_{N}^{\prime}=\pi_{N}\circ\iota_{N}$ in the diagram (19), so that

[TABLE]

Recall that

[TABLE]

where

[TABLE]

One component of $\operatorname{Cov}_{N}$ is the moduli space $\operatorname{Cov}_{N}^{\mathrm{cyc}}$ of cyclic $N$-fold coverings $E^{\prime}\to E$. There are natural identifications

[TABLE]

where

[TABLE]

In the first isomorphism, the ${\Gamma}_{0}(N)$-orbit of $\tau$ corresponds to the isomorphism class of the pair of lattices

[TABLE]

In the second, the ${\Gamma}_{0}(N)^{\mathrm{op}}$-orbit of $\tau$ corresponds to the pair

[TABLE]

So the diagram

[TABLE]

commutes, where the vertical arrows are the identifications described above and where the top map is induced by $g_{N}:\tau\to N\tau$.

Denote the projection ${\Gamma}_{0}(N)^{\mathrm{op}}{\backslash\negthickspace\backslash}{\mathfrak{h}}\to{\mathrm{SL}}_{2}({\mathbb{Z}}){\backslash\negthickspace\backslash}{\mathfrak{h}}$ by $\pi^{\mathrm{op}}$. Observe that $\pi^{\mathrm{op}}_{N}\circ\iota_{N}=\pi^{\prime}_{N}$.

Recall the definition of $\hat{{\mathfrak{h}}}$ from (9) in Section 2. When $N$ is a prime number $p$, every covering is cyclic. This, the relation (20) and the fact that $g_{p}\in{\mathrm{GL}}_{2}({\mathbb{Q}})^{+}$ acts on $\hat{{\mathfrak{h}}}$ establishes:

Proposition 9.3**.**

The operator $T_{p}$ is realized by the unramified correspondence

[TABLE]

where the top map is induced by $g_{p}:\hat{{\mathfrak{h}}}\to\hat{{\mathfrak{h}}}$ and $\phi$ and $\psi$ are induced by group inclusion. It is self dual in the sense that $T_{N}$ equals its adjoint $T_{N}^{\vee}$.

Remark 9.4*.*

The Cartan involution of ${\mathrm{SL}}_{2}({\mathbb{R}})$ takes $g\in{\mathrm{SL}}_{2}({\mathbb{R}})$ to its inverse transpose $g^{-T}$. The automorphism

[TABLE]

of ${\mathrm{SL}}_{2}({\mathbb{R}})$ restricts to an isomorphism ${\Gamma}_{0}(N)\to{\Gamma}_{0}(N)^{\mathrm{op}}$. It also induces the orbifold isomorphism

[TABLE]

that is covered by $\tau\mapsto-1/N\tau$. It takes the cyclic isogeny $E^{\prime}\to E$ to the dual isogeny $\operatorname{Pic}^{0}E\to\operatorname{Pic}^{0}E^{\prime}$.

9.2. Proof of the second identity

Fix a prime number $p$ and a positive integer $n$. We now establish the second relation in Theorem 9.1; the relation that relates $T_{p^{n}}\circ T_{p}$ to $T_{p^{n+1}}$ and $T_{p^{n-1}}$.

The correspondence $T_{p^{n}}\circ T_{p}$ is the unramified correspondence

[TABLE]

where the maps are defined by

[TABLE]

The middle square is a pullback along $\pi^{\mathrm{op}}:Y_{0}(p)\to{\mathcal{M}}_{1,1}$ and $\pi^{\prime}:\operatorname{Cov}_{p^{n}}\to{\mathcal{M}}_{1,1}$. To establish the formula, we need to relate it to $\operatorname{Cov}_{p^{n\pm 1}}$.

Note that for all non-zero integers $N$, the lattices $\Lambda$ and $N\Lambda$ are isomorphic, and thus determine the same (orbi) point in ${\mathcal{M}}_{1,1}$. A sublattice $\Lambda^{\prime\prime}$ of a lattice $\Lambda$ of index $p^{n+1}$ is either contained in $p\Lambda$ or is not. Consequently, we can decompose $\operatorname{Cov}_{p^{n+1}}$ as the disjoint union

[TABLE]

where $\operatorname{Cov}_{p^{n+1}}^{0}$ is the moduli space of pairs of lattices $(\Lambda,\Lambda^{\prime\prime})$, where $\Lambda^{\prime\prime}$ has index $p^{n+1}$ and $\Lambda^{\prime\prime}\not\subseteq p\Lambda$. The other component consists of all index $p^{n-1}$ sublattices $\Lambda^{\prime\prime}$ of $p\Lambda$. These have index $p^{n+1}$ in $\Lambda$.

The next lemma is the analogue for correspondences of the usual argument used to compute $T_{p}\circ T_{p^{n}}$ on the level of points.

Lemma 9.5**.**

There is a natural isomorphism of the pullback $Y_{0}(p)\times_{{\mathcal{M}}_{1,1}}\operatorname{Cov}_{p^{n}}$ in (23) with $\operatorname{Cov}_{p^{n+1}}^{0}\sqcup\pi^{\ast}\operatorname{Cov}_{p^{n-1}}$. The diagram

[TABLE]

is a pullback square.

Proof.

The pullback $Y_{0}(p)\times_{{\mathcal{M}}_{1,1}}\operatorname{Cov}_{p^{n}}$ consists of all triples $(\Lambda\supset\Lambda^{\prime}\supset\Lambda^{\prime\prime})$, where $\Lambda^{\prime}$ has index $p$ in $\Lambda$ and $\Lambda^{\prime\prime}$ has index $p^{n}$ in $\Lambda^{\prime}$. This can be decomposed into two subspaces, one consisting of the triples where $\Lambda^{\prime\prime}$ is contained in $p\Lambda$ and there other where it not. If $\Lambda^{\prime\prime}\not\subseteq p\Lambda$, then the image of $\Lambda^{\prime\prime}$ in $\Lambda/p\Lambda$ has index $p$. This implies that there is a unique sublattice $\Lambda^{\prime}$ of $\Lambda$ index $p$ that contains $\Lambda^{\prime\prime}$. This implies that one component is $\operatorname{Cov}_{p^{n+1}}^{0}$. The other component consists of all isomorphism classes of sequences

[TABLE]

of lattices. But this component is clearly the pullback of $\operatorname{Cov}_{p^{n-1}}$, the space of all $(p\Lambda\supseteq\Lambda^{\prime\prime})$ along the projection $\pi:Y_{0}(p)\to{\mathcal{M}}_{1,1}$. ∎

As a consequence, the unramified correspondence $T_{p^{n}}\circ T_{p}$ is the sum of the two correspondences

[TABLE]

The decomposition in (24) implies that the first is $T_{p^{n+1}}-T_{p^{n-1}}$. The second correspondence is the composite

[TABLE]

which is $T_{p^{n-1}}\circ(\pi_{\ast}\pi^{\ast})=T_{p^{n-1}}+T_{p^{n-1}}\circ\mathbf{e}_{p}$. This completes the proof of the identity.

The commuting of $\mathbf{e}_{p}$ and $\mathbf{e}_{q}$ can be proved directly or deduced from the fact that

[TABLE]

and that $T_{p^{n}}$ commutes with $T_{q^{m}}$ when $p\neq q$. This completes the proof of the theorem.

10. The Action of $T_{p}$ on ${\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}}))$

In this section we explicitly compute the action of $T_{p}$ on the elliptic and parabolic elements of $\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}}))$ and compute the minimal polynomial $m_{p}(x)$ of $\mathbf{e}_{p}$. We also make some general comments about computing the value of $T_{p}$ on hyperbolic elements.

Throughout this section $p$ is a fixed prime number. Recall that ${\mathbb{H}}$ is the local system over ${\mathcal{M}}_{1,1}$ whose fiber over the point corresponding to the elliptic curve $E$ is $H_{1}(E;{\mathbb{Z}})$. Denote its reduction mod $p$ by ${\mathbb{H}}_{{\mathbb{F}}_{p}}$ and its projectivization by ${\mathbb{P}}({\mathbb{H}}_{{\mathbb{F}}_{p}})$. The projection

[TABLE]

is a covering projection that is isomorphic to the covering $Y_{0}(p)\to{\mathcal{M}}$. The point $L\in{\mathbb{P}}({\mathbb{H}}_{{\mathbb{F}}_{p}})$ corresponds to the covering $E^{\prime}\to E$ of $E$ where $H_{1}(E^{\prime})$ is the inverse image of the one dimensional subspace $L$ of $H(E;{\mathbb{F}}_{p})$.

Another way to think of the fiber of $Y_{0}(p)\to{\mathcal{M}}$ over $x_{o}$ and the right ${\mathrm{SL}}_{2}({\mathbb{Z}})$-action on it is to note that the fiber is

[TABLE]

where $B$ is the upper triangular Borel subgroup; the stabilizer of $(0,1)\in({\mathbb{F}}_{p})^{2}$.

The right action of $\gamma\in{\mathrm{SL}}_{2}({\mathbb{Z}})$ on $\pi^{-1}(x_{o})$ is given by matrix multiplication on the right:

[TABLE]

The following result is elementary and well-known. Since it plays a key role, we provide a short proof.

Proposition 10.1**.**

The distinct conjugates of ${\Gamma}_{0}(p)$ in ${\mathrm{SL}}_{2}({\mathbb{Z}})$ are

[TABLE]

where $\gamma_{0}=\operatorname{id}$ and

[TABLE]

Proof.

The stabilizer in ${\mathrm{SL}}_{2}({{\mathbb{F}}_{p}})$ of $[0,1]\in{\mathbb{P}}^{1}({{\mathbb{F}}_{p}})$ is the Borel $B=B({{\mathbb{F}}_{p}})$. Since

[TABLE]

stabilizer of $[1,j]\in{\mathbb{P}}^{1}({{\mathbb{F}}_{p}})$ is $\gamma_{j}^{-1}B\gamma_{j}$. This proves the result as the conjugates of ${\Gamma}_{0}(p)$ in ${\mathrm{SL}}_{2}({\mathbb{Z}})$ are the inverse images of the conjugates of $B$ in ${\mathrm{SL}}_{2}({{\mathbb{F}}_{p}})$. ∎

10.1. The general formula

Suppose that $\alpha\in{\mathrm{SL}}_{2}({\mathbb{Z}})$. We will abuse notation and also denote its image in $\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}}))$ by $\alpha$. The set ${\mathscr{S}}_{\alpha}$ of $\langle\alpha\rangle$ orbits on the fiber of $Y_{0}(p)\to{\mathcal{M}}_{1,1}$ is ${\mathbb{P}}^{1}({{\mathbb{F}}_{p}})/\langle\alpha\rangle$.

Let $d_{j}$ be the size of the $\langle\alpha\rangle$ that contains $[0,1]\gamma_{j}$. Choose a set of orbit representatives $[0,1]\gamma_{k}$ and let $S_{\alpha}$ be the corresponding set of indices $k\in\{0,\dots,p\}$. The following formulas are immediate consequences of the formula (12) and the discussion in Section 4.2, Proposition 9.3 and the fact that if $j$ and $k$ are in the same $\alpha$ orbit, then $\gamma_{j}\alpha^{d_{j}}\gamma_{j}^{-1}$ and $\gamma_{k}\alpha^{d_{k}}\gamma_{k}^{-1}$ are conjugate in ${\Gamma}_{0}(p)$.

Proposition 10.2**.**

With this notation

[TABLE]

[TABLE]

and

[TABLE]

All $m_{k}$ equal one if and only if $\alpha$ acts trivially on ${\mathbb{P}}^{1}({{\mathbb{F}}_{p}})$. That is, when $\pm\alpha\in{\Gamma}(p)$.

Corollary 10.3**.**

If $\pm\alpha\in{\Gamma}(p)$, then

[TABLE]

and

[TABLE]

Moreover $\mathbf{e}_{p}(\alpha)=p\alpha$ if and only if $\pm\alpha\in{\Gamma}(p)$.

10.2. Computing $\mathbf{e}_{p}$

Suppose that $\alpha\in{\mathrm{SL}}_{2}({\mathbb{Z}})$. In this section we will regard $\alpha$ as acting on ${\mathbb{P}}^{1}({{\mathbb{F}}_{p}})$. This action is well defined up to conjugacy and therefore has a well defined cycle type. We already know how to compute $\pi^{\ast}(\alpha)$ when $\alpha$ acts trivially. We now suppose that $\alpha$ acts non-trivially on ${\mathbb{P}}^{1}({\mathbb{F}}_{p})$.

To understand $\pi^{\ast}\alpha$, we need to understand the cycle decomposition of $\alpha$ acting on ${\mathbb{P}}^{1}({{\mathbb{F}}_{p}})$. This can be understood using linear algebra. The first observation is that the fixed points of $\alpha$ correspond to eigenspaces of ${\overline{\alpha}}$, its reduction mod $p$. This means that there are at most 2 fixed points.

To understand this better, consider the characteristic polynomial

[TABLE]

of $\alpha$, where $t$ is its trace. It has discriminant $\Delta=t^{2}-4$. There are 3 cases, namely:

[TABLE]

which correspond to $\Delta$ being 0 mod $p$, a non-zero square or a non-square mod $p$. In the first case, $t\equiv\pm 2$ mod $p$ so that the characteristic polynomial of $\alpha$ is $(x\pm 1)^{2}$ mod $p$.

10.2.1. $\Delta\equiv 0\bmod p$ and $\pm\alpha\notin{\Gamma}(p)$

Since the characteristic polynomial of $\alpha$ is $(x\pm 1)^{2}$ mod $p$ and since $\pm\alpha\notin{\Gamma}(p)$, it is conjugate to

[TABLE]

mod $p$, where $u\not\equiv 0\bmod p$. So $\alpha$ fixes a unique point of ${\mathbb{P}}^{1}({{\mathbb{F}}_{p}})$ (viz, its eigenspace). As a permutation of ${\mathbb{P}}^{1}({{\mathbb{F}}_{p}})$ it is the product of a 1-cycle and a $p$-cycle. Consequently, $\pi^{\ast}(\alpha)$ is the sum of two loops. If the eigenspace of $\alpha$ is $[0,1]\gamma_{j}$, then $\alpha\in\gamma_{j}^{-1}{\Gamma}_{0}(p)\gamma_{j}$ and $\alpha^{p}\in\gamma_{k}^{-1}{\Gamma}_{0}(p)\gamma_{k}$ for any $k\neq j$. In this case

[TABLE]

and

[TABLE]

10.2.2. $\Delta$ is non-zero and not a square mod $p$

In this case $\alpha$ has no eigenvectors in $H_{{\mathbb{F}}_{p}}$, so $\alpha$ has no fixed points. Since $\Delta$ is not a square mod $p$,

[TABLE]

is a field of order $p^{2}$. Denote the Galois involution of ${\Bbbk}$ by $y\mapsto\bar{y}$.

The group ${\Bbbk}^{\times}/{\mathbb{F}}_{p}^{\times}$ is cyclic of order $p+1$. Let $r,\bar{r}\in{\Bbbk}^{\times}$ be the eigenvalues of $\alpha$.

Proposition 10.4**.**

If $m$ is the order of $r$ in ${\Bbbk}^{\times}/{\mathbb{F}}_{p}^{\times}$, then $m>1$ and $\alpha$ acts on ${\mathbb{P}}^{1}({{\mathbb{F}}_{p}})$ as a product of $(p+1)/m$ disjoint $m$-cycles. Consequently

[TABLE]

Proof.

Fix an eigenvector $v\in H_{\Bbbk}$ of $\alpha$ with eigenvalue $r$. Then $\bar{v}$ is an eigenvector with eigenvalue $\bar{r}$. We can identify $H_{{\mathbb{F}}_{p}}$ with the Galois invariants

[TABLE]

The ${{\mathbb{F}}_{p}}$-linear map ${\Bbbk}\to H_{{\mathbb{F}}_{p}}$ that takes $y\in{\Bbbk}$ to $yv+\bar{y}\bar{v}$ is an isomorphism of ${\mathbb{F}}_{p}$ vector spaces. It therefore induces a bijection

[TABLE]

Since $v\alpha=rv$, the automorphism of ${\mathbb{P}}^{1}({{\mathbb{F}}_{p}})$ induced by $\alpha$ corresponds to the multiplication by $r$ map ${\Bbbk}^{\times}/{{\mathbb{F}}_{p}}^{\times}\to{\Bbbk}^{\times}/{{\mathbb{F}}_{p}}^{\times}$. The result follows. ∎

10.2.3. $\Delta$ odd and $p=2$

This case is similar to the previous one in that $\alpha$ will not have eigenvectors in $H_{{\mathbb{F}}_{2}}$. This corresponds to the case where we identify ${\mathbb{P}}^{1}({\mathbb{F}}_{2})$ with ${\mathbb{P}}({\mathbb{F}}_{4})$ and $\alpha$ acts as multiplication by a generator of ${\mathbb{F}}_{4}^{\times}$. That is, it acts as a 3-cycle on ${\mathbb{P}}^{1}({\mathbb{F}}_{2})$.

10.2.4. $\Delta$ a non-zero square mod $p$, with $p$ odd

In this case, $\alpha$ has two distinct eigenvalues in ${{\mathbb{F}}_{p}}$ and thus 2 distinct fixed points in ${\mathbb{P}}^{1}({{\mathbb{F}}_{p}})$. Since $\alpha$ acts non-trivially on ${\mathbb{P}}^{1}({{\mathbb{F}}_{p}})$, the eigenvalues cannot be $\pm 1$.

Proposition 10.5**.**

If $\alpha$ has eigenvalue $\lambda\in{{\mathbb{F}}_{p}}^{\times}$, then $\alpha$ acts on ${\mathbb{P}}^{1}({{\mathbb{F}}_{p}})$ as a product of two 1-cycles and $(p-1)/m$ disjoint $m$-cycles, where $m>1$ is the order of $\lambda^{2}$ in ${{\mathbb{F}}_{p}}^{\times}$. Consequently,

[TABLE]

Proof.

Here $\alpha$ is conjugate to

[TABLE]

This fixes $0,\infty\in{\mathbb{P}}^{1}({{\mathbb{F}}_{p}})$. Identify their complement with ${\mathbb{G}_{m}}({{\mathbb{F}}_{p}})$. Then $\alpha$ acts on it by multiplication by $\lambda^{2}$. Since this action is faithful, it will act on ${\mathbb{G}_{m}}({{\mathbb{F}}_{p}})$ with $(p-1)/m$ orbits of length $m$. Since $\lambda\neq\pm 1$, $m>1$. ∎

10.3. The minimal polynomial of $\mathbf{e}_{p}$

As we have seen in Section 10.2, each $\gamma\in{\mathrm{SL}}_{2}({\mathbb{Z}})$ acts on ${\mathbb{P}}^{1}({{\mathbb{F}}_{p}})$ with either 0, 1, 2 or $p+1$ fixed points and it acts on the complement of these as a product of disjoint cycles, all of the same length. This makes it easy to compute the minimal polynomial of $\mathbf{e}_{p}$.101010That $\mathbf{e}_{p}$ satisfies a polynomial equation and the approach to proving it were suggested by an anonymous reviewer.

Lemma 10.6**.**

If $\alpha$ acts on ${\mathbb{P}}^{1}({{\mathbb{F}}_{p}})$ with $f$ fixed points and $d$ disjoint $m$-cycles, then the ideal of polynomials $h(x)$ with the property that $h(\pi_{\ast}\pi^{\ast})(\alpha)=0$ is generated by

[TABLE]

Proof.

If $d=0$, then $\alpha$ acts trivially on ${\mathbb{P}}^{1}({{\mathbb{F}}_{p}})$, so that $\pi_{\ast}\pi^{\ast}(\alpha)=(p+1)\alpha$. This satisfies the polynomial $x-p-1$. Suppose now that $d>0$. Then

[TABLE]

These do not satisfy any linear polynomial, but do satisfy

[TABLE]

That is, it satisfies the polynomial $(x-p-1)(x-f)$, but not any polynomial of lower degree. ∎

We can now compute the minimal polynomial of $\mathbf{e}_{p}$.

Proposition 10.7**.**

The minimal polynomial $m_{p}(x)$ of $\mathbf{e}_{p}$ is

[TABLE]

Proof.

The minimal polynomial of the restriction of $\mathbf{e}_{p}$ to the conjugacy class $\alpha$ is $m_{\alpha}(x):=h_{\alpha}(x+1)$. The minimal polynomial of $\mathbf{e}_{p}$ is the lowest common multiple of these.

If $p=2$ and $\alpha$ acts non-trivially on ${\mathbb{P}}^{1}({{\mathbb{F}}_{p}})$, then $f\in\{0,1\}$. Thus $m_{2}(x)$ is the lcm of $(x-2)$, $x(x-2)$ and $(x+1)(x-2)$, which is $m_{2}(x)$.

If $p$ is odd, then $f\in\{0,1,2,p+1\}$. The previous lemma and the results in Section 10.2 imply that $m_{p}(x)$ is the lcm of $(x-p)$, $(x+1)(x-p)$, $x(x-p)$ and $(x-1)(x-p)$, so that $m_{p}(x)=x(x^{2}-1)(x-p)$. ∎

10.4. Computing $T_{p}(\alpha)$ on parabolic and elliptic elements

Here we compute the action of $T_{p}$ on parabolic and elliptic elements of ${\mathrm{SL}}_{2}({\mathbb{Z}})$.

10.4.1. Parabolic elements

The horocycle about the cup of ${\mathcal{M}}_{1,1}$ corresponds to the conjugacy class $\sigma_{o}$ of the matrix

[TABLE]

Proposition 10.8**.**

Suppose ${\epsilon}\in\{\pm 1\}$. For all $n\in{\mathbb{Z}}$ we have

[TABLE]

and

[TABLE]

Proof.

The modular curve $Y_{0}(p)$ has two cusps and the projection $Y_{0}(p)\to{\mathcal{M}}$ has orbifold degree $p$ in the neighbourhood of one cusp and is a local isomorphism in a neighbourhood of the other. These cusps correspond to the conjugacy classes of

[TABLE]

in ${\Gamma}_{0}(p)$. Denote their classes in $\boldsymbol{\lambda}({\Gamma}_{0}(p))$ by $\sigma_{o}$ and $\rho_{o}$, respectively. Then

[TABLE]

where $g=\gcd(n,p)$. The first formula follows as $\pi_{\ast}(\rho_{o})=\sigma_{o}^{p}$. Since $g_{p}\rho_{o}g_{p}^{-1}=\sigma_{o}$ and $g_{p}\sigma_{o}g_{p}^{-1}=\sigma_{o}^{p}$ in $\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}}))$, we have

[TABLE]

∎

Example 10.9**.**

These formulas imply that

[TABLE]

In particular, $T_{p}$ and $\mathbf{e}_{p}$ do not commute. This implies that $T_{p}$ and $T_{p^{2}}$ do not commute either since

[TABLE]

10.4.2. Elliptic elements

We’ll refer to the conjugacy class of a torsion element as a torsion (conjugacy) class. Hecke operators take torsion classes to sums of torsion classes. For this reason, the action of the $T_{p}$ on them is easily computed.

The order of a torsion class is defined to be the order of any one of its elements.

Proposition 10.10**.**

If $p$ is a prime number, and $\gamma\in\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}}))$ is a torsion class, then $T_{p}=\operatorname{id}+\mathbf{e}_{p}$ and

[TABLE]

Proof.

First observe that since there is only one conjugacy class of subgroups of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ of order 1, 2, 3, 4 and 6, the conjugacy class of an element of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ of finite order is determined by the angle through which it rotates the tangent space of ${\mathfrak{h}}$ at any one of its fixed points. Since $g_{p}:{\mathfrak{h}}\to{\mathfrak{h}}$ is conformal, this implies that if $\mu\in{\Gamma}_{0}(p)$, then $g_{p}\mu g_{p}^{-1}\in{\mathrm{SL}}_{2}({\mathbb{Z}})$ and that $\mu$ and $g_{p}\mu g_{p}^{-1}$ are conjugate in ${\mathrm{SL}}_{2}({\mathbb{Z}})$. The definition of $T_{p}$ on ${\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}}))$ then implies that $T_{p}=\pi_{\ast}\pi^{\ast}=\operatorname{id}+\mathbf{e}_{p}$ on the finite order conjugacy classes of ${\mathrm{SL}}_{2}({\mathbb{Z}})$.

Regarding the computation of $T_{p}$: the first case is immediate as $\pm\operatorname{id}$ acts trivially on ${\mathbb{P}}^{1}({{\mathbb{F}}_{p}})$.

The first interesting case is classes of order 4. There are two conjugacy classes of them in ${\mathrm{SL}}_{2}({\mathbb{Z}})$. They are switched by multiplication by $-1$. Since the minimal polynomial of an element of order 4 of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ is $x^{2}-1$, every element ${\tilde{\gamma}}$ of order 4 of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ has trace 0 and discriminant $\Delta=-4$. Since ${\tilde{\gamma}}^{2}=-\operatorname{id}$, it acts on ${\mathbb{P}}^{1}({\mathbb{F}}_{p})$ as a product of 1- and 2-cycles. When $p=2$, $\Delta\equiv 0$ and ${\tilde{\gamma}}$ has one fixed point. Consequently

[TABLE]

When $p\equiv 1\bmod 4$, $\Delta$ is a non-zero square mod $p$, so that ${\tilde{\gamma}}$ acts on ${\mathbb{P}}^{1}({\mathbb{F}}_{p})$ with exactly 2 fixed points (its eigenspaces). In this case,

[TABLE]

And when $p\equiv 3\bmod 4$, $\Delta$ is not a square mod $p$, and so ${\tilde{\gamma}}$ acts on ${\mathbb{P}}^{1}({\mathbb{F}}_{p})$ as a product of $(p+1)/2$ disjoint 2-cycles. Consequently

[TABLE]

Next we consider classes of order 3 and 6. In ${\mathrm{SL}}_{2}({\mathbb{Z}})$, there are two conjugacy classes of elements of order 3, and two of order 6; $\gamma$ has order 3 if and only if $-\gamma$ has order 6. Each of these classes is determined by the order of a representative ${\tilde{\gamma}}\in{\mathrm{SL}}_{2}({\mathbb{Z}})$ and the angle of rotation it induces in the tangent space of any of its fixed points.

The minimal polynomial of an element ${\tilde{\gamma}}$ of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ of order 3 is $x^{2}+x+1$. It thus has trace $-1$ and discriminant $\Delta=-3$. When $p=3$, $\Delta\equiv 0$, which implies that ${\tilde{\gamma}}$ acts on ${\mathbb{P}}^{1}({\mathbb{F}}_{3})$ with 1 fixed point and one 3-cycle. In this case,

[TABLE]

Quadratic reciprocity implies that when $p>3$, $-3$ is a square mod $p$ if and only if $p\equiv 1\bmod 3$. In this case, ${\tilde{\gamma}}$ acts on ${\mathbb{P}}^{1}({\mathbb{F}}_{p})$ with 2 fixed points (eigenspaces) and $(p-1)/3$ orbits of length 3, so that

[TABLE]

When $p\equiv-1\bmod 3$, the eigenvalues of ${\tilde{\gamma}}$ lie in a quadratic extension of ${{\mathbb{F}}_{p}}$, and the action of ${\tilde{\gamma}}$ on ${\mathbb{P}}^{1}({{\mathbb{F}}_{p}})$ has no fixed points. So, in this case,

[TABLE]

Since the action of $\gamma$ and $-\gamma$ on ${\mathbb{P}}^{1}({{\mathbb{F}}_{p}})$ are identical, the analysis for elements of order 6 is similar and is omitted. ∎

The previous result implies that, unlike in the general case, $T_{p}$ and $\mathbf{e}_{p}$ commute when restricted to the finite order conjugacy classes. This allows us to give a quick computation of the action $T_{p^{n}}$ (and thus all $T_{N}$) on the torsion classes. It is an immediate consequence of Theorem 9.1 and Proposition 10.10.

Corollary 10.11**.**

If $\gamma\in\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}}))$ is a torsion conjugacy class, then

[TABLE]

11. The Hecke Action in Higher Level, Higher Rank, etc

One can also define Hecke operators in higher level. Since we are mainly focused on level 1 and to avoid technical complications, we will not say much. However, it is worth mentioning that if $Y$ is one of $Y_{0}(m)$, $Y_{1}(m)$ or $Y(m)$ (definitions recalled in Section 2) and if $\gcd(m,N)=1$, then we have the unramified correspondence

[TABLE]

and we can define $T_{N}:{\mathbb{Z}}\lambda(Y)\to{\mathbb{Z}}\lambda(Y)$ as $\pi^{\prime}_{\ast}\circ\pi^{\ast}$. The point here is that, since $m$ and $N$ are relative prime, an $N$-fold covering $E^{\prime}\to E$ of elliptic curves induces an isomorphism $E^{\prime}[m]\to E[m]$ of $m$-torsion subgroups. This ensures that the projection $\pi^{\prime}$ is well defined and allows the definition of $T_{N}$.

One case of potential importance in the study of multiple zeta values is where $Y=Y(2)$, the moduli stack of elliptic curves with a full level 2 structure. The associated coarse moduli space is the thrice punctured sphere ${{\mathbb{P}}^{1}-\{0,1,\infty\}}$. So for all odd primes $p$, we have Hecke operators

[TABLE]

Remark 11.1*.*

It is clear that one can similarly define an action of Hecke operators on ${\mathbb{Z}}\boldsymbol{\lambda}({\Gamma})$ in more general situations, such as where ${\Gamma}$ is a lattice, such as ${\mathrm{GL}}_{n}({\mathbb{Z}})$ or ${\mathrm{Sp}}_{g}({\mathbb{Z}})$. More generally, one can define Hecke-like operators

[TABLE]

where ${\Gamma}_{g}$ denotes the mapping class group associated to a closed surfaces of genus $g$ and where $f:S\to T$ is the topological model of an unramified covering of compact oriented surfaces of genus $g$ by one of genus $h$.

12. The Hecke Action on ${\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}}))$ and its characters

Fix a prime number $p$. To better understand $T_{p^{n}}:{\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}}))\to{\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}}))$ we write

[TABLE]

where

[TABLE]

and $\ell$ ranges over the prime numbers $\neq p$. The second identity of Theorem 9.1 reduces the task of understanding the action of $T_{p^{n}}$ to understanding the actions of $T_{p}$ and $\mathbf{e}_{p}$.

There are natural inclusions and projections

[TABLE]

Since ${\mathrm{SL}}_{2}({\mathbb{Z}}_{p^{\prime}})$ is contained in the closure of ${\Gamma}_{0}(p)$ in ${\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$, it follows that both $T_{p}$ and $\mathbf{e}_{p}$ commute with both maps in (25).

The product decomposition further implies that

[TABLE]

which yields a canonical isomorphism

[TABLE]

Recall the definition of $g_{p}$ from equation (22).

Proposition 12.1**.**

If $\alpha\in\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}}_{p}))$ and $\alpha^{\prime}\in\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}}_{p^{\prime}}))$, then

[TABLE]

In particular,

[TABLE]

Suppose that ${\Bbbk}$ is a commutative ring. Endow it with the discrete topology. As in Section 7, ${\mathscr{C}\!\ell}_{\Bbbk}({\mathrm{SL}}_{2}({\mathbb{Z}}))$ denotes the group

[TABLE]

of continuous ${\Bbbk}$-valued class functions on ${\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$. The Hecke operator $T_{N}$ and the operator $\mathbf{e}_{p}$ induce dual operators

[TABLE]

The next result is an immediate consequence of Theorem 9.1.

Proposition 12.2**.**

When $m$ and $n$ are relatively prime, the dual Hecke operators ${\check{T}}_{n}$ and ${\check{T}}_{m}$ commute. When $n\geq 1$ we have

[TABLE]

The ring of class functions on ${\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$ decomposes:

[TABLE]

Combining this with Proposition 12.1, we obtain:

Proposition 12.3**.**

For all $\chi\in{\mathscr{C}\!\ell}_{\Bbbk}({\mathrm{SL}}_{2}({\mathbb{Z}}_{p}))$ and $\chi^{\prime}\in{\mathscr{C}\!\ell}_{\Bbbk}({\mathrm{SL}}_{2}({\mathbb{Z}}_{p^{\prime}}))$, we have

[TABLE]

where $(g_{p}^{\ast}\chi^{\prime})(\mu):=\chi^{\prime}(g_{p}\mu g_{p}^{-1})$.

Remark 12.4*.*

The results of this section can be easily generalized to groups such as ${\mathrm{GL}}_{n}({\widehat{{\mathbb{Z}}}})$.

Part III Relative Unipotent Completions of Modular Groups

The primary goal of this section is to explain the various incarnations of the relative completion of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ that we will be working with and to explain the various structures on them, such as mixed Hodge structures and Galois actions. As we shall see in the next part, these structures pass to the rings of class of functions on each, which correspond under the canonical comparison maps.

This discussion necessitates a discussion of the modular curve, both as a stack over ${\mathbb{Q}}$ and as a complex analytic orbifold. Even though this material is surely very well known, much of it classically, we give a quick review of it to fix notation and normalizations, and also for the convenience of the reader. An expanded version of some of the topics can be found in Section 4 of [17]. Unlike in the previous part, in this part ${\mathcal{M}}_{1,1}$ will be regarded as a stack over ${\mathbb{Q}}$ and ${\mathcal{M}}_{1,1}^{\mathrm{an}}$ will denote the corresponding complex analytic orbifold.

13. The Modular Curve as an Orbifold and as a Stack

13.1. The modular curve as a stack over ${\mathbb{Q}}$

The moduli space ${\mathcal{M}}_{1,{\vec{1}}/{\mathbb{Q}}}$ of smooth elliptic curves together with a non-zero abelian differential is the scheme $\operatorname{Spec}{\mathbb{Q}}[u,v,D^{-1}]$, where $D$ is (up to a factor of 4) the discriminant $u^{3}-27v^{2}$ of the plane cubic

[TABLE]

In other words, ${\mathcal{M}}_{1,{\vec{1}}}$ is the complement in ${\mathbb{A}}^{2}_{\mathbb{Q}}$ of the cusp $D=0$. The point $(u,v)$ corresponds to the elliptic curve defined by the equation above together with the abelian differential $dx/y$. All points of $D-\{0\}$ correspond to the nodal cubic. The multiplicative group ${\mathbb{G}_{m}}$ acts on ${\mathbb{A}}^{2}_{\mathbb{Q}}-\{0\}$ by

[TABLE]

This action takes the point of ${\mathcal{M}}_{1,{\vec{1}}}$ that corresponds to $(E,{\omega})$ to the point that corresponds to $(E,t{\omega})$, where $E$ is an elliptic curve (possibly nodal) and ${\omega}$ is a translation invariant differential on $E$.

The moduli stack ${\mathcal{M}}_{1,1/{\mathbb{Q}}}$ is the stack quotient

[TABLE]

and its Deligne–Mumford compactification ${\overline{{\mathcal{M}}}}_{1,1}$ is

[TABLE]

The universal elliptic curve over ${\mathbb{A}}^{2}_{\mathbb{Q}}-\{0\}$ is the subscheme of

[TABLE]

defined by $y^{2}z=4x^{3}-uxz^{2}-vz^{3}$. The ${\mathbb{G}_{m}}$ action lifts to this curve:

[TABLE]

The stack quotient is the universal elliptic curve ${\mathcal{E}}_{/{\mathbb{Q}}}$ over ${\overline{{\mathcal{M}}}}_{1,1/{\mathbb{Q}}}$.

The relative dualizing sheaf of the universal elliptic curve over ${\mathbb{A}}^{2}_{\mathbb{Q}}-\{0\}$ is the trivial line bundle $({\mathbb{A}}^{2}_{\mathbb{Q}}-\{0\})\times{\mathbb{A}}_{\mathbb{Q}}^{1}$. The point $(u,v,s)$ corresponds to the section that takes the elliptic curve (26) to $sdx/y$. Denote it by ${\widetilde{{\mathcal{L}}}}$. The ${\mathbb{G}_{m}}$ action on ${\mathbb{A}}^{2}_{\mathbb{Q}}$ lifts to ${\widetilde{{\mathcal{L}}}}$:

[TABLE]

Observe that $u$ is a ${\mathbb{G}_{m}}$-invariant section of ${\widetilde{{\mathcal{L}}}}^{4}$ and that $v$ is a ${\mathbb{G}_{m}}$-invariant section of ${\widetilde{{\mathcal{L}}}}^{6}$.

The line bundle ${\mathcal{L}}$ over ${\overline{{\mathcal{M}}}}_{1,1/{\mathbb{Q}}}$ is the stack quotient of ${\widetilde{{\mathcal{L}}}}$ by this ${\mathbb{G}_{m}}$ action. Note that $u$ descends to a section of ${\mathcal{L}}^{4}$ over ${\overline{{\mathcal{M}}}}_{1,1}$ and $v$ to a section of ${\mathcal{L}}^{6}$. These correspond to appropriate normalizations (see below) of the Eisenstein series of weights 4 and 6, respectively.

13.2. The modular curve as a complex analytic orbifold

The complex analytic orbifold ${\mathcal{M}}_{1,1}^{\mathrm{an}}$ associated to ${\mathcal{M}}_{1,1}$ is the orbifold quotient

[TABLE]

of the upper half plane ${\mathfrak{h}}$ by the standard action of ${\mathrm{SL}}_{2}({\mathbb{Z}})$. For $\tau\in{\mathfrak{h}}$, set $\Lambda_{\tau}={\mathbb{Z}}\oplus{\mathbb{Z}}\tau$. The point $\tau\in{\mathfrak{h}}$ corresponds to the elliptic curve

[TABLE]

together with the symplectic basis $\mathbf{a},\mathbf{b}$ of $\Lambda_{\tau}\cong H_{1}(E_{\tau};{\mathbb{Z}})$, where $\mathbf{a}$ and $\mathbf{b}$ correspond to the elements of the symplectic basis $1,\tau$ of $\Lambda_{\tau}\cong H_{1}(E_{\tau};{\mathbb{Z}})$.

The quotient of ${\mathfrak{h}}$ by the group

[TABLE]

is a punctured disk ${\mathbb{D}}^{\ast}$ with coordinate $q=e^{2\pi i\tau}$. Since the stabilizer of $\infty$ in ${\mathrm{SL}}_{2}({\mathbb{Z}})$ is

[TABLE]

there is an orbifold map $C_{2}{\backslash\negthickspace\backslash}{\mathbb{D}}^{\ast}\to{\mathcal{M}}_{1,1}^{\mathrm{an}}$, where $C_{2}=\{\pm 1\}$ acts trivially on ${\mathbb{D}}^{\ast}$. The analytic orbifold ${\overline{{\mathcal{M}}}}_{1,1}^{\mathrm{an}}$ is the orbifold obtained by glueing a copy of $C_{2}{\backslash\negthickspace\backslash}{\mathbb{D}}$ onto ${\mathcal{M}}_{1,1}^{\mathrm{an}}$ via this map.

The line bundle ${\mathcal{L}}^{\mathrm{an}}\to{\mathcal{M}}_{1,1}^{\mathrm{an}}$ is the quotient of ${\mathbb{C}}\times{\mathfrak{h}}$ by the action

[TABLE]

Its restriction to ${\mathbb{D}}^{\ast}$ is trivial and thus extends naturally to ${\mathbb{D}}$. This is the canonical extension of ${\mathcal{L}}^{\mathrm{an}}$ to ${\overline{{\mathcal{M}}}}_{1,1}^{\mathrm{an}}$.

13.3. The space of lattices and ${\mathcal{M}}_{1,{\vec{1}}}^{\mathrm{an}}$

Denote the set of lattices111111This can be identified with ${\mathrm{SL}}_{2}({\mathbb{Z}})\backslash{\mathrm{GL}}_{2}^{+}({\mathbb{R}})$. The right ${\mathbb{C}}^{\ast}$ action is induced by right multiplication, where ${\mathbb{C}}^{\ast}$ is regarded as a subgroup of ${\mathrm{GL}}_{2}^{+}({\mathbb{R}})$ in the standard way. in ${\mathbb{C}}$ by ${\mathscr{R}}$. The group ${\mathbb{C}}^{\ast}$ acts on it; $z\in{\mathbb{C}}^{\ast}$ takes the lattice $\Lambda$ to $z\Lambda$. The map

[TABLE]

is a ${\mathbb{C}}^{\ast}$-equivariant covering map, where ${\mathbb{C}}^{\ast}$ acts on ${\mathbb{C}}^{\ast}\times{\mathfrak{h}}$ by multiplication on the first factor. The covering group is ${\mathrm{SL}}_{2}({\mathbb{Z}})$, which acts on ${\mathbb{C}}^{\ast}\times{\mathfrak{h}}$ by

[TABLE]

This map descends to a ${\mathbb{C}}^{\ast}$ equivariant bijection ${\mathscr{R}}\cong{\mathrm{SL}}_{2}({\mathbb{Z}})\backslash({\mathbb{C}}^{\ast}\times{\mathfrak{h}})$, which gives ${\mathscr{R}}$ the structure of a complex manifold. This descends to an analytic orbifold isomorphism ${\mathcal{M}}_{1,1}^{\mathrm{an}}={\mathrm{SL}}_{2}({\mathbb{Z}}){\backslash\negthickspace\backslash}{\mathfrak{h}}\cong{\mathscr{R}}{/\negthickspace/}{\mathbb{C}}^{\ast}$.

13.4. The comparison

The goal of this section is to recall the comparison map between the complex analytic and algebraic constructions of ${\mathcal{M}}_{1,1}$. More precisely, we prove the following result. The proof is classical, but we include a sketch of it as later we need the precise isomorphism.

Throughout this subsection, we regard ${\mathcal{M}}_{1,{\vec{1}}}({\mathbb{C}})$ as the complex manifold ${\mathbb{C}}^{2}-D^{-1}(0)$.

Proposition 13.1**.**

The map ${\mathscr{R}}\to{\mathcal{M}}_{1,{\vec{1}}}({\mathbb{C}})$ defined by taking a lattice $\Lambda$ to $({\mathbb{C}}/\Lambda,2\pi idz)$ is a ${\mathbb{C}}^{\ast}$ equivariant biholomorphism. It induces an orbifold isomorphism ${\mathcal{M}}_{1,1}^{\mathrm{an}}\to{\mathcal{M}}_{1,1}({\mathbb{C}})$.

The proof uses Eisenstein series and the Weierstrass $\wp$ function. Suppose that $k\geq 2$. The (normalized) Eisenstein series of weight $2k$ is defined by the series

[TABLE]

where $\sigma_{k}(n)=\sum_{d|n}d^{k}$. It converges absolutely to a modular form of weight $2k$. The Weierstrass $\wp$-function associated to a lattice $\Lambda$ in ${\mathbb{C}}$ is defined by

[TABLE]

For $\tau\in{\mathfrak{h}}$, set $\wp_{\tau}(z)=\wp_{\Lambda_{\tau}}(z)$. The function ${\mathbb{C}}\to{\mathbb{P}}^{2}({\mathbb{C}})$ defined by

[TABLE]

induces an embedding of ${\mathbb{C}}/\Lambda_{\tau}$ into ${\mathbb{P}}^{2}({\mathbb{C}})$ as the plane cubic with affine equation

[TABLE]

where

[TABLE]

The abelian differential $dx/y$ pulls back to the 1-form $2\pi idz$ on ${\mathbb{C}}/\Lambda_{\tau}$.

The holomorphic map ${\mathbb{C}}^{\ast}\times{\mathfrak{h}}\to{\mathbb{C}}^{2}-D^{-1}(0)$ defined by

[TABLE]

is surjective (as every smooth elliptic curve over ${\mathbb{C}}$ is of the form ${\mathbb{C}}/\Lambda)$ and is constant on ${\mathrm{SL}}_{2}({\mathbb{Z}})$ the orbits of ${\mathbb{C}}^{\ast}\times{\mathfrak{h}}$. It therefore induces a holomorphic map

[TABLE]

which is a biholomorphism as one can construct a holomorphic inverse by taking the point that corresponds to the pair $(E,{\omega})$ to its period lattice

[TABLE]

14. Fundamental Groups and the Galois Action on ${\mathrm{SL}}_{2}({\mathbb{Z}})^{\wedge}$

Since every elliptic curve has the non-trivial automorphism that takes each point to its inverse, the moduli space ${\mathcal{M}}_{1,1/{\overline{{\mathbb{Q}}}}}$ does not have a function field.121212The étale fundamental group of a smooth curve over ${\overline{{\mathbb{Q}}}}$ is isomorphic to the Galois group of the maximal algebraic extension of its function field that is unramified at each of its points. For this reason, we need to explain why the absolute Galois group $\operatorname{Gal}({\overline{{\mathbb{Q}}}}/{\mathbb{Q}})$ acts on the profinite completion ${\mathrm{SL}}_{2}({\mathbb{Z}})^{\wedge}$ of its orbifold fundamental group. To do this, we exploit the facts that ${\mathrm{SL}}_{2}({\mathbb{Z}})$ is a quotient of $B_{3}$, the braid group on 3-strings, and that ${\mathcal{M}}_{1,{\vec{1}}/{\overline{{\mathbb{Q}}}}}$ is a smooth scheme, so that its étale fundamental group is the profinite completion of its topological fundamental group, which is well-known to be $B_{3}$.

14.1. Braid groups

Recall that the braid group $B_{n}$ is the fundamental group of the space of monic polynomials of degree $n$ with complex coefficients that have non-vanishing discriminant. This space of polynomials retracts onto the space of polynomials

[TABLE]

with distinct roots that sum to 0. The center of the braid group is generated by the “full twist”.

The space ${\mathcal{M}}_{1,{\vec{1}}}({\mathbb{C}})={\mathbb{C}}^{2}-D^{-1}(0)$ is the space of polynomials $4x^{3}-ux-v$. By setting $T=4^{1/3}x$, it follows that its fundamental group is isomorphic to $B_{3}$. Fix a base point $(u_{o},v_{o})$ of ${\mathcal{M}}_{1,{\vec{1}}}({\mathbb{C}})$. Let $E_{o}$ be the corresponding elliptic curve. The monodromy action of $B_{3}$ on $H_{1}(E_{o};Z)$ defines a homomorphism $B_{3}\to{\mathrm{SL}}(H_{1}(E_{o};{\mathbb{Z}}))$.

The following is well known. A proof can be found in [12, §8].

Proposition 14.1**.**

The braid group $B_{3}$ is isomorphic to the fundamental group of the complement of the trefoil knot in $S^{3}$. It has presentation

[TABLE]

The center of $B_{3}$ is infinite cyclic and generated by the full twist $s^{2}$. For a suitable choice of symplectic basis of $H_{1}(E_{o};{\mathbb{Z}})$, the monodromy homomorphism $B_{3}\to{\mathrm{SL}}_{2}({\mathbb{Z}})$ is defined by

[TABLE]

It is surjective. The full twist is mapped to $-\operatorname{id}$, so that the kernel is generated by the square of a full twist.

Corollary 14.2**.**

The group $B_{3}$ is an extension

[TABLE]

where the kernel is generated by the square of a full twist.

The pure braid group $P_{n}$ is the kernel of the natural homomorphism $B_{n}\to\Sigma_{n}$ onto the permutation group of the roots of the polynomial corresponding to the base point. It is the fundamental group of the complement of the “discriminant” divisor $\prod_{j<k}(\lambda_{j}-\lambda_{k})$ in the hyperplane $\lambda_{1}+\dots+\lambda_{n}=0$ in ${\mathbb{C}}^{n}$.

Define ${\mathrm{PSL}}_{2}({\mathbb{Z}})[2]$ to be ${\mathrm{SL}}_{2}({\mathbb{Z}})[2]/(-\operatorname{id})$. Using the fact the 2-torsion points of an elliptic curve $y^{2}=f(x)$ correspond to the roots of the cubic polynomial $f(x)$ and the fact that $\Sigma_{3}$ is isomorphic to ${\mathrm{SL}}_{2}({\mathbb{F}}_{2})$, one can easily show that the composite

[TABLE]

is the natural homomorphism to $\Sigma_{3}$. It follows that $P_{3}$ is an extension

[TABLE]

were the kernel is generated by a full twist. This is the geometric incarnation of the ${\mathbb{C}}^{\ast}$ torsor

[TABLE]

From this (or otherwise), it follows that ${\mathrm{PSL}}_{2}({\mathbb{Z}})[2]$ is isomorphic to the fundamental group of ${{\mathbb{P}}^{1}-\{0,1,\infty\}}$ and is therefore free of rank 2. The section $t\mapsto(0,1,t)$ induces a splitting on fundamental groups.

Corollary 14.3**.**

The sequence (30) is split exact, which implies that $P_{3}$ is isomorphic to the product $F_{2}\times{\mathbb{Z}}$ of a free group of rank 2 with the infinite cyclic group generated by a full twist.

Remark 14.4*.*

One can define Hecke operators on ${\mathscr{R}}$ as in [34, VII.§5]. The action of $T_{N}$ on ${\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}}))$ lifts to an action of $T_{N}$ on ${\mathbb{Z}}\boldsymbol{\lambda}(B_{3})$. The action of $t\in{\mathbb{C}}^{\ast}$ on ${\mathscr{R}}$ corresponds to Serre’s operator $R_{t}$. The relations (5) lift, but will also involve the operator $R_{p}$. Note, however, that each $R_{t}$ acts trivially on $\boldsymbol{\lambda}(B_{3})$.

14.2. Profinite completion

In this section, ${\Gamma}^{\wedge}$ will denote the profinite completion of the discrete group ${\Gamma}$.

Proposition 14.5**.**

The profinite completion

[TABLE]

of the exact sequence (29) is exact.

Proof.

It is elementary to show that profinite completion is right exact. To prove left exactness here, it suffices to prove that the kernel of $B_{3}^{\wedge}\to{\mathrm{PSL}}_{2}({\mathbb{Z}})^{\wedge}$ is topologically generated by a full twist. Since $P_{3}$ has finite index in $B_{3}$, it suffices to show that the kernel of $P_{3}^{\wedge}\to{\mathrm{PSL}}_{2}({\mathbb{Z}})[2]^{\wedge}$ is topologically generated by a full twist. But this follows immediately from Corollary 14.3 as the sequence (30) is split exact and as the profinite completion of a product is the product of the completions. ∎

14.3. Base points

In order to specify the Galois action on the étale/orbifold fundamental group of ${\mathcal{M}}_{1,1/{\overline{{\mathbb{Q}}}}}$, we need to specify a base point. We will also specify the corresponding base point in the Betti case. In both cases, we will have occasion to use tangential base points as defined by Deligne in [7, §15].

14.3.1. The Betti case

One can use any orbifold map $b$ from a simply connected space $B$ to ${\mathcal{M}}_{1,1}^{\mathrm{an}}$ as a base point. A diagram

[TABLE]

in which $B$ and $B^{\prime}$ are simply connected determines an isomorphism $\phi_{\ast}:\pi_{1}({\mathcal{M}}_{1,1}^{\mathrm{an}},b)\to\pi_{1}({\mathcal{M}}_{1,1}^{\mathrm{an}},b^{\prime})$.

Standard and useful choices of base points include:

1. (i)

the quotient map $p:{\mathfrak{h}}\to{\mathcal{M}}_{1,1}^{\mathrm{an}}$, 2. (ii)

the map $\{\tau\}\to{\mathfrak{h}}\to{\mathcal{M}}_{1,1}^{\mathrm{an}}$, where $\tau\in{\mathfrak{h}}$, 3. (iii)

the map $i(y_{o},\infty)\to{\mathfrak{h}}\to{\mathcal{M}}_{1,1}^{\mathrm{an}}$ from a segment of the imaginary axis.

The corresponding fundamental groups will be denoted $\pi_{1}({\mathcal{M}}_{1,1}^{\mathrm{an}},p)$, $\pi_{1}({\mathcal{M}}_{1,1}^{\mathrm{an}},\tau)$, and $\pi_{1}({\mathcal{M}}_{1,1},\partial/\partial q)$, respectively.131313Note that the image of $i(y_{o},\infty)$ in the $q$-disk is the segment $(0,e^{-2\pi y_{o}})$ of the positive real axis, which lies in the direction of the tangent vector $\partial/\partial q$ under the standard identification of the real and holomorphic tangent spaces. The inclusions $i(y_{o},\infty)\to{\mathfrak{h}}$ and $\{\tau\}\to{\mathfrak{h}}$ induce natural isomorphisms

[TABLE]

The action of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ on ${\mathfrak{h}}$ induces a natural isomorphism $\pi_{1}({\mathcal{M}}_{1,1}^{\mathrm{an}},p)\to{\mathrm{SL}}_{2}({\mathbb{Z}})$ and therefore an isomorphism

[TABLE]

14.3.2. The étale case

For each choice of geometric point ${\overline{x}}$ of ${\mathcal{M}}_{1,{\vec{1}}/{\mathbb{Q}}}$, one can define $\pi_{1}^{\mathrm{\acute{e}t}}({\mathcal{M}}_{1,{\vec{1}}/{\mathbb{Q}}},{\overline{x}})$, which is an extension

[TABLE]

If ${\overline{x}}$ lies above $x\in{\mathcal{M}}_{1,{\vec{1}}}({\mathbb{Q}})$, we get a splitting of this sequence and therefore a natural action of the absolute Galois group $\operatorname{Gal}({\overline{{\mathbb{Q}}}}/{\mathbb{Q}})$ on the geometric étale fundamental group $\pi_{1}^{\mathrm{\acute{e}t}}({\mathcal{M}}_{1,{\vec{1}}/{\overline{{\mathbb{Q}}}}},{\overline{x}})$ of ${\mathcal{M}}_{1,{\vec{1}}}$. By [9, XII, Cor. 5.2], the geometric étale fundamental group is canonically isomorphic to the profinite completion of the topological fundamental group of ${\mathcal{M}}_{1,{\vec{1}}}({\mathbb{C}})$, so that we have isomorphisms141414Here and below, we are regarding the ${\mathbb{Q}}$-point $x$ as lying in ${\mathcal{M}}_{1,{\vec{1}}}({\mathbb{C}})$, so that it can be used as a base point for the topological fundamental group.

[TABLE]

We can use the same base point ${\overline{x}}$ as a base point of ${\mathcal{M}}_{1,1/{\mathbb{Q}}}$. For simplicity, we suppose that the elliptic curve corresponding to ${\overline{x}}$ has automorphism group $\{\pm\operatorname{id}\}$. The ${\mathbb{G}_{m}}$ orbit $\boldsymbol{O}_{\overline{x}}$ of ${\overline{x}}$ in ${\mathcal{M}}_{1,{\vec{1}}}$ is isomorphic to ${\mathbb{G}_{m}}_{/{\overline{{\mathbb{Q}}}}}$ and has geometric étale fundamental group isomorphic to ${\widehat{{\mathbb{Z}}}}(1)$. It is topologically generated by a full twist. The étale double covering $\boldsymbol{O}_{\overline{x}}^{\prime}$ of the orbit also has fundamental group ${\widehat{{\mathbb{Z}}}}(1)$ and is topologically generated by the square of a full twist. Define the étale fundamental group of ${\mathcal{M}}_{1,1/{\mathbb{Q}}}$ by

[TABLE]

and its geometric étale fundamental group by

[TABLE]

so that there is an exact sequence

[TABLE]

There is a natural exact sequence

[TABLE]

Proposition 14.5 implies that there is a natural isomorphism

[TABLE]

14.3.3. The Tate curve as base point

This is the étale analogue of $\partial/\partial q$. It also gives a natural lift of $\partial/\partial q$ to ${\mathcal{M}}_{1,{\vec{1}}}^{\mathrm{an}}$. It is needed at various points, including the discussion of certain étale local systems over ${\mathcal{M}}_{1,1}$ in Section 15.1.1 below.

The Tate elliptic curve [36, Chapt. V] is defined by $Y^{2}+XY=X^{3}+a_{4}(q)X+a_{6}(q)$, where $a_{4}(q)$ and $a_{6}(q)$ are in ${\mathbb{Z}}[[q]]$. It has discriminant $\Delta(q)$, the normalized cusp form of weight 12. Since $\Delta\equiv q$ mod $q^{2}$, its pullback to ${\mathbb{Z}}[q]/(q^{2})$ is a smooth elliptic curve with good reduction at all primes $p$. This curve can be regarded as the fiber of the universal elliptic curve over $\partial/\partial q$.

After a change of variables [36, V§1], the pullback of the Tate curve to ${\mathbb{Q}}[[q]]$ has affine equation $y^{2}=4x^{2}+g_{2}(q)x+g_{3}(q)$. It corresponds to a map

[TABLE]

which restricts to a map

[TABLE]

where ${\Bbbk}(\!(q)\!)$ denotes the ring of formal Laurent series in the indeterminate $q$ with coefficients in the field ${\Bbbk}$. When ${\Bbbk}$ is a field of characteristic zero, its algebraic closure is ${\overline{{\Bbbk}}}(\!(q^{1/N}:N\geq 1)\!)$, the field of Puiseux series in $q$ which is generated by compatible $N$th roots $q^{1/N}$ of $q$.

For all ${\mathbb{Q}}$-algebras, the map above extends to a “formal geometric point”

[TABLE]

of ${\mathcal{M}}_{1,{\vec{1}}/{\Bbbk}}$. The right automorphisms of the associated fiber functor from finite étale covers to the category of finite sets151515See convention in Section 2. is $\pi_{1}^{\mathrm{\acute{e}t}}({\mathcal{M}}_{1,{\vec{1}}/{\Bbbk}},{\vec{\mathsf{v}}})$.

Since $\operatorname{Gal}({\overline{{\mathbb{Q}}}}(\!(q^{1/N}:N\geq 1)\!))/{\mathbb{Q}}(\!(q^{1/N}:N\geq 1)\!)$ is isomorphic to $\operatorname{Gal}({\overline{{\mathbb{Q}}}}/{\mathbb{Q}})$, there is a split exact sequence

[TABLE]

and a natural $\operatorname{Gal}({\overline{{\mathbb{Q}}}}/{\mathbb{Q}})$ action on $\pi_{1}^{\mathrm{\acute{e}t}}({\mathcal{M}}_{1,{\vec{1}}/{\overline{{\mathbb{Q}}}}},{\vec{\mathsf{v}}})$.

The corresponding analytic construction is to consider the map ${\mathbb{D}}^{\ast}\to{\mathcal{M}}_{1,{\vec{1}}}^{\mathrm{an}}$ defined on the $q$-disk by $q\mapsto(g_{2}(q),g_{3}(q))$. The lift of $\partial/\partial q$ is the restriction of this map to the positive real axis. We will denote it by ${\vec{\mathsf{v}}}^{\mathrm{an}}$. It projects to the base point $\partial/\partial q$ of ${\mathcal{M}}_{1,1}^{\mathrm{an}}$ defined above.

Proposition 14.6**.**

There is a natural isomorphism

[TABLE]

Consequently, there is a natural $\operatorname{Gal}({\overline{{\mathbb{Q}}}}/{\mathbb{Q}})$ action on the profinite completion of $\pi_{1}({\mathcal{M}}_{1,{\vec{1}}}^{\mathrm{an}},{\vec{\mathsf{v}}}^{\mathrm{an}})^{\wedge}$.

Proof.

It suffices to give a sequence of maps of “base points” that interpolates between ${\vec{\mathsf{v}}}$ and ${\vec{\mathsf{v}}}^{\mathrm{an}}$ and thus between their fiber functors. Denote by ${\mathbb{C}}\{\!\{q\}\!\}$ the ring of power series in $q$ that correspond to germs at the origin of holomorphic functions on the $q$-disk. For each $N\geq 1$, set

[TABLE]

This is a compatible set of $N$th roots of $q$. Adjoin them to ${\mathbb{C}}\{\!\{q\}\!\}$ to obtain the algebraically closed field

[TABLE]

of Puiseux series. It imbeds naturally into the ring of germs at $q=0$ of continuous functions on the positive real axis of ${\mathbb{D}}$. The fiber functor ${\vec{\mathsf{v}}}^{\mathrm{an}}$ associated to pulling back finite étale covers of ${\mathcal{M}}_{1,{\vec{1}}}^{\mathrm{an}}$ to the positive real axis of ${\mathbb{D}}^{\ast}$ is isomorphic to the fiber functor $b$ associated to base changing the covers to ${\mathbb{C}}\{\!\{q^{1/N}:N\geq 1\}\!\}$. This gives an isomorphism

[TABLE]

Algebraically closed base change, [9, XIII,Prop. 4.6], gives the isomorphism

[TABLE]

To complete the proof, observe that the fiber functor $b$ on the category of finite étale coverings of ${\mathcal{M}}_{1,{\vec{1}}/{\overline{{\mathbb{Q}}}}}$ is isomorphic to the fiber functor obtained by base changing covers to ${\overline{{\mathbb{Q}}}}(\!(q^{1/N}:N\geq 1)\!)$ via the maps

[TABLE]

of algebraically closed fields. This gives isomorphisms

[TABLE]

∎

Remark 14.7*.*

This construction is a minor variant of those used by Ihara and Matsumoto [20] and Nakamura [26] to study the $\operatorname{Gal}({\overline{{\mathbb{Q}}}}/{\mathbb{Q}})$ action on the profinite braid groups $B_{n}^{\wedge}$. Nakamura studied the $\operatorname{Gal}({\overline{{\mathbb{Q}}}}/{\mathbb{Q}})$ action on $B_{3}^{\wedge}$ in detail in [27].

Corollary 14.8**.**

The $\operatorname{Gal}({\overline{{\mathbb{Q}}}}/{\mathbb{Q}})$ action on $\pi_{1}({\mathcal{M}}_{1,{\vec{1}}/{\overline{{\mathbb{Q}}}}},{\vec{\mathsf{v}}})$ induces a $\operatorname{Gal}({\overline{{\mathbb{Q}}}}/{\mathbb{Q}})$ action on ${\mathrm{SL}}_{2}({\mathbb{Z}})^{\wedge}$ via the isomorphisms

[TABLE]

Remark 14.9*.*

Each choice of an elliptic curve $E/{\mathbb{Q}}$ and a symplectic isomorphism

[TABLE]

of the Tate module of $E$ with ${\widehat{{\mathbb{Z}}}}^{2}$ gives a natural action of $\operatorname{Gal}({\overline{{\mathbb{Q}}}}/{\mathbb{Q}})$ on ${\widehat{{\mathbb{Z}}}}^{2}$ and thus on ${\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$. This descends to a Galois action on ${\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}}))$, which is independent of the choice of $E$. The quotient map ${\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}})^{\wedge})\to{\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}}))$ is Galois equivariant. The fact that the Tate module $H_{\widehat{{\mathbb{Z}}}}$ of $E_{\partial/\partial q}$ is ${\widehat{{\mathbb{Z}}}}(0)\oplus{\widehat{{\mathbb{Z}}}}(1)$ implies that the Galois action on ${\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}}))$ factors through the cyclotomic character $\chi:G_{\mathbb{Q}}\to{\widehat{{\mathbb{Z}}}}^{\times}$.

14.4. Hecke correspondences are Galois equivariant

We can now establish the Galois equivariance of the Hecke operators.

Theorem 14.10**.**

For each geometric base point $b$ of ${\mathcal{M}}_{1,1/{\mathbb{Q}}}$, the action of the absolute Galois group $\operatorname{Gal}({\overline{{\mathbb{Q}}}}/{\mathbb{Q}})$ on $\pi_{1}^{\mathrm{\acute{e}t}}({\mathcal{M}}_{1,1/{\overline{{\mathbb{Q}}}}},b)$ induces an action of $\operatorname{Gal}({\overline{{\mathbb{Q}}}}/{\mathbb{Q}})$ on ${\mathbb{Z}}\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}})^{\wedge})$. This action does not depend on the choice of the base point $b$. The operators $\mathbf{e}_{p}$ ($p$ prime) and the Hecke operators

[TABLE]

are $\operatorname{Gal}({\overline{{\mathbb{Q}}}}/{\mathbb{Q}})$-equivariant.

Proof.

The first two assertions follow easily from the results above. To prove the last assertion, it suffices to show that $T_{p}$ and $\mathbf{e}_{p}$ are Galois equivariant. The relations in Theorem 9.1 combined with Theorem 6.3 and Proposition 6.4 imply that, to do this, we need only show that $Y_{0}(p)$ and its two projections to ${\mathcal{M}}_{1,1}$ are defined over ${\mathbb{Q}}$. But this follows from the fact that $Y_{0}(p)_{/{\mathbb{Q}}}$ is the étale covering of ${\mathcal{M}}_{1,1/{\mathbb{Q}}}$ corresponding to the inverse image of the upper triangular Borel subgroup $B$ of ${\mathrm{GL}}_{2}({\mathbb{F}}_{p})$ in $\pi_{1}^{\mathrm{\acute{e}t}}({\mathcal{M}}_{1,1/{\mathbb{Q}}},b)$ under the homomorphism $\pi_{1}^{\mathrm{\acute{e}t}}({\mathcal{M}}_{1,1/{\mathbb{Q}}},b)\to{\mathrm{GL}}_{2}({\mathbb{F}}_{p}).$ It is geometrically connected as the canonical homomorphism $\pi_{1}(Y_{0}(p)_{/{\mathbb{Q}}},\ast)\to\operatorname{Gal}({\overline{{\mathbb{Q}}}}/{\mathbb{Q}})$ is surjective. ∎

15. Local systems and Connections on the Modular Curve

In preparation for defining the version of relative completion of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ we shall need when discussing the Hecke action on iterated integrals, we introduce some local systems over ${\mathcal{M}}_{1,1}$ and the corresponding connections.

15.1. Local systems

Let $A$ be a commutative ring. A local system ${\mathbb{V}}$ of $A$-modules over ${\mathcal{M}}_{1,1}^{\mathrm{an}}$ is a local system ${\widetilde{{\mathbb{V}}}}$ over ${\mathfrak{h}}$ endowed with a left ${\mathrm{SL}}_{2}({\mathbb{Z}})$ action such the projection ${\widetilde{{\mathbb{V}}}}\to{\mathfrak{h}}$ is ${\mathrm{SL}}_{2}({\mathbb{Z}})$ equivariant. Since ${\mathfrak{h}}$ is simply connected, ${\widetilde{{\mathbb{V}}}}$ is isomorphic to $V\times{\mathfrak{h}}$ as a local system, were $V$ is an $A$-module which is endowed with a natural left ${\mathrm{SL}}_{2}({\mathbb{Z}})$ action. The fundamental group $\pi_{1}({\mathcal{M}}_{1,1}^{\mathrm{an}},p)$ acts on $V$ on the right via $\gamma:v\to\gamma^{-1}v$. Consequently, local systems over ${\mathcal{M}}_{1,1}^{\mathrm{an}}$ correspond to right ${\mathrm{SL}}_{2}({\mathbb{Z}})$-modules.

Alternatively, a local system ${\mathbb{V}}$ over ${\mathcal{M}}_{1,1}^{\mathrm{an}}$ corresponds to a local system ${\mathbb{V}}^{\prime}$ over ${\mathcal{M}}_{1,{\vec{1}}}({\mathbb{C}})$ endowed with a ${\mathbb{G}_{m}}$ action. The restriction of ${\mathbb{V}}^{\prime}$ to each ${\mathbb{G}_{m}}$ orbit in ${\mathcal{M}}_{1,{\vec{1}}}({\mathbb{C}})$ is required to be trivial.161616Note that the action of the isotropy group at a point can act non-trivially on the fiber of ${\mathbb{V}}^{\prime}$ over it. The fiber is naturally a right $B_{3}$ module on which the square of the full twist acts trivially. As above, such local systems correspond to ${\mathrm{SL}}_{2}({\mathbb{Z}})$-modules.

When ${\Bbbk}$ is a field of characteristic zero, the category of local systems of finite dimensional ${\Bbbk}$-modules over ${\mathcal{M}}_{1,1}^{\mathrm{an}}$ is a ${\Bbbk}$-linear neutral tannakian category.

15.1.1. The local system ${\mathbb{H}}$

Denote the basis of $H_{1}(E_{\tau})$ that corresponds to the basis $1,\tau$ of $\Lambda_{\tau}\cong H_{1}(E_{\tau};{\mathbb{Z}})$ by $\mathbf{a},\mathbf{b}$. This basis is symplectic with respect to the intersection form $\langle{\phantom{x}},{\phantom{x}}\rangle$.

The local system ${\mathbb{H}}_{A}$ over ${\mathcal{M}}_{1,1}^{\mathrm{an}}$ is the local system whose fiber over the moduli point of an elliptic curve $E$ is $H_{1}(E;A)$. The corresponding local system $\widetilde{{\mathbb{H}}}$ over ${\mathfrak{h}}$ is naturally isomorphic to the trivial local system

[TABLE]

It is convenient to set

[TABLE]

which is the first homology group $H_{1}({\mathcal{E}}_{\mathfrak{h}};A)$ of the universal elliptic curve over ${\mathfrak{h}}$ and can also be regarded as the first homology of the fiber over the universal elliptic curve over $\partial/\partial q$.

The element

[TABLE]

of ${\mathrm{SL}}_{2}({\mathbb{Z}})$, which we identify with $\pi_{1}({\mathcal{M}}_{1,1}^{\mathrm{an}},\partial/\partial q)$ via (31), acts on the fiber $A\mathbf{a}\oplus A\mathbf{b}$ on the right by

[TABLE]

The dual local system ${\mathbb{H}}^{\vee}_{A}$ is the local system $R^{1}f_{\ast}A$ associated to the universal elliptic curve $f:{\mathcal{E}}\to{\mathcal{M}}_{1,1}^{\mathrm{an}}$. It has fiber $H^{1}(E;A)$ over the moduli point of $E$. Denote the basis of $H^{1}(E_{\tau})$ dual to the basis $\mathbf{a},\mathbf{b}$ of $H_{1}(E_{\tau})$ by $\mathbf{a}^{\vee},\mathbf{b}^{\vee}$. The natural left action of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ on $H^{1}(E_{\tau})$ is

[TABLE]

Poincaré duality induces the isomorphism ${\mathbb{H}}_{A}\to{\mathbb{H}}^{\vee}_{A}$ that takes $u$ to $\langle u,{\phantom{x}}\rangle$. It identifies the frame $\begin{pmatrix}\mathbf{b}^{\vee}&\mathbf{a}^{\vee}\end{pmatrix}$ of $H^{1}(E_{\tau})$ with $\begin{pmatrix}\mathbf{a}&-\mathbf{b}\end{pmatrix}$ of $H_{1}(E_{\tau})$. Under this identification, the natural left ${\mathrm{SL}}_{2}({\mathbb{Z}})$ action on $H_{A}$ is given by

[TABLE]

This is the same as the left action $x\mapsto x\gamma^{-1}$ obtained from the right action on $H_{A}$.

15.1.2. The étale local system ${\mathbb{H}}_{{\mathbb{Q}}_{\ell}}$ over ${\mathcal{M}}_{1,1/{\mathbb{Q}}}$

Fix a prime number $\ell$. There are natural isomorphisms

[TABLE]

This acts on $H_{{\mathbb{Q}}_{\ell}}$, the fiber of ${\mathbb{H}}_{{\mathbb{Q}}_{\ell}}$ over $\partial/\partial q$. It is well known (see, [27, §4], for example) that, as a Galois module,

[TABLE]

where ${\mathbb{Q}}(0)$ is spanned by $\mathbf{b}$ and ${\mathbb{Q}}(1)$ is spanned by $\mathbf{a}$. Consequently, $\pi_{1}^{\mathrm{\acute{e}t}}({\mathcal{M}}_{1,1/{\mathbb{Q}}},\partial/\partial q)$ acts on $H_{{\mathbb{Q}}_{\ell}}$ on the right via (32) and the homomorphism

[TABLE]

where the last map takes $\sigma\in\operatorname{Gal}({\overline{{\mathbb{Q}}}}/{\mathbb{Q}})$ to

[TABLE]

and $\chi_{\ell}$ is the $\ell$-adic cyclotomic character.

15.2. Connections over the modular curve

We recall the connection associated to the local system ${\mathbb{H}}$ over ${\mathcal{M}}_{1,1}$ and introduce the local systems ${\mathbb{V}}_{N}$ and their associated connections ${\mathcal{V}}_{N}$ which will be used in the construction of the (large) relative completions of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ in Section 17.

15.2.1. The analytic version

A vector bundle ${\mathcal{V}}$ over ${\mathcal{M}}_{1,1}^{\mathrm{an}}$ is, by definition, a vector bundle ${\widetilde{{\mathcal{V}}}}$ over ${\mathfrak{h}}$ endowed with an ${\mathrm{SL}}_{2}({\mathbb{Z}})$ action such that the projection ${\widetilde{{\mathcal{V}}}}\to{\mathfrak{h}}$ is ${\mathrm{SL}}_{2}({\mathbb{Z}})$ equivariant. A connection on ${\mathcal{V}}$ is an ${\mathrm{SL}}_{2}({\mathbb{Z}})$ invariant connection on ${\mathcal{V}}$.

15.2.2. The algebraic version

Suppose that ${\Bbbk}$ is a field of characteristic 0. A vector bundle on ${\mathcal{M}}_{1,1/{\Bbbk}}$ is a vector bundle ${\widetilde{{\mathcal{V}}}}$ over ${\mathcal{M}}_{1,{\vec{1}}/{\mathbb{Q}}}$ whose restriction to each fiber has a trivialization that is ${\mathbb{G}_{m}}$-invariant. (The isotropy group of a point in a ${\mathbb{G}_{m}}$ orbit may act non-trivially on the fiber over it.)

A connection on ${\mathcal{V}}$ is a ${\mathbb{G}_{m}}$-invariant connection $\nabla$ on the vector bundle ${\widetilde{{\mathcal{V}}}}$ over ${\mathcal{M}}_{1,{\vec{1}}}$ which is trivial on each ${\mathbb{G}_{m}}$ orbit. The connection on ${\mathcal{V}}$ has regular singularities at infinity if the bundle ${\widetilde{{\mathcal{V}}}}$ extends to a vector bundle ${\overline{{\mathcal{V}}}}$ over ${\mathbb{A}}^{2}-\{0\}$ and

[TABLE]

where $D$ is the discriminant divisor $u^{3}-27v^{2}$. The restriction of ${\overline{{\mathcal{V}}}}$ to the discriminant divisor in ${\mathbb{A}}^{2}-\{0\}$ is trivialized by the ${\mathbb{G}_{m}}$-action. We define the fiber of ${\mathcal{V}}$ over the cusp of the modular curve to be the vector space of ${\mathbb{G}_{m}}$-invariant sections of ${\overline{{\mathcal{V}}}}$ over $D$.

15.2.3. The connection ${\mathcal{H}}$

This is the connection associated to the local system ${\mathbb{H}}$. The analytic version ${\mathcal{H}}^{\mathrm{an}}$ is the trivial connection $\nabla=d$ on the trivial bundle $H_{\mathbb{C}}\times{\mathfrak{h}}\to{\mathfrak{h}}$. It is invariant under the left ${\mathrm{SL}}_{2}({\mathbb{Z}})$ action. Equivalently, it is the natural ${\mathrm{SL}}_{2}({\mathbb{Z}})$ invariant connection on ${\mathbb{H}}\otimes_{\mathbb{Q}}{\mathcal{O}}_{\mathfrak{h}}$ over ${\mathfrak{h}}$.

The algebraic version ${\mathcal{H}}$ is the connection

[TABLE]

on the restriction $\overline{{\mathcal{H}}}$ of the trivial bundle ${\mathcal{O}}_{{\mathbb{A}}^{2}_{/{\mathbb{Q}}}}S\oplus{\mathcal{O}}_{{\mathbb{A}}^{2}_{/{\mathbb{Q}}}}T$ to ${\mathbb{A}}^{2}_{{\mathbb{Q}}}-\{0\}$, where $\alpha=2udv-3vdu$ and $D=u^{3}-27v^{2}$. It is defined over ${\mathbb{Q}}$ and invariant under the ${\mathbb{G}_{m}}$ action defined by $t\cdot S=t^{-1}S$ and $t\cdot T=tT$. The extended bundle has a Hodge filtration

[TABLE]

The connection has fiber $H^{\mathrm{DR}}={\mathbb{Q}}S\oplus{\mathbb{Q}}T$ over the cusp and residue $-S\frac{\partial}{\partial T}\in\operatorname{End}_{\mathbb{Q}}H^{\mathrm{DR}}$. After tensoring with ${\mathcal{O}}_{{\mathbb{C}}^{2}-\{0\}}^{\mathrm{an}}$, this connection is isomorphic to ${\mathcal{H}}^{\mathrm{an}}$. Combined with the ${\mathbb{Q}}$ structure on ${\mathbb{H}}$, it defines a polarized variation of Hodge structure over ${\mathcal{M}}_{1,1}$. Details can be found in [14, §19]. (See also [17, Prop. 5.2].)

15.3. Distinguished local systems and connections over ${\mathcal{M}}_{1,1}$

In this section we construct the various incarnations of the semi-simple local systems over ${\mathcal{M}}_{1,1}$ we shall need in the construction of the relative completion of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ in Section 17. These are obtained by pulling back $S^{m}{\mathbb{H}}$ to its full level $N$ covering and then pushing the result forward to ${\mathcal{M}}_{1,1}$. We will construct the Betti, complex analytic de Rham and algebraic de Rham versions. These are needed to construct the various incarnations of the relative completion of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ we shall need and the comparison isomorphisms between them.

Suppose that $N$ is a positive integer. Recall from Section 2 that ${\Gamma}(N)$ denotes the full level $N$ subgroup of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ and that $Y(N)^{\mathrm{an}}$ is the modular curve ${\Gamma}(N)\backslash{\mathfrak{h}}$. Its points correspond to isomorphism classes of elliptic curves $E$ over ${\mathbb{C}}$ together with a symplectic isomorphism $H_{1}(E,{\mathbb{Z}}/N)\overset{\sim}{\to}({\mathbb{Z}}/N)^{2}$, where $({\mathbb{Z}}/N)^{2}$ is given the standard inner product.

Let $\pi:Y(N)^{\mathrm{an}}\to{\mathcal{M}}_{1,1}^{\mathrm{an}}$ denote the projection. It is Galois with automorphism group ${\mathrm{SL}}_{2}({\mathbb{Z}}/N)$. Denote the trivial rank 1 local system of ${\Bbbk}$ vector spaces over ${\mathcal{M}}_{1,1}^{\mathrm{an}}$ by ${\Bbbk}_{\mathcal{M}}$. Denote the trivial rank 1 connection on ${\mathcal{M}}_{1,1}^{\mathrm{an}}$ by $({\mathcal{O}}_{\mathcal{M}}^{\mathrm{an}},d)$, or simply by ${\mathcal{O}}_{\mathcal{M}}^{\mathrm{an}}$.

Proposition 15.1**.**

Suppose that ${\Bbbk}$ is a field of characteristic zero.

1. (i)

If ${\mathbb{V}}$ is a local system of ${\Bbbk}$ vector spaces on ${\mathcal{M}}_{1,1}^{\mathrm{an}}$, then there is a natural isomorphism

[TABLE]

It is characterized by the property that the composition of the unit ${\mathbb{V}}\to\pi_{\ast}\pi^{\ast}{\mathbb{V}}$ of the adjunction with this isomorphism is the tensor product of the unit ${\Bbbk}_{\mathcal{M}}\to\pi_{\ast}\pi^{\ast}{\Bbbk}_{\mathcal{M}}$ with ${\mathbb{V}}$. As a $\pi_{1}({\mathcal{M}}_{1,1},b)$ module, the fiber of $\pi_{\ast}\pi^{\ast}{\mathbb{V}}$ of the base point $b$ is naturally isomorphic to the right ${\mathrm{SL}}_{2}({\mathbb{Z}})$ module

[TABLE]

where $V_{b}$ denotes the fiber of ${\mathbb{V}}$ over $b$ and where $\operatorname{Hom}_{{\Gamma}(N)}$ denotes right ${\Gamma}(N)$ module homomorphisms. 2. (ii)

If ${\mathcal{V}}$ is a connection over ${\mathcal{M}}_{1,1}^{\mathrm{an}}$, then there is a canonical isomorphism

[TABLE]

It is characterized by the property that the composition of the unit ${\mathcal{V}}\to\pi_{\ast}\pi^{\ast}{\mathcal{V}}$ of the adjunction with this isomorphism is the tensor product of the unit ${\mathcal{O}}^{\mathrm{an}}_{\mathcal{M}}\to\pi_{\ast}\pi^{\ast}{\mathcal{O}}^{\mathrm{an}}_{\mathcal{M}}$ with ${\mathcal{V}}$. 3. (iii)

If ${\Bbbk}$ is a subfield of ${\mathbb{C}}$ and ${\mathcal{V}}={\mathbb{V}}\otimes_{\Bbbk}{\mathcal{O}}_{\mathcal{M}}^{\mathrm{an}}$, then there is a natural isomorphism

[TABLE]

which is compatible with the isomorphisms (35) and (36) above. Consequently, there are canonical isomorphisms

[TABLE]

The proof is an exercise, but with one caveat — one has to work ${\mathrm{SL}}_{2}({\mathbb{Z}})$ equivariantly on ${\mathfrak{h}}$ or ${\mathbb{G}_{m}}$-equivariantly on ${\mathcal{M}}_{1,{\vec{1}}}$. This means that local sections of the sheaf ${\mathcal{O}}_{\mathcal{M}}^{\mathrm{an}}$ are identified with either functions defined locally on the upper half plane, or else ${\mathbb{G}_{m}}$ invariant functions on a saturated open subset of ${\mathbb{C}}^{2}-D$, where $D$ is the discriminant divisor $u^{3}-27v^{2}$. The last isomorphism follows from Shapiro’s Lemma.

Corollary 15.2**.**

The connection associated to $\pi_{\ast}\pi^{\ast}S^{m}{\mathbb{H}}$ is naturally isomorphic to $\pi_{\ast}\pi^{\ast}S^{m}{\mathcal{H}}^{\mathrm{an}}$. There are natural compatible isomorphisms

[TABLE]

For each $m\geq 0$, the monodromy representation factors through the representation

[TABLE]

which is the inclusion on the first factor and reduction mod $N$ on the second.

Remark 15.3*.*

There is also an $\ell$-adic étale version of these results. But, due to a slight of hand, we will not need them.

15.3.1. The modular curve $Y(N)/{\mathbb{Q}}$

Fix a positive integer $N$. Before discussing the algebraic de Rham theory, we need to recall a few facts about the modular curve $Y(N)$. It is an irreducible curve over ${\mathbb{Q}}$ except when $N=1,2$ when it is a smooth stack over ${\mathbb{Q}}$. Maps $S\to Y(N)$ from schemes $S/{\mathbb{Q}}$ classify elliptic curves $E/S$ together with isomorphisms

[TABLE]

under which the Weil pairing $H^{0}(S,E[N])^{\otimes 2}\to\boldsymbol{\mu}_{N}(S)$ corresponds to the standard symplectic inner product on $({\mathbb{Z}}_{S}/N)^{\oplus 2}$. The projection $\pi:Y(N)\to Y$ that forgets the framings is a ${\mathrm{GL}}_{2}({\mathbb{Z}}/N)$ torsor.

The base point ${\vec{\mathsf{v}}}:\operatorname{Spf}{\overline{{\mathbb{Q}}}}(\!(q^{1/N}:N\geq 1)\!)\to{\mathcal{M}}_{1,1/{\mathbb{Q}}}$ lifts to $Y(N)$. All lifts are conjugate under the ${\mathrm{GL}}_{2}({\mathbb{Z}}/N)$ action. Fix one lift ${\vec{\mathsf{v}}}_{N}$. We have the commutative diagram

[TABLE]

where $\chi_{N}$ denotes the cyclotomic character mod $N$ and the central vertical map is the monodromy representation of $Y(N)\to{\mathcal{M}}_{1,1/{\mathbb{Q}}}$ on the fiber over ${\vec{\mathsf{v}}}$. It follows that if $K$ is an algebraically closed field, then

[TABLE]

where each component $Y_{\nu}$ is absolutely irreducible and defined over ${\mathbb{Q}}(\boldsymbol{\mu}_{N})$. Taking $K={\mathbb{C}}$, we see that each $Y_{\nu}^{\mathrm{an}}$ is isomorphic to $Y(N)^{\mathrm{an}}$.

The homomorphism171717On the fiber over ${\vec{\mathsf{v}}}_{N}$, which is isomorphic to $H_{{\mathbb{Z}}/N}$, this lift coincides with the action of $\operatorname{Gal}({\overline{{\mathbb{Q}}}}/{\mathbb{Q}})$. This follows from (32) with $A={\mathbb{Z}}/N$ and (33) as $\mathbf{a}$ spans a copy of ${\mathbb{Z}}(1)$ and $\mathbf{b}$ a copy of ${\mathbb{Z}}(0)$.

[TABLE]

defines a section of the surjection ${\mathrm{GL}}_{2}({\mathbb{Z}}/N)\to({\mathbb{Z}}/N)^{\times}$. Composing it with the action of ${\mathrm{GL}}_{2}({\mathbb{Z}}/N)$ on $Y(N)$ defines an action of $\operatorname{Gal}({\mathbb{Q}}(\boldsymbol{\mu}_{N})/{\mathbb{Q}})$ on $Y(N)$. These actions for $N\geq 1$ are compatible in the sense that when $N|M$, the diagram

[TABLE]

commutes.

15.3.2. Algebraic DR version

In order that the de Rham incarnation of relative completion is defined over ${\mathbb{Q}}$, we need to show that the connections $\pi_{\ast}\pi^{\ast}S^{m}{\mathcal{H}}^{\mathrm{an}}$ over ${\mathcal{M}}_{1,1}^{\mathrm{an}}$ have a ${\mathbb{Q}}$-de Rham incarnation.

Proposition 15.4**.**

There is a connection ${\mathscr{P}}_{N}$ over ${\mathcal{M}}_{1,1/{\mathbb{Q}}}$ of rank $d_{N}=|{\mathrm{SL}}_{2}({\mathbb{Z}}/N)|$ and a canonical isomorphism

[TABLE]

These connections are compatible in the sense that if $N|M$, there is a surjective morphism of connections

[TABLE]

where $\pi_{M,N}:Y(M)\to Y(N)$ denotes the canonical projection.

Proof.

The action of ${\mathrm{GL}}_{2}({\mathbb{Z}}/N)$ on $Y(N)$ lifts to an action of it on $\pi^{\ast}{\mathcal{O}}_{\mathcal{M}}$. This descends to a fiber preserving action of ${\mathrm{GL}}_{2}({\mathbb{Z}}/N)$ on $\pi_{\ast}\pi^{\ast}{\mathcal{O}}_{\mathcal{M}}$ which respects its natural connection. This is a vector bundle of rank $|{\mathrm{GL}}_{2}({\mathbb{Z}}/N)|$ whose fibers are isomorphic to the regular representation ${\mathbb{Q}}[{\mathrm{GL}}_{2}({\mathbb{Z}}/N)]$ of ${\mathrm{GL}}_{2}({\mathbb{Z}}/N)$. Define

[TABLE]

It has rank $d_{N}=|{\mathrm{GL}}_{2}({\mathbb{Z}}/N)|/|({\mathbb{Z}}/N)^{\times}|$. Since the connection on $\pi_{\ast}\pi^{\ast}{\mathcal{O}}_{\mathcal{M}}$ is $({\mathbb{Z}}/N)^{\times}$ invariant, it restricts to a natural connection on ${\mathscr{P}}_{N}$. Both ${\mathscr{P}}_{N}$ and its connection are defined over ${\mathbb{Q}}$.

Denote the projection $Y_{\nu}^{\mathrm{an}}\to{\mathcal{M}}_{1,1}^{\mathrm{an}}$ associated to an embedding $\nu:{\mathbb{Q}}(\boldsymbol{\mu}_{N})\to{\mathbb{C}}$ by $\pi_{\nu}$. Then there are natural isomorphisms

[TABLE]

For each $\nu$, the restriction of the projection

[TABLE]

to $[\pi_{\ast}\pi^{\ast}{\mathcal{O}}_{\mathcal{M}}^{\mathrm{an}}]^{{({\mathbb{Z}}/N)^{\times}}}$ induces an isomorphism of connections

[TABLE]

∎

Corollary 15.5**.**

For each $m\geq 0$ and $N\geq 1$, there is a connection ${\mathscr{S}}^{m}_{N}$ over ${\mathcal{M}}_{1,1/{\mathbb{Q}}}$ such that

[TABLE]

The comparison isomorphisms between $\pi_{\ast}\pi^{\ast}S^{m}{\mathbb{H}}$ and $\pi_{\ast}\pi^{\ast}S^{m}{\mathcal{H}}^{\mathrm{an}}$ from Corollary 15.2 define a polarized variation of Hodge structure of weight $m$ over ${\mathcal{M}}_{1,1}^{\mathrm{an}}$. When $N|M$, the surjection (40) is a morphism of variations of MHS.

Proof.

The connection ${\mathscr{S}}^{m}_{N}$ is defined to be ${\mathscr{P}}_{N}\otimes_{{\mathcal{O}}_{\mathcal{M}}}S^{m}{\mathcal{H}}$. The Hodge filtration (34) extends to $S^{m}{\mathcal{H}}$ and hence to ${\mathscr{S}}^{m}_{N}$. Since the comparison of ${\mathbb{H}}$ and ${\mathcal{H}}$ define a polarized VHS over ${\mathcal{M}}_{1,1}$, it follows from standard Hodge theory that the comparison isomorphisms from Corollary 15.2 define a polarized variation of Hodge structure of weight $m$ over ${\mathcal{M}}_{1,1}^{\mathrm{an}}$. ∎

16. Relative Unipotent Completion in the Abstract

This is a terse review of relative completion. Since we need to define it in several related contexts, we use Saad’s efficient abstract setup [32, §5.2.1] (see also, [4, §12]) and establish several basic results about relative completion in this context.

Suppose that ${\Bbbk}$ is a field of characteristic zero and that ${\mathsf{C}}$ is a ${\Bbbk}$-linear neutral tannakian category with fiber functor ${\omega}:{\mathsf{C}}\to{\mathsf{Vec}}_{\Bbbk}$. Suppose that ${\mathsf{S}}$ is a full tannakian subcategory of ${\mathsf{C}}$, all of whose objects are semi-simple. Define ${\mathscr{F}}({\mathsf{C}},{\mathsf{S}})$ to be the full subcategory of ${\mathsf{C}}$ that consists of all objects ${\mathbb{V}}$ of ${\mathsf{C}}$ that admit a filtration

[TABLE]

where each graded quotient ${\mathbb{V}}_{j}/{\mathbb{V}}_{j-1}$ is isomorphic to an object of ${\mathsf{S}}$. This is a tannakian subcategory of ${\mathsf{C}}$. The restriction of ${\omega}$ to ${\mathscr{F}}({\mathsf{C}},{\mathsf{S}})$ is a fiber functor. Denote its tannakian fundamental group with respect to ${\omega}$ by $\pi_{1}({\mathsf{C}},{\mathsf{S}},{\omega})$:

[TABLE]

Since every object of ${\mathsf{S}}$ is semi-simple, $\pi_{1}({\mathsf{S}},{\omega})$ is a (pro)reductive ${\Bbbk}$-group.181818We’ll usually drop the pro and simply say that $\pi_{1}({\mathsf{S}},{\omega})$ is reductive. The important fact for us is that every representation of $\pi_{1}({\mathsf{S}},{\omega})$ be completely reducible. Since ${\mathsf{S}}$ is a full subcategory of ${\mathsf{C}}$, there is a faithfully flat homomorphism $\pi_{1}({\mathsf{C}},{\mathsf{S}};{\omega})\to\pi_{1}({\mathsf{S}},{\omega})$ of affine ${\Bbbk}$-groups. The kernel is the maximal prounipotent normal subgroup of ${\mathcal{G}}$. Denote it by ${\mathcal{U}}$ so that ${\mathcal{G}}$ is an extension

[TABLE]

For the time being, we set ${\mathcal{G}}=\pi_{1}({\mathsf{C}},{\mathsf{S}};{\omega})$ and $S=\pi_{1}({\mathsf{S}},{\omega})$. Denote the Lie algebra of ${\mathcal{U}}$ by ${\mathfrak{u}}$. It is pronilpotent. Suppose that ${\mathbb{V}}$ is an object of ${\mathscr{F}}({\mathsf{C}},{\mathsf{S}})$. Set $V={\omega}({\mathbb{V}})$. It is a left ${\mathcal{G}}$-module. There are natural isomorphisms

[TABLE]

where ${\mathbf{1}}$ denotes the unit object of ${\mathsf{C}}$. The first follows from Tannaka duality, the second is the definition of cohomology of algebraic groups, and the third follows from the analogue for affine groups of the Hochschild–Serre spectral sequence of the extension (41), the natural isomorphism $H^{j}({\mathfrak{u}},V)\cong H^{j}({\mathcal{U}},V)$, and the fact that $S$ is reductive.

The inclusion functor ${\mathscr{F}}({\mathsf{C}},{\mathsf{S}})\to{\mathsf{C}}$ is exact and therefore induces homomorphisms

[TABLE]

for all objects $A$ and $B$ of ${\mathscr{F}}({\mathsf{C}},{\mathsf{S}})$ and all $j\geq 0$.

Proposition 16.1**.**

The homomorphism (43) is an isomorphism when $j=0,1$ and an injection when $j=2$.

Proof.

The case $j=0$ follows from the fact that ${\mathscr{F}}({\mathsf{C}},{\mathsf{S}})$ is a full subcategory of ${\mathsf{C}}$. The case $j=1$ follows from fullness and the fact that ${\mathscr{F}}({\mathsf{C}},{\mathsf{S}})$ is closed under extensions. We prove the case $j=2$ using Yoneda’s description [39] of Ext groups. Suppose that

[TABLE]

represents an element of $\operatorname{Ext}^{j}_{{\mathscr{F}}({\mathsf{C}},{\mathsf{S}})}(A,B)$. By [39, p. 575], it represents 0 in $\operatorname{Ext}^{2}_{\mathsf{C}}(A,B)$ if and only if there is an object $M$ of ${\mathsf{C}}$ with a filtration

[TABLE]

in ${\mathsf{C}}$ and an isomorphism of 2-extensions

[TABLE]

To prove injectivity it suffices, by Yoneda’s criterion, to show that $M$ is an object of ${\mathscr{F}}({\mathsf{C}},{\mathsf{S}})$. Set $C=\operatorname{coker}\{B\to E\}$. This is an object of ${\mathscr{F}}({\mathsf{C}},{\mathsf{S}})$. Since $M_{1}$ is an extension of $C$ by $B$, it is also in ${\mathscr{F}}({\mathsf{C}},{\mathsf{S}})$, and since $M$ is an extension of $A$ by $M_{1}$, it is a in ${\mathscr{F}}({\mathsf{C}},{\mathsf{S}})$, as required. ∎

Combining this with (42), where $A={\mathbf{1}}$ and $B={\mathbb{V}}$, we obtain the following results which is useful for proving comparison theorems.

Corollary 16.2**.**

For all objects ${\mathbb{V}}$ of ${\mathscr{F}}({\mathsf{C}},{\mathsf{S}})$, the homomorphism

[TABLE]

induced by the inclusion ${\mathscr{F}}({\mathsf{C}},{\mathsf{S}})\to{\mathsf{C}}$, where $V={\omega}({\mathbb{V}})$, is an isomorphism when $j\leq 1$ and an injection when $j=2$.

The following useful criterion for the freeness of ${\mathfrak{u}}$ is an immediate consequence of the isomorphisms (42) and the well-know fact that a pronilpotent Lie algebra ${\mathfrak{n}}$ is free if and only $H^{2}({\mathfrak{n}})$ vanishes. (See [17, §18].)

Proposition 16.3**.**

The Lie algebra ${\mathfrak{u}}$ is a free pronilpotent Lie algebra if and only if $\operatorname{Ext}^{2}({\mathbf{1}},{\mathbb{V}})$ vanishes for all simple objects ${\mathbb{V}}$ of ${\mathsf{S}}$. Equivalently, ${\mathfrak{u}}$ is free if and only if $H^{2}({\mathfrak{u}})=0$. ∎

16.0.1. Extension of scalars

Suppose that ${\mathbb{K}}$ is an extension field of ${\Bbbk}$. Let ${\omega}_{\mathbb{K}}$ be the fiber functor ${\mathsf{C}}\to{\mathsf{Vec}}_{\mathbb{K}}$ defined by ${\omega}_{\mathbb{K}}({\mathbb{V}})={\omega}({\mathbb{V}})\otimes_{\Bbbk}{\mathbb{K}}$. The following observation is useful when establishing comparison and base change isomorphisms. It can be proved using matrix entries.

Proposition 16.4**.**

There is a natural isomorphism

[TABLE]

We shall denote the extension $G\times_{\Bbbk}{\mathbb{K}}$ of an affine ${\Bbbk}$-group $G$ to ${\mathbb{K}}$ by $G_{\mathbb{K}}$. The prounipotent radical of ${\mathcal{G}}_{\mathbb{K}}$ is ${\mathcal{U}}_{\mathbb{K}}$ and its reductive quotient $S_{\mathbb{K}}$. The Lie algebra of ${\mathcal{U}}_{\mathbb{K}}$ is ${\mathfrak{u}}\otimes_{\Bbbk}{\mathbb{K}}$, which we denote by ${\mathfrak{u}}_{\mathbb{K}}$.

16.1. Computation of $H^{1}({\mathfrak{u}})$

As above, ${\mathbb{K}}$ is an extension field of ${\Bbbk}$. Suppose that $V$ is a simple left $S_{\mathbb{K}}$-module. Its endomorphism ring is thus a division algebra naturally isomorphic to $\operatorname{End}_{S_{\mathbb{K}}}V$. We shall denote it by $D_{V}$. The ${\mathbb{K}}$-algebra $\operatorname{End}_{D_{V}}V$ is right $S_{\mathbb{K}}$-module via precomposition and a left $S_{\mathbb{K}}$-module via post composition. Denote its dual by $\operatorname{End}_{D_{V}}^{\vee}V$. It has commuting left and right $S_{\mathbb{K}}$ actions induced by pre and post composition by $S_{\mathbb{K}}$. Note that the map $S_{\mathbb{K}}\to\operatorname{Aut}_{D_{V}}V$ induces a map

[TABLE]

into the coordinate ring of $S_{\mathbb{K}}$. It is a right and left $S_{\mathbb{K}}$-module homomorphism.

Suppose that $\{V_{\alpha}\}$ is a complete set of representatives of the isomorphism classes of simple $S_{\mathbb{K}}$-modules. Set $D_{\alpha}=D_{V_{\alpha}}$. The next result is an analogue of the Peter–Weyl theorem. It follows from Tannaka duality (by taking matrix entries). Alternatively, it can be proved by first reducing to the case where ${\mathbb{K}}$ is algebraically closed using the Artin–Wedderburn Theorem.

Proposition 16.5**.**

If $\{V_{\alpha}\}$ is a complete set of representatives of the isomorphism classes of simple $S_{\mathbb{K}}$-modules, then the map

[TABLE]

is an isomorphism of right and left $S_{\mathbb{K}}$-modules.

Suppose that $V$ is a left $S_{\mathbb{K}}$-module. The right action of $S_{\mathbb{K}}$ on $\operatorname{End}_{D_{V}}V$ gives the space $\operatorname{Hom}_{S_{\mathbb{K}}}(\operatorname{End}_{D_{V}}V,V)$ of left $S_{\mathbb{K}}$-invariant maps the structure of a left $S_{\mathbb{K}}$-module.

Lemma 16.6**.**

If $V$ is a simple $S_{\mathbb{K}}$-module, then the map

[TABLE]

that takes $v\in V$ to $\phi\mapsto\phi(v)$ is an isomorphism of left $S_{\mathbb{K}}$-modules.

Proof.

Tensor both sides with an algebraic closure of ${\mathbb{K}}$ and apply the Artin–Wedderburn Theorem. ∎

The conjugation action $g:g\mapsto ghg^{-1}$ of ${\mathcal{G}}$ on itself induces a left ${\mathcal{G}}$ action on ${\mathfrak{u}}$ and a right ${\mathcal{G}}$ action on $H^{\bullet}({\mathfrak{u}})$. This descends to a right $S$ action on $H^{\bullet}({\mathfrak{u}})$.

We will regard the coordinate ring ${\mathcal{O}}(S_{\mathbb{K}})$ as a left ${\mathcal{G}}_{\mathbb{K}}$-module via right multiplication. We therefore have the cohomology group

[TABLE]

The right action of $S$ on ${\mathcal{O}}(S)$ gives this the structure of a right $S_{\mathbb{K}}$-module. Likewise, we regard each $\operatorname{End}_{D_{\alpha}}^{\vee}V_{\alpha}$ as a left ${\mathcal{G}}_{\mathbb{K}}$-module and a right $S_{\mathbb{K}}$-module.

Proposition 16.7**.**

There is are natural right $S_{\mathbb{K}}$-module isomorphisms

[TABLE]

In particular, if each $D_{\alpha}$ is isomorphic to ${\mathbb{K}}$ (i.e, each $V_{\alpha}$ is absolutely irreducible), then there is a right $S_{\mathbb{K}}$-module isomorphism

[TABLE]

Proof.

This follows directly from Proposition 16.5 and the isomorphisms (42) using Lemma 16.6. ∎

16.2. Criterion for isomorphism

The following criterion for when certain homomorphisms of affine groups to be an isomorphism is needed to establish comparison isomorphisms. It is well-known. A proof can be deduced from the discussion in [17, §18]. Suppose that $G$ and $G^{\prime}$ are affine ${\mathbb{K}}$ group that are extensions of a reductive group $R$ by prounipotent groups $U$ and $U^{\prime}$, respectively. Denote the Lie algebras of $U$ and $U^{\prime}$ by ${\mathfrak{u}}$ and ${\mathfrak{u}}^{\prime}$.

Proposition 16.8**.**

Suppose that $\phi:G\to G^{\prime}$ is a homomorphism that commutes with the projections to $R$. Then $\phi$ is an isomorphism if and only if the induced map $H^{j}({\mathfrak{u}}^{\prime})\to H^{j}({\mathfrak{u}})$ is an isomorphism when $j=1$ and injective when $j=2$. Equivalently, $\phi$ is an isomorphism if and only if, for all $R$-modules $V$, the induced map $H^{j}(G^{\prime},V)\to H^{j}(G,V)$ is an isomorphism when $j=1$ and injective when $j=2$.

16.3. Variant: completion with constrained $\operatorname{Ext}^{1}$

This section is needed in Section 24 where it is used to define certain natural filtrations on the coordinate ring of the relative completion of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ that is defined in Section 17. It can also be used to define the weighted and crystalline completions of arithmetic fundamental groups [16]. The construction is inspired by [8, Def. 1.4].

Let ${\mathsf{C}}$, ${\mathsf{S}}$ and ${\omega}$ be as above. Suppose that $\{{\mathbb{V}}_{\alpha}\}$ be a collection of simple objects of ${\mathsf{S}}$ and that ${\mathsf{E}}$ is a collection of subspaces of $E_{\alpha}$ of $\operatorname{Ext}^{1}_{\mathsf{C}}({\mathbf{1}},{\mathbb{V}}_{\alpha})$. Define the category ${\mathscr{F}}({\mathsf{C}},{\mathsf{S}};{\mathsf{E}})$ to be the full subcategory of ${\mathscr{F}}({\mathsf{C}},{\mathsf{S}})$ consisting of those objects ${\mathbb{V}}$ for which the 1-extensions that occur in every subquotient of ${\mathbb{V}}$ are isomorphic to sums of elements of $\cup_{\alpha}E_{\alpha}$ tensored with an object of ${\mathsf{S}}$. It is tannakian.

Proposition 16.9**.**

If ${\mathfrak{u}}$ is the Lie algebra of the prounipotent radical of $\pi_{1}({\mathscr{F}}({\mathsf{C}},{\mathsf{S}};{\mathsf{E}}^{\prime}),{\omega})$, then there is an $S({\Bbbk})$-module isomorphism

[TABLE]

where $V_{\alpha}={\omega}({\mathbb{V}}_{\alpha})$. ∎

If ${\mathsf{E}}^{\prime}\supseteq{\mathsf{E}}$, then, ${\mathscr{F}}({\mathsf{C}},{\mathsf{S}};{\mathsf{E}}^{\prime})$ is a full tannakian subcategory of ${\mathscr{F}}({\mathsf{C}},{\mathsf{S}};{\mathsf{E}}^{\prime})$, so that

[TABLE]

is faithfully flat. In particular,

[TABLE]

17. Relative Unipotent Completion of the Modular Group

In this section we construct various incarnations (Betti, étale and de Rham) of a relative unipotent completion ${\mathcal{G}}$ of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ that is more general and larger than the one defined and studied in previous works such as [13, 4, 17]. This enlargement has the property that its category of representations is closed under restriction to and pushforward from all congruence subgroups, which is needed to guarantee that the Hecke operators act on its ring ${\mathscr{C}\!\ell}({\mathcal{G}})$ of class functions.

We will view every finite group $G$ as the affine algebraic group scheme over ${\mathbb{Q}}$ in the standard way; its coordinate ring is the algebra of functions $G\to{\mathbb{Q}}$. By taking inverse limits, we will regard every profinite group as an affine ${\mathbb{Q}}$ group scheme. Taking the group of ${\Bbbk}$ rational points of the algebraic group recovers the original (pro)finite group.

17.1. The Betti incarnation

For each positive integers $N$, define

[TABLE]

to be the homomorphism that is the inclusion on the first factor and is reduction mod $N$ on the second. It has Zariski dense image in the ${\mathbb{Q}}$ group ${\mathrm{SL}}_{2}\times{\mathrm{SL}}_{2}({\mathbb{Z}}/N)$. Consequently, the inverse limit

[TABLE]

of the $\rho_{N}$ has Zariski dense image.

Suppose that ${\Bbbk}$ is a field of characteristic zero. Denote the category of representations ${\mathrm{SL}}_{2}({\mathbb{Z}})$ in finite dimensional ${\Bbbk}$ vector spaces by ${\mathsf{C}}_{\Bbbk}$. Each object of ${\mathsf{C}}_{\Bbbk}$ is regarded as a left ${\mathrm{SL}}_{2}({\mathbb{Z}})$-module. Let ${\mathsf{S}}_{\Bbbk}$ be the full subcategory of ${\mathsf{C}}_{\Bbbk}$ consisting of those representations that factor through $\rho$. These categories are neutral tannakian. The functor ${\omega}_{\Bbbk}$ that takes a representation to its underlying vector space is a fiber functor.

Definition 17.1**.**

The (Betti incarnation of the) relative completion ${\mathcal{G}}^{B}$ of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ is is the affine ${\mathbb{Q}}$-group defined by

[TABLE]

It is an extension

[TABLE]

where ${\mathcal{U}}^{B}$ is prounipotent. There is a canonical Zariski dense homomorphism

[TABLE]

whose composition with the quotient map ${\mathcal{G}}^{B}({\mathbb{Q}})\to({\mathrm{SL}}_{2}\times{\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}}))({\mathbb{Q}})$ is the representation (44).

Proposition 17.2**.**

The functor $\underline{{\phantom{x}}}\otimes_{\mathbb{Q}}{\Bbbk}:{\mathscr{F}}({\mathsf{C}}_{\mathbb{Q}},{\mathsf{S}}_{\mathbb{Q}})\to{\mathscr{F}}({\mathsf{C}}_{\Bbbk},{\mathsf{S}}_{\Bbbk})$ induces an isomorphism $\pi_{1}({\mathscr{F}}({\mathsf{C}}_{\Bbbk},{\mathsf{S}}_{\Bbbk}),{\omega}_{\Bbbk})\to{\mathcal{G}}_{\mathbb{Q}}^{B}\times_{\mathbb{Q}}{\Bbbk}$.

We will denote ${\mathcal{G}}^{B}\times_{\mathbb{Q}}{\Bbbk}$ by ${\mathcal{G}}^{B}_{\Bbbk}$. The proposition implies that this is unambiguous.

Proof.

The functor in the statement induces a homomorphism

[TABLE]

so that one has the following diagram

[TABLE]

Applying cohomology we obtain, for all objects ${\mathbb{V}}$ of ${\mathscr{F}}({\mathsf{C}}_{\mathbb{Q}},{\mathsf{S}}_{\mathbb{Q}})$, the commutative diagram

[TABLE]

where $V={\omega}_{\Bbbk}({\mathbb{V}})$. Standard results about group cohomology imply that the bottom arrow is an isomorphism for all $j$. Corollary 16.2 implies that the vertical arrows are isomorphisms when $j=0,1$ and injective when $j=2$. This implies that the top arrow is an isomorphism when $j=0,1$ and injective when $j=2$. Proposition 16.8 now implies that

[TABLE]

is an isomorphism. The result follows Proposition 16.4, which implies that ${\mathcal{G}}^{B}\times_{\mathbb{Q}}{\Bbbk}$ is isomorphic to $\pi_{1}({\mathscr{F}}({\mathsf{C}}_{\mathbb{Q}},{\mathsf{S}}_{\mathbb{Q}}),{\omega}_{\Bbbk})$. ∎

Suppose that ${\overline{{\Bbbk}}}$ is an algebraically closed field. For each irreducible character $\chi:{\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})\to{\overline{{\Bbbk}}}$ of ${\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$, choose an irreducible left ${\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$ module $V_{\chi}$. We will regard its dual $V_{\chi}^{\vee}$ as a right ${\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$ module.

Recall that ${\Gamma}(N)$ denotes the full level $N$ subgroup of ${\mathrm{SL}}_{2}({\mathbb{Z}})$. For each $m\geq 0$, $H^{1}({\Gamma}(N),S^{m}H)\otimes{\Bbbk}$ is a left ${\mathrm{SL}}_{2}({\mathbb{Z}}/N)$ module. Recall that for a character $\psi$ of ${\mathrm{SL}}_{2}({\mathbb{Z}}/N)$, $H^{1}({\Gamma}(N),S^{m}H)_{\psi}$ denotes the $\psi$-isotypical summand of $H^{1}({\Gamma}(N),S^{m}H)\otimes{\Bbbk}$.

Proposition 17.3**.**

There are canonical right ${\mathrm{SL}}_{2}\times{\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$ isomorphisms

[TABLE]

where $\chi$ ranges over the irreducible characters of ${\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$ and $\psi$ over the irreducible characters of ${\mathrm{SL}}_{2}({\mathbb{Z}}/N)$. The cohomology of ${\mathfrak{u}}^{B}$ vanishes in degree 2.

Remark 17.4*.*

Later it will be convenient to make this into a left $S$-module isomorphism. This is easily achieved by instead taking the action of ${\mathcal{G}}$ on $H^{\bullet}({\mathfrak{u}})$ to be the one induced by the “right conjugation” action: $g:u\mapsto g^{-1}ug$.

Proof.

The first statement is an immediate consequence of Proposition 16.7. To prove the second statement, first observe that ${\mathrm{SL}}_{2}({\mathbb{Z}})$ is virtually free and that this implies that $H^{j}({\mathrm{SL}}_{2}({\mathbb{Z}}),M)$ vanishes for all $j>1$ and all divisible ${\mathrm{SL}}_{2}({\mathbb{Z}})$ modules $M$. Proposition 16.1 now implies the vanishing of $H^{2}({\mathfrak{u}})$ via the isomorphisms (42). ∎

The next result follows from Proposition 16.3.

Corollary 17.5**.**

The Lie algebra ${\mathfrak{u}}^{B}$ is a free pronilpotent Lie algebra.

17.2. The $\ell$-adic étale incarnation and the Galois action

Fix a prime number $\ell$. The homomorphism (44) extends to a continuous homomorphism

[TABLE]

Denote the category of continuous (left) representations of ${\mathrm{SL}}_{2}({\mathbb{Z}})^{\wedge}$ in finite dimensional ${{\mathbb{Q}}_{\ell}}$ vector spaces by ${\mathsf{C}}^{\mathrm{cts}}_{{\mathbb{Q}}_{\ell}}$. Denote the full subcategory of ${\mathsf{C}}^{\mathrm{cts}}_{{\mathbb{Q}}_{\ell}}$ consisting of those representations that factor through $\rho^{\mathrm{\acute{e}t}}_{\ell}$. Set

[TABLE]

where ${\omega}_{{\mathbb{Q}}_{\ell}}$ takes a representation to its underlying ${{\mathbb{Q}}_{\ell}}$ vector space. The homomorphism $\rho^{\mathrm{\acute{e}t}}_{\ell}$ lifts to a canonical Zariski dense homomorphism

[TABLE]

The restriction functor

[TABLE]

is an exact tensor functor and thus induces a homomorphism ${\mathcal{G}}^{B}_{{\mathbb{Q}}_{\ell}}\to{\mathcal{G}}^{\mathrm{\acute{e}t}}_{{\mathbb{Q}}_{\ell}}$.

Proposition 17.6**.**

The homomorphism ${\mathcal{G}}^{\mathrm{\acute{e}t}}_{{\mathbb{Q}}_{\ell}}\to{\mathcal{G}}^{B}_{{\mathbb{Q}}_{\ell}}$ induced by restriction is an isomorphism of ${{\mathbb{Q}}_{\ell}}$ groups. Moreover, the diagram

[TABLE]

commutes.

Proof.

Suppose that $V$ is an object of ${\mathscr{F}}({\mathsf{C}}^{\mathrm{cts}}_{{\mathbb{Q}}_{\ell}},{\mathsf{S}}^{\mathrm{cts}}_{{\mathbb{Q}}_{\ell}})$. There are natural isomorphisms

[TABLE]

where $H^{j}_{\mathrm{cts}}$ denotes continuous group cohomology. We have the commutative diagram

[TABLE]

whose horizontal maps are induced by restriction. By [35, p. 16], ${\mathrm{SL}}_{2}({\mathbb{Z}})$ is a “good group”, which implies that the bottom arrow is an isomorphism for all $j\geq 0$. Corollary 16.2 implies that the vertical arrows are isomorphisms when $j\leq 1$ and are injective when $j=2$. It follows that the top arrow is an isomorphism when $j\leq 1$ and injective when $j=2$. The result now follows from Proposition 16.8. ∎

Recall from Corollary 14.8 that the absolute Galois group $\operatorname{Gal}({\overline{{\mathbb{Q}}}}/{\mathbb{Q}})$ acts on ${\mathrm{SL}}_{2}({\mathbb{Z}})^{\wedge}$ via the isomorphism $\pi_{1}({\mathcal{M}}_{1,1/{\overline{{\mathbb{Q}}}}},\partial/\partial q)\cong{\mathrm{SL}}_{2}({\mathbb{Z}})^{\wedge}$. This Galois action induces one on ${\mathrm{SL}}_{2}\times_{\mathbb{Q}}{{\mathbb{Q}}_{\ell}}$ and also on $SL_{2}({\widehat{{\mathbb{Z}}}})$ via its action on the Tate module. With these actions, the homomorphism

[TABLE]

is Galois equivariant. The functoriality of relative completion yields the following result.

Corollary 17.7**.**

For each prime number $\ell$, the absolute Galois group acts on ${\mathcal{G}}^{\mathrm{\acute{e}t}}_{{\mathbb{Q}}_{\ell}}$ and the natural representation

[TABLE]

is $\operatorname{Gal}({\overline{{\mathbb{Q}}}}/{\mathbb{Q}})$ equivariant.

Remark 17.8*.*

The weight filtration on ${\mathcal{O}}({\mathcal{G}}^{\mathrm{\acute{e}t}}_{{\mathbb{Q}}_{\ell}})$ can be constructed using the fact that it is a module over the weighted completion (as defined in [16]) of ${\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$ relative to the natural homomorphism to the ${{\mathbb{Q}}_{\ell}}$ points of ${\mathrm{GL}}_{2}\times{\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$ and the central cocharacter $c:{\mathbb{G}_{m}}\to{\mathrm{GL}}_{2}$ defined by $c(t)=t^{-1}\operatorname{id}$. This weight filtration agrees, via the comparison isomorphism, with the one on ${\mathcal{G}}^{B}$ that is constructed in [11] using Hodge theory. The details are omitted as we will not use the weight filtration on ${\mathcal{O}}({\mathcal{G}}^{\mathrm{\acute{e}t}}_{{\mathbb{Q}}_{\ell}})$ in this paper.

17.3. Construction via local systems

Since local systems correspond to representations of the fundamental group, we can also define ${\mathcal{G}}^{B}$ and ${\mathcal{G}}_{{\mathbb{Q}}_{\ell}}^{{\mathrm{\acute{e}t}}_{\ell}}$ as tannakian fundamental groups of categories of certain local systems. One advantage of doing this is that we are able to define completions of torsors of paths in modular curves. Here we quickly sketch the constructions using local systems in place of representations.

17.3.1. The Betti case

Define ${\mathsf{C}}_{\Bbbk}^{\mathrm{loc}}$ to be the category of ${\Bbbk}$-local systems over ${\mathcal{M}}_{1,1}^{\mathrm{an}}$ and let ${\mathsf{S}}_{\Bbbk}^{\mathrm{loc}}$ be the full tannakian subcategory of ${\mathsf{C}}^{\mathrm{loc}}_{\Bbbk}$ generated by the local systems $\pi_{N\,\ast}\pi_{N}^{\ast}S^{m}{\mathbb{H}}_{\Bbbk}$ over ${\mathcal{M}}_{1,1}$ where $\pi_{N}:Y(N)^{\mathrm{an}}\to{\mathcal{M}}_{1,1}^{\mathrm{an}}$ is the projection. Recall that with our conventions, the fundamental group of a space acts on the right of the fiber over the base point. Since ${\mathsf{C}}_{\Bbbk}$ is a category of left ${\mathrm{SL}}_{2}({\mathbb{Z}})$ modules, it will be equivalent to $({\mathsf{C}}_{\Bbbk}^{\mathrm{loc}})^{\mathrm{op}}$

So, restricting to the fiber at the base point $\partial/\partial q$ of ${\mathcal{M}}_{1,1}$ defines a covariant tensor functor

[TABLE]

and, by restriction, a functor $({\mathsf{S}}^{\mathrm{loc}}_{\Bbbk})^{\mathrm{op}}\to{\mathsf{S}}_{\Bbbk}$. Both are equivalences of categories.

Each orbifold base point $x$ of ${\mathcal{M}}_{1,1}$ defines a fiber functor

[TABLE]

Define

[TABLE]

When $x=\partial/\partial q$, there is a canonical isomorphism ${\mathcal{G}}^{B}_{\Bbbk}\cong{\mathcal{G}}^{B}_{x,x/{\Bbbk}}$. For orbifold base points $x,y,z$ we have a morphism ${\mathcal{G}}^{B}_{x,y}\times_{\Bbbk}{\mathcal{G}}^{B}_{y,z}\to{\mathcal{G}}^{B}_{x,z}$. This product is associative in the obvious sense.

17.3.2. The étale case

The étale setup is similar. Details are left to the reader.

17.4. The de Rham incarnation

Denote the complete modular curve of full level $N$ by $X(N)/{\mathbb{Q}}$ and its divisor of cusps by $C_{N}$. We have $Y(N)=X(N)-C_{N}$. We will call a connection $({\mathcal{V}},\nabla)$ over ${\mathcal{M}}_{1,1/{\mathbb{Q}}}$ virtually nilpotent if there is an $N\geq 1$ such that its pullback $\pi_{N}^{\ast}{\mathcal{V}}$ to $Y(N)/{\mathbb{Q}}$ has an extension

[TABLE]

whose residue at each geometric point of $C_{N}$ is nilpotent. We will call such pullbacks locally nilpotent and call $({\overline{{\mathcal{V}}}}_{N},\nabla)$ the canonical extension of $({\mathcal{V}},\nabla)$.191919This is the ${\mathbb{Q}}$-form of Deligne’s canonical extension of the corresponding connection over $Y(N)^{\mathrm{an}}$ to $X(N)^{\mathrm{an}}$.

The virtually nilpotent connections over ${\mathcal{M}}_{1,1/{\mathbb{Q}}}$ form a ${\mathbb{Q}}$-linear tannakian category that we shall denote by ${\mathsf{C}}^{\mathrm{DR}}$. The connections ${\mathscr{S}}^{m}_{N}$ constructed in Corollary 15.5 are virtually nilpotent for all $m\geq 0$ and $N\geq 1$ and are thus objects of ${\mathsf{C}}^{\mathrm{DR}}$. They are, in fact, semi-simple objects. Denote the full tannakian subcategory of ${\mathsf{C}}^{\mathrm{DR}}$ they generate by ${\mathsf{S}}^{\mathrm{DR}}$. To define ${\mathcal{G}}^{\mathrm{DR}}$, we need to construct a fiber functor ${\omega}^{\mathrm{DR}}:{\mathsf{C}}^{\mathrm{DR}}\to{\mathsf{Vec}}_{\mathbb{Q}}$. We do this by restricting to the cusps of $X(N)$ and appealing to Galois descent.

17.4.1. The fiber functor

Suppose that ${\mathcal{V}}$ is an object of ${\mathsf{C}}^{\mathrm{DR}}$ whose pullback to $Y(N)$ is locally nilpotent. The natural action of ${\mathrm{GL}}_{2}({\mathbb{Z}}/N)$ on $Y(N)$ lifts to an action on the connection $\pi_{N}^{\ast}{\mathcal{V}}$ and, by the uniqueness of the canonical extension, to $({\overline{{\mathcal{V}}}}_{N},\nabla)$.

Now extend scalars to ${\mathbb{Q}}(\boldsymbol{\mu}_{N})$ and set $G_{N}=\operatorname{Gal}({{\mathbb{Q}}(\boldsymbol{\mu}_{N})}/{\mathbb{Q}})$. It acts on $X(N)\times_{\mathbb{Q}}{{\mathbb{Q}}(\boldsymbol{\mu}_{N})}$ and semi-linearly on $\pi_{N}^{\ast}{\mathcal{V}}\otimes_{\mathbb{Q}}{{\mathbb{Q}}(\boldsymbol{\mu}_{N})}$. It therefore acts semi-linearly on the canonical extension ${\overline{{\mathcal{V}}}}_{N}\otimes_{\mathbb{Q}}{{\mathbb{Q}}(\boldsymbol{\mu}_{N})}$ and also on its restriction

[TABLE]

to the set of cusps $C_{N}({{\mathbb{Q}}(\boldsymbol{\mu}_{N})})$. Here $V_{p}$ denotes the fiber of ${\overline{{\mathcal{V}}}}_{N}\otimes_{\mathbb{Q}}{\mathbb{Q}}(\boldsymbol{\mu}_{N})$ over $P\in C_{N}({\mathbb{Q}}(\boldsymbol{\mu}_{N}))$. The action of ${\mathrm{GL}}_{2}({\mathbb{Z}}/N)$ on $X(N)$ induces a ${{\mathbb{Q}}(\boldsymbol{\mu}_{N})}$-linear action on this vector space. We would like to take ${\mathrm{GL}}_{2}({\mathbb{Z}}/N)$-invariants to get a fiber functor. This does not work as the stabilizer of $P$ in ${\mathrm{GL}}_{2}({\mathbb{Z}}/N)$ may act non-trivially on $V_{P}$. We circumvent this problem by twisting the ${\mathrm{GL}}_{2}({\mathbb{Z}}/N)$ action as follows.

Denote the stabilizer of $P$ in ${\mathrm{GL}}_{2}({\mathbb{Z}}/N)$ by $U_{P}$. It is conjugate to the cyclic subgroup of ${\mathrm{GL}}_{2}({\mathbb{Z}}/N)$ consisting of upper triangular unipotent matrices and their negatives. The group $G_{N}$ acts on ${\mathrm{GL}}_{2}({\mathbb{Z}}/N)$ and $\sigma\in G_{N}$ will conjugate $U_{P}$ into $U_{\sigma(P)}$. Consequently, ${\mathrm{GL}}_{2}({\mathbb{Z}}/N)$ acts ${{\mathbb{Q}}(\boldsymbol{\mu}_{N})}$-linearly on

[TABLE]

and the Galois group $G_{N}$ acts semi-linearly. For each cusp $P$, evaluation at the identity of $U_{P}$ induces an isomorphism

[TABLE]

Lemma 17.9**.**

The set

[TABLE]

is a ${\mathbb{Q}}$ vector space of dimension equal to the rank of ${\mathcal{V}}_{N}$. For each $P\in C_{N}({{\mathbb{Q}}(\boldsymbol{\mu}_{N})})$, the projection (46) induces a natural isomorphism

[TABLE]

Proof.

For each cusp $P$, the projection

[TABLE]

is an isomorphism as $U_{P}$ is the isotropy group of $P$. This implies that the LHS is a ${{\mathbb{Q}}(\boldsymbol{\mu}_{N})}$ vector space of dimension equal to the rank of ${\overline{{\mathcal{V}}}}_{N}$. But this vector space has a semi-linear action by $G_{N}$. So, by Galois descent its set of $G_{N}$-invariants is a ${\mathbb{Q}}$-form. ∎

Denote the residue of the connection on ${\overline{{\mathcal{V}}}}_{N}$ at $P\in C_{N}({{\mathbb{Q}}(\boldsymbol{\mu}_{N})})$ by $L_{P}$. It is a nilpotent element of $\operatorname{End}_{{\mathbb{Q}}(\boldsymbol{\mu}_{N})}V_{P}$ which commutes with the action of $U_{P}$.

Proposition 17.10**.**

There is a nilpotent element $L\in\operatorname{End}_{\mathbb{Q}}V$ that corresponds to $L_{P}\in\operatorname{End}_{{\mathbb{Q}}(\boldsymbol{\mu}_{N})}V_{P}$ under the isomorphism (47). There is also a subgroup $U$ of $\operatorname{Aut}_{\mathbb{Q}}V$ that corresponds to the subgroup $U_{P}$ of $\operatorname{Aut}_{{\mathbb{Q}}(\boldsymbol{\mu}_{N})}V_{P}$ under this isomorphism. It commutes with $L$.

Proof.

Galois descent implies that

[TABLE]

The element $(L_{P})_{P\in C_{N}}$ is ${\mathrm{GL}}_{2}({\mathbb{Z}}/N)\rtimes G_{N}$ invariant and is therefore an element $L$ of $\operatorname{End}_{\mathbb{Q}}V$. Similarly, each element of the family $(U_{P})_{P\in C_{N}}$ is invariant, and thus defines an element of $\operatorname{Aut}_{\mathbb{Q}}V$. Denote the collection of these elements by $U$. The nilpotent endomorphism $L$ of $V$ and the subgroup $U$ of $\operatorname{Aut}V$ correspond to $L_{P}$ and $U_{P}$ under the isomorphism (47). ∎

Proposition 17.11**.**

The assignment of $V$ to the object ${\mathcal{V}}$ of ${\mathsf{C}}^{\mathrm{DR}}$ is a well defined fiber functor

[TABLE]

In particular, it does not depend on the choice of $N$. ∎

Define ${\mathcal{G}}^{\mathrm{DR}}_{\mathbb{Q}}$ by

[TABLE]

17.4.2. A comparison isomorphism

The comparison homomorphism

[TABLE]

is induced by the tensor functor ${\mathsf{C}}^{\mathrm{DR}}\to{\mathsf{C}}_{\mathbb{C}}$ that takes an object ${\mathcal{V}}$ of ${\mathsf{C}}^{\mathrm{DR}}$ to the monodromy representation

[TABLE]

of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ in the fiber over $\partial/\partial q$, which is naturally isomorphic to ${\omega}^{\mathrm{DR}}({\mathcal{V}})\otimes_{\mathbb{Q}}{\mathbb{C}}$. This functor restricts to a functor ${\mathsf{S}}^{\mathrm{DR}}\to{\mathsf{S}}_{\mathbb{C}}$. We therefore have a tensor functor

[TABLE]

It commutes with the fiber functors ${\omega}^{\mathrm{DR}}$ and ${\omega}$. We therefore have a homomorphism

[TABLE]

which induces the comparison homomorphism (48).

We now sketch a proof that the comparison homomorphism is, in fact, an isomorphism. Let $H={\omega}^{\mathrm{DR}}({\mathcal{H}})$ be the fiber of ${\mathcal{H}}$ over the cusp. The first step is to note that

[TABLE]

which implies that the reductive quotient of ${\mathcal{G}}^{\mathrm{DR}}$ is ${\mathrm{SL}}(H^{\mathrm{DR}})\times{\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$.

Next observe that if ${\mathcal{V}}$ is an object of ${\mathsf{C}}^{\mathrm{DR}}$ whose pullback to $X(N)$ is nilpotent, then

[TABLE]

where we take $N\geq 3$ to ensure that $X(N)$ is a scheme. This vanishes when $j>1$, which implies that the prounipotent radical of ${\mathcal{G}}^{\mathrm{DR}}$ is free by Proposition 16.3.

To prove that the comparison homomorphism is an isomorphism, it suffices by Proposition 16.8 to show that for all simple objects ${\mathscr{S}}^{m}_{N}$ of ${\mathsf{S}}^{\mathrm{DR}}$, the induced mapping

[TABLE]

is an isomorphism. But this follows as

[TABLE]

where $\nu:{{\mathbb{Q}}(\boldsymbol{\mu}_{N})}\to{\mathbb{C}}$ is a fixed embedding and because the right hand group is

[TABLE]

by Shapiro’s Lemma.

17.5. The mixed Hodge structure

The existence of a mixed Hodge structure on the relative completion of $\pi_{1}({\mathcal{M}}_{1,1},b)$ for any base point $b\in{\mathfrak{h}}$ follows from the main result of [11]. Here our preferred base point is $\partial/\partial q$. In this case, the MHS is the associated limit MHS. A proof of the existence of this limit MHS can be found in [13, §7].

Theorem 17.12**.**

The coordinate ring ${\mathcal{O}}({\mathcal{G}})$ of the relative completion of $\pi_{1}({\mathcal{M}}_{1,1},\partial/\partial q)$ has a natural MHS.

The Hodge filtration constructed using smooth forms. It would be better to construct a Hodge filtration on ${\mathcal{O}}({\mathcal{G}}^{\mathrm{DR}})$ and use the comparison theorem with ${\mathcal{O}}({\mathcal{G}}^{B})$ to get the MHS. This is a topic for another day.

18. Variants

There are many natural variants of this construction. Here we enumerate a few.

1. (i)

One can replace ${\mathrm{SL}}_{2}({\mathbb{Z}})$ by a congruence subgroup ${\Gamma}$ and the proreductive group ${\mathrm{SL}}_{2}\times{\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$ by ${\mathrm{SL}}_{2}\times G$, where $G$ is the image of ${\Gamma}$ in ${\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$. If, for example, ${\Gamma}={\Gamma}(N)$, the full level $N$ subgroup, ${\mathcal{G}}^{\mathrm{DR}}$ will only be defined over ${\mathbb{Q}}(\boldsymbol{\mu}_{N})$. 2. (ii)

One can also replace ${\mathrm{SL}}_{2}({\mathbb{Z}})$ by ${\mathrm{PSL}}_{2}({\mathbb{Z}})$. This has the effect of ignoring all modular forms of odd weight. 3. (iii)

Another useful construction is to complete with respect to the inclusion ${\mathrm{SL}}_{2}({\mathbb{Z}})\to{\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$. In this the completion is an extension

[TABLE]

where ${\mathcal{U}}$ is the inverse limit of the unipotent fundamental groups $\pi_{1}^{\mathrm{un}}(Y_{\Gamma}^{\mathrm{an}},\ast)$ of all modular curves. The coordinate ring of ${\mathcal{U}}$ consists of all closed iterated integrals of (not necessarily holomorphic) modular forms of weight 2. 4. (iv)

The previous construction can be restricted to a congruence subgroup and also its quotient by $\pm\operatorname{id}$. One case which may be useful is to take ${\Gamma}$ to be the level 2 subgroup ${\Gamma}(2)/(\pm\operatorname{id})$ of ${\mathrm{PSL}}_{2}({\mathbb{Z}})$. This is the fundamental group of ${{\mathbb{P}}^{1}-\{0,1,\infty\}}$. Since all multizeta values occur as periods of ${{\mathbb{P}}^{1}-\{0,1,\infty\}}$, the Hecke action in this case may shed light on the connection between MZVs and periods of modular forms.

Part IV Class Functions from Relative Completions

In this part, we show that the Hecke operators act on the class functions ${\mathcal{G}}\to{\Bbbk}$, where ${\mathcal{G}}$ is the relative completion of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ defined in the previous part. We also show that the space of such class functions is large and give a description of its elements.

To show that the Hecke operators $T_{N}$ act on conjugation invariant iterated integrals of modular forms, we need to understand how iterated integrals behave under pullback and pushforward under finite coverings. It is clear that pullbacks of iterated integrals that are constant on conjugacy classes are again iterated integrals that are class functions. However, it is not so clear that pushforwards of such iterated integrals along finite unramified coverings are again iterated integrals, possibly twisted by characters of a finite quotient of ${\mathrm{SL}}_{2}({\mathbb{Z}})$. The following very simple example illustrates the shape of the general pushforward formula and why we are forced to consider the large relative completions of the type defined in the previous part if we want the Hecke operators to act on the corresponding class functions.

Example 18.1**.**

Consider the covering $\pi:{\mathbb{C}}^{\ast}\to{\mathbb{C}}^{\ast}$ defined by $z=\pi(w)=w^{d}$. Its automorphism group is canonically $\boldsymbol{\mu}_{d}$, the group of $d$th roots of unity in ${\mathbb{C}}$, which we consider to be the affine group whose set of ${\mathbb{Q}}$-points is $\boldsymbol{\mu}_{d}$. Denote the characteristic function of $\zeta\in\boldsymbol{\mu}_{d}$ by ${\mathbf{1}}_{\zeta}$ and its order by $o(\zeta)$. Corollary 4.3 and the iterated integral identity

[TABLE]

imply that

[TABLE]

where $\rho:\pi_{1}(S^{1},1)\to\boldsymbol{\mu}_{d}$ is the monodromy homomorphism. In particular, the pushforward is an iterated integral with coefficients in the coordinate ring of the automorphism group of the covering.

19. Class Functions

We extend the definition of the class functions of a discrete or profinite group given in Section 7 to affine groups in the obvious way. Suppose that ${\Bbbk}$ is a field and that $G$ is an affine ${\Bbbk}$-group with coordinate ring ${\mathcal{O}}(G)$. The ring ${\mathscr{C}\!\ell}(G)$ of class functions on $G$ is defined to be the ring of conjugation invariant elements of ${\mathcal{O}}(G)$. For a vector space $V$ over ${\Bbbk}$, we define

[TABLE]

This is the space of $V$-valued class functions.

By the ring of class functions ${\mathscr{C}\!\ell}({\mathcal{G}})$ on the relative completion ${\mathcal{G}}$ of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ constructed in Section 17, we will mean the collection of the class functions on each of its realizations — Betti, de Rham and $\ell$-adic. Set

[TABLE]

These are vector spaces over ${\mathbb{Q}}$, ${\mathbb{Q}}$ and ${{\mathbb{Q}}_{\ell}}$, respectively. There are comparison isomorphisms

[TABLE]

for each prime number $\ell$. These are ring isomorphisms.

Theorem 19.1**.**

The ring of class functions ${\mathscr{C}\!\ell}({\mathcal{G}})$ on the completion ${\mathcal{G}}$ of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ has a natural ind ${\mathbb{Q}}$ mixed Hodge structure. For each prime $\ell$, there is a natural action of $G_{\mathbb{Q}}$ on ${\mathscr{C}\!\ell}({\mathcal{G}})^{\mathrm{\acute{e}t}}_{{\mathbb{Q}}_{\ell}}$. Neither structure depends on the choice of a base point of ${\mathcal{M}}_{1,1}$. The Hodge and Galois structures are compatible with the ring structure.

Proof.

This follows from Corollary 17.7, Theorem 17.12 and the theorem of the fixed part. ∎

20. Pushforward of Class Functions

In this section we prove a result that implies that the Hecke algebra ${\widehat{{\mathbb{T}}}}^{\mathrm{op}}$ acts on the class functions of ${\mathscr{C}\!\ell}({\mathcal{G}})$. We will assume familiarity with the setup and notation introduced in Section 3 and also with the discussion of the pullback map from Section 4.2.

20.1. The setup

Suppose that $\pi:(Y,y_{o})\to(X,x_{o})$ is a finite covering of connected, pointed topological spaces. Denote the image of the monodromy representation

[TABLE]

by $G$. Denote the image of

[TABLE]

by $H$.

Fix a field ${\Bbbk}$ of characteristic zero. We will regard $G$ as an affine ${\Bbbk}$-group in the standard way. Suppose that ${\mathcal{G}}_{X}$ is an affine ${\Bbbk}$-group and that ${\mathcal{G}}_{X}\to G$ is a surjective (i.e., faithfully flat) homomorphism. Denote the inverse image of $H$ by ${\mathcal{G}}_{Y}$. It is also a ${\Bbbk}$ group. Suppose that we have a commutative diagram

[TABLE]

where ${\mathbb{K}}$ is an extension of ${\Bbbk}$ and $\theta_{X}$ and $\theta_{Y}$ are Zariski dense. So we have isomorphisms of right $G$-sets

[TABLE]

We will also assume that ${\mathcal{G}}_{X}({\Bbbk})\to G$ is surjective. Equivalently, each coset of ${\mathcal{G}}_{Y}$ in ${\mathcal{G}}_{X}$ has a ${\Bbbk}$ rational point.

Example 20.1**.**

Our principal example is where $X={\mathcal{M}}_{1,1}$, $Y=Y_{0}(p)$ and $\pi$ is the projection $Y_{0}(p)\to{\mathcal{M}}_{1,1}$ induced by the inclusion of ${\Gamma}_{0}(p)$ into ${\mathrm{SL}}_{2}({\mathbb{Z}})$. The group ${\mathcal{G}}_{X}$ will be one of the incarnations of the relative completion of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ constructed in Section 17. More precisely, it will ${\mathcal{G}}^{B}$, ${\mathcal{G}}^{\mathrm{DR}}_{\mathbb{Q}}$ or ${\mathcal{G}}^{{\mathrm{\acute{e}t}}_{\ell}}_{{\mathbb{Q}}_{\ell}}$ for some $\ell$. In the first two cases ${\Bbbk}$ will be ${\mathbb{Q}}$ and in the third it is ${{\mathbb{Q}}_{\ell}}$. The field ${\mathbb{K}}$ will be ${\mathbb{Q}}$ in the first case, ${\mathbb{C}}$ in the second and ${{\mathbb{Q}}_{\ell}}$ in the third. In all cases, $G={\mathrm{PSL}}_{2}({{\mathbb{F}}_{p}})$ and $H$ is its upper triangular Borel subgroup. The group ${\mathcal{G}}_{Y}$ will be the Zariski closure of $\pi_{1}(Y,y_{o})$ in ${\mathcal{G}}_{X}({\mathbb{K}})$. In all cases it is defined over ${\Bbbk}$ and the homomorphism ${\mathcal{G}}_{X}({\Bbbk})\to{\mathrm{PSL}}_{2}({{\mathbb{F}}_{p}})$ will be surjective.

If $y\in\pi^{-1}(x_{o})$, then $\pi_{1}(Y,y)=\gamma^{-1}\pi_{1}(Y,y_{o})\gamma$, where $\gamma\in\pi_{1}(X,x_{o})$ is the projection of a path in $Y$ from $y_{o}$ to $y$. Its Zariski closure in ${\mathcal{G}}_{X}$ will be $\theta_{X}(\gamma)^{-1}{\mathcal{G}}_{Y}\theta_{X}(\gamma)$. If $g\in{\mathcal{G}}_{X}({\Bbbk})$ projects to $\rho(\gamma)\in G$, then

[TABLE]

This implies that the Zariski closure of $\pi_{1}(Y,y)$ is defined over ${\Bbbk}$.

Conjugation by $g$ induces a map

[TABLE]

of ${\Bbbk}$-algebras. This map depends only on the coset $Hg\in H\backslash G$. The following result is key to proving that the Hecke operators are morphisms of MHS.

Lemma 20.2**.**

In the setting of Example 20.1, the mapping $\Phi_{g}$ is a morphism of MHS.

Proof.

This is not immediately clear as the conjugation by $g$ map from ${\mathcal{O}}({\mathcal{G}}_{X})$ to itself is not a morphism of MHS. To prove the result, we use the fact that ${\mathcal{O}}(G)$ is a Hodge structure of type $(0,0)$ and strictness (i.e., exactness of $\operatorname{Gr}^{W}_{\bullet}$) to construct an algebra splitting

[TABLE]

of $\rho^{\ast}:{\mathcal{O}}(G)\to{\mathcal{O}}({\mathcal{G}}_{X})$. This implies that we can lift $\rho(g)\in G$ to $g^{\prime}\in F^{0}W_{0}{\mathcal{G}}({\mathbb{C}})$, which implies that $\Phi_{g^{\prime}}$ preserves the Hodge and weight filtrations. Since $\Phi_{g}=\Phi_{g^{\prime}}$, it also preserves these filtrations. Since we may choose $g$ to be an element of ${\mathcal{G}}^{B}({\mathbb{Q}})$, $\Phi_{g}$ is also defined over ${\mathbb{Q}}$. It is thus an isomorphism of MHS. ∎

For each $a\in G$ and each $g\in G$, we set

[TABLE]

Equivalently, it is the length of the orbit of $Hg$ under the right action of the subgroup $\langle a\rangle$ of $G$.

Proposition 20.3** (pushforward formula).**

The pushforward map $\pi_{\ast}:{\mathscr{C}\!\ell}({\mathcal{G}}_{Y})\to{\mathscr{C}\!\ell}({\mathcal{G}}_{X})$ is well defined and is given by the formula

[TABLE]

Proof.

The map given by the formula in the statement is clearly defined over ${\Bbbk}$. To prove the assertion, it suffices to show that the formula holds on the image of $\theta_{X}$ after extending scalars to ${\mathbb{K}}$. Suppose that $F\in{\mathscr{C}\!\ell}({\mathcal{G}}_{Y})$, that $\alpha\in\pi_{1}(X,x_{o})$ and that $\rho(\alpha)=a$. For each $g\in G$, choose $\gamma_{g}\in\pi_{1}(X,x_{o})$ such that $\rho(\gamma_{g})=g$. Observe that $\Phi_{g}=\Phi_{\theta_{X}(\gamma_{g})}$. Since

[TABLE]

we have

[TABLE]

On the other hand, since $\alpha^{d_{a}(g)}\in\gamma_{g}^{-1}\pi_{1}(Y,y_{o})\gamma_{g}$, we have

[TABLE]

∎

This formula and Lemma 20.2 imply that the pushforward map is compatible with the de Rham and Hodge structures in the case of interest.

Corollary 20.4**.**

In the setting of Example 20.1, the pushforward map

[TABLE]

on the various incarnations of ${\mathcal{G}}_{X}$ and ${\mathcal{G}}_{Y}$ correspond under the comparison maps. They also induce a morphism of MHS on the Betti incarnation.

Proof.

This follows directly from the formula in Proposition 20.2. The Hodge statement follows from Lemma 20.2, which implies that each $\Phi_{g}$ is a morphism, as well as the fact that the Adams operators are morphisms of MHS, which is a direct consequence of [11, Thm. 13.3]. ∎

21. Hecke Action on ${\mathscr{C}\!\ell}({\mathcal{G}})$

Armed with the pushforward formula (49), we can now prove that the dual Hecke operators act on ${\mathscr{C}\!\ell}({\mathcal{G}})$ and that this action respects the mixed Hodge structure and commutes with the $\ell$-adic Galois action, one of the main results of this paper. Recall that the dual Hecke algebra ${\widehat{{\mathbb{T}}}}^{\mathrm{op}}$ is the opposite ring of ${\widehat{{\mathbb{T}}}}$, which was defined in the introduction.

Theorem 21.1**.**

The Hecke algebra ${\widehat{{\mathbb{T}}}}^{\mathrm{op}}$ acts on ${\mathscr{C}\!\ell}({\mathcal{G}})$. Each element of ${\widehat{{\mathbb{T}}}}^{\mathrm{op}}$ acts as a morphism of MHS and commutes with the action of $\operatorname{Gal}({\overline{{\mathbb{Q}}}}/{\mathbb{Q}})$. The Hecke action restricts to a Hecke action on the totally holomorphic elements of ${\mathscr{C}\!\ell}({\mathcal{G}})$.

Proof.

Choose a base point $\tau\in{\mathfrak{h}}$ whose stabilizer is $\{\pm\operatorname{id}\}$. The existence of the dual action of $T_{N}$ on ${\mathscr{C}\!\ell}({\mathcal{G}})$ satisfying the equations (6) follows from the pushforward formula, Proposition 20.3. That it acts as a morphism of MHS follows from Corollary 20.4. The Galois equivariance is an immediate consequence of Theorem 14.10.

The fact that $T_{N}$ preserves the totally holomorphic iterated integrals follows from the pushforward formula. ∎

22. Constructing Class Functions

It is not immediately clear that the ring ${\mathscr{C}\!\ell}({\mathcal{G}})$ of a general affine group is large or interesting. In this section we give a general description of all elements of ${\mathscr{C}\!\ell}({\mathcal{G}})$ when ${\mathcal{G}}$ is an affine group scheme whose prounipotent radical is free. This description will apply to the relative completions ${\mathcal{G}}$ of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ constructed in Section 17.

We first work in the following abstract setting: Assume that ${\Bbbk}$ is a field of characteristic zero and that ${\mathcal{G}}$ is an affine ${\Bbbk}$-group that is an extension

[TABLE]

of a proreductive group $R$ by a prounipotent group ${\mathcal{U}}$. In addition, we assume that the Lie algebra ${\mathfrak{u}}$ of ${\mathcal{U}}$ is free as a pronilpotent Lie algebra.

22.1. A reduction

It is convenient to first reduce to the case where $R$ is an algebraic group and $H_{1}({\mathcal{U}})$ is finite dimensional.

Proposition 22.1**.**

Every extension (50) is the inverse limit of extensions

[TABLE]

where $R_{\alpha}$ is reductive, the Lie algebra ${\mathfrak{u}}_{\alpha}$ of ${\mathcal{U}}_{\alpha}$ is free on a finite dimensional representation $V_{\alpha}$ of $R_{\alpha}$ and

[TABLE]

as a pronilpotent Lie algebra in the category of pro $R$-modules.

The significance of this result for us is that

[TABLE]

The proof is somewhat technical and can be skipped on a first reading.

Sketch of Proof.

A general version of Levi’s Theorem implies that the extension (50) is split.202020This can be proved using the fact that ${\mathcal{G}}$ is the inverse limit of affine algebraic groups, to which one can apply the usual version of Levi’s theorem, and use Zorn’s Lemma to prove that there is a maximal quotient of ${\mathcal{G}}$ on which there is a splitting. One then shows, as usual, that the maximal quotient has to be ${\mathcal{G}}$ itself. The choice of a splitting makes the Lie algebra ${\mathfrak{u}}$ of ${\mathcal{U}}$ into a pronilpotent Lie algebra in the category of pro $R$-modules. The choice of a continuous $R$-invariant splitting of the abelianization map

[TABLE]

induces a Lie algebra homomorphism ${\mathbb{L}}(H_{1}({\mathfrak{u}}))\to{\mathfrak{u}}$, which is continuous if we give ${\mathbb{L}}({\mathbb{H}}_{1}({\mathfrak{u}}))$ the natural topology. Since ${\mathfrak{u}}$ is complete, this homomorphism extends to a continuous homomorphism

[TABLE]

which is surjective as both Lie algebras are pronilpotent and the homomorphism induces an isomorphism on abelianizations. It is injective as ${\mathfrak{u}}$ is free. We conclude that there are isomorphisms

[TABLE]

To complete the proof, write $H_{1}({\mathfrak{u}})=\varprojlim V_{\alpha}$, where each $V_{\alpha}$ is a finite dimensional $R$-module. We can write $R$ as the inverse limit of reductive affine algebraic groups $R_{\alpha}$, where $R_{\alpha}$ acts on $V_{\alpha}$. Then

[TABLE]

where ${\mathcal{U}}_{\alpha}$ is the quotient of ${\mathcal{U}}$ whose Lie algebra is ${\mathbb{L}}(V_{\alpha})^{\wedge}$. ∎

22.2. A special case

The results of the previous section reduce the problem of understanding ${\mathscr{C}\!\ell}({\mathcal{G}})$ to the case where $R$ is a reductive (and thus algebraic) group and where the abelianization $H_{1}({\mathfrak{u}})$ of ${\mathfrak{u}}$ is finite dimensional. We examine this case in this section. For convenience, we denote $H_{1}({\mathfrak{u}})$ by $V$.

Every such extension is split and the splitting is unique up to conjugation by an element of ${\mathcal{U}}({\Bbbk})$. Fix a splitting. It determines a left action of $R$ on ${\mathfrak{u}}$ and an isomorphism

[TABLE]

Elements of ${\mathcal{G}}(K)$, where $K$ is an extension of ${\Bbbk}$, will be identified with pairs $(u,r)\in{\mathcal{U}}(K)\times R(K)$ with multiplication

[TABLE]

where $r:u\to r\cdot u$ denotes the left conjugation action of $R$ on ${\mathcal{U}}$. Since the map ${\mathcal{U}}\times R\to{\mathcal{G}}$ defined by $(u,r)\mapsto ur$ is an isomorphism of affine schemes, we will sometimes denotes $(u,r)$ by $ur$.

The choice of an $R$-invariant splitting $s:V\to{\mathfrak{u}}$ of ${\mathfrak{u}}\to H_{1}({\mathfrak{u}})$ induces a continuous $R$-invariant homomorphism ${\mathbb{L}}(V)^{\wedge}\to{\mathfrak{u}}$ which induces an isomorphism on $H_{1}$ and is thus surjective. Since ${\mathfrak{u}}$ is free, it is an isomorphism as both Lie algebras are pronilpotent.

22.2.1. The coordinate ring of ${\mathcal{G}}$

A basic reference for this section is Appendix A of Quillen’s paper [29]. Denote the tensor algebra on $V$ by $T(V)$ and its degree completion by $T(V)^{\wedge}$. It is a topological algebra that is complete in the topology defined by the powers of the closed ideal $I$ generated by $V$. It has additional structure; it is a complete Hopf algebra with diagonal

[TABLE]

defined by $\Delta v=1\otimes v+v\otimes 1$ for all $v\in V$. Its space of primitive elements is ${\mathbb{L}}(V)^{\wedge}$ and ${\mathcal{U}}$ is isomorphic to the set of group-like elements of $T(V)^{\wedge}$.

Since ${\mathbb{L}}(V)^{\wedge}$ is a topological Lie algebra, its enveloping algebra is a topological Hopf algebra. The following result summarizes several well-known facts.

Proposition 22.2** ([29, Appendix A]).**

The inclusion ${\mathbb{L}}(V)^{\wedge}\to T(V)^{\wedge}$ induces a complete Hopf algebra isomorphism of the completed universal enveloping algebra of ${\mathbb{L}}(V)^{\wedge}$ with $T(V)^{\wedge}$. Consequently, there is a complete coalgebra isomorphism

[TABLE]

Moreover, if we identify ${\mathfrak{u}}$ with ${\mathbb{L}}(V)^{\wedge}$, then the exponential mapping

[TABLE]

restricts to an isomorphism of affine schemes ${\mathfrak{u}}\to{\mathcal{U}}$.

The last statement of the proposition implies that the coordinate ring of ${\mathcal{U}}$ is the ring of continuous polynomials on ${\mathbb{L}}(V)^{\wedge}$. The previous result implies that this is just the continuous dual of $T(V)^{\wedge}$, which is the graded dual

[TABLE]

of $T(V)$, where $\check{V}$ denotes the dual of $V$. Multiplication is given by the shuffle product

[TABLE]

which is defined by

[TABLE]

where each $\varphi_{j}\in\check{V}$ and $\sigma$ ranges over the shuffles of type $(a,b)$.

Corollary 22.3**.**

The chosen splitting of ${\mathcal{G}}\to R$ induces an algebra isomorphism of the coordinate ring of ${\mathcal{G}}$ with

[TABLE]

22.2.2. Computation of ${\mathscr{C}\!\ell}({\mathcal{G}})$

To compute ${\mathscr{C}\!\ell}({\mathcal{G}})$ first observe that

[TABLE]

where ${\mathcal{G}}$ (and thus $R$ and ${\mathcal{U}}$ by restriction) acts on ${\mathcal{O}}({\mathcal{G}})$ on the left by conjugation:

[TABLE]

This convention will hold throughout this section.

We regard ${\mathcal{O}}({\mathcal{U}})$ as an ind-scheme over ${\Bbbk}$. Fix an algebraic closure ${\overline{{\Bbbk}}}$ of ${\Bbbk}$. Elements of ${\mathcal{O}}({\mathcal{U}})\otimes{\mathcal{O}}(R)$ correspond to morphisms $R\to{\mathcal{O}}({\mathcal{U}})$: $f:R\to{\mathcal{O}}({\mathcal{U}})$ and $F\in{\mathcal{O}}({\mathcal{U}})\otimes{\mathcal{O}}(R)$ correspond if and only if

[TABLE]

for all $u\in{\mathcal{U}}({\overline{{\Bbbk}}})$, $r\in R({\overline{{\Bbbk}}})$.

Lemma 22.4**.**

Let $R$ act on itself by conjugation: $r:t\mapsto rtr^{-1}$. Under the correspondence above, elements of \big{[}{\mathcal{O}}({\mathcal{U}})\otimes{\mathcal{O}}(R)\big{]}^{R} correspond to $R$-invariant morphisms $R\to{\mathcal{O}}({\mathcal{U}})$.

Proof.

Suppose that $f:R\to{\mathcal{O}}({\mathcal{U}})$ and $F\in{\mathcal{O}}({\mathcal{U}})\otimes{\mathcal{O}}(R)$ correspond, that $r,t\in R({\overline{{\Bbbk}}})$ and that $u\in{\mathcal{U}}({\overline{{\Bbbk}}})$. On the one hand we have

[TABLE]

and on the other, we have

[TABLE]

Since this holds for all $u\in{\mathcal{U}}({\overline{{\Bbbk}}})$, it follows that for all $t\in R({\overline{{\Bbbk}}})$, $F=t^{-1}F$ if and only if $f(trt^{-1})=t\cdot f(r)$. ∎

The ring \big{[}{\mathcal{O}}({\mathcal{U}})\otimes{\mathcal{O}}(R)\big{]}^{R} is graded by degree in $V$:

[TABLE]

Elements of the summand \big{[}\check{V}^{\otimes m}\otimes{\mathcal{O}}(R)\big{]}^{R} correspond to $R$-invariant functions $R\to\check{V}^{\otimes m}$

We now compute the ${\mathcal{U}}$-invariants that are in \big{[}{\mathcal{O}}({\mathcal{U}})\otimes{\mathcal{O}}(R)\big{]}^{R}. For this, it is useful to introduce the infinite cyclic group $\Sigma$ generated by the symbol $\sigma$. It acts on the set $(V^{\otimes m}\otimes_{\Bbbk}K)\times R(K)$, where $K$ is an extension of ${\Bbbk}$, via the formula

[TABLE]

and on $[\check{V}^{\otimes m}\otimes{\mathcal{O}}(R)]^{R}$ via the dual action.

Lemma 22.5**.**

If $F\in[\check{V}^{\otimes m}\otimes{\mathcal{O}}(R)]^{R}$, then $\sigma^{m}F=F$, so that the $\Sigma$ action factors through an action of its cyclic quotient $C_{m}:=\Sigma/\langle\sigma^{m}\rangle$. Moreover, the degree $m$ summand of ${\mathscr{C}\!\ell}({\mathcal{G}})$ consists of the $R$-invariant functions that are also $C_{m}$ invariant:

[TABLE]

Proof.

Suppose that $v_{1},\dots,v_{m}\in V$ $r\in R({\overline{{\Bbbk}}})$ and that $F\in[\check{V}^{\otimes m}\otimes{\mathcal{O}}(R)]^{R}$. Then

[TABLE]

so that $\sigma^{m}F=F$. Since

[TABLE]

in $(T(V)^{\wedge}\otimes{\overline{{\Bbbk}}})\rtimes R({\overline{{\Bbbk}}})$, we see that a function $F\in[\check{V}^{\otimes m}\otimes{\mathcal{O}}(R)]^{R}$ is ${\mathcal{U}}$-invariant (and hence ${\mathcal{G}}$-invariant) if and only if for all $v_{1},\dots,v_{m}\in V$ and $r\in R({\overline{{\Bbbk}}})$ we have

[TABLE]

That is, if and only if $\sigma F=F$. ∎

Corollary 22.6**.**

The choice of a splitting (51) determines a graded ring isomorphism

[TABLE]

where

[TABLE]

The product ${\mathscr{C}\!\ell}_{m}({\mathcal{G}})\otimes{\mathscr{C}\!\ell}_{n}({\mathcal{G}})\to{\mathscr{C}\!\ell}_{m+n}({\mathcal{G}})$ is the extension

[TABLE]

of the shuffle product (52), where $\varphi_{j}^{\prime}\in\check{V}^{\otimes m}$, $\varphi_{k}\in\check{V}^{\otimes n}$, $\psi_{j}^{\prime},\psi_{k}^{\prime\prime}\in{\mathcal{O}}(R)$.

While the grading of ${\mathscr{C}\!\ell}({\mathcal{G}})$ depends on the splitting of the projection ${\mathcal{G}}\to R$, the filtration

[TABLE]

does not, where

[TABLE]

However, since the splitting $s:R\to{\mathcal{G}}$ is unique up to conjugation by an element of ${\mathcal{U}}({\Bbbk})$, the projection

[TABLE]

induced by $s$ does not depend on $s$. This means that there is a canonical decomposition

[TABLE]

Remark 22.7*.*

Several comments are in order:

1. (i)

The prounipotent case (i.e., $R$ trivial) is well known, even when ${\mathfrak{u}}$ is not free. See [21] for the surface group case. The general case is almost identical. The ring ${\mathscr{C}\!\ell}({\mathcal{U}})$ is the continuous dual $\operatorname{Hom}^{\mathrm{cts}}_{\Bbbk}(|U{\mathfrak{u}}|,{\Bbbk})$ of the cyclic quotient

[TABLE]

of the enveloping algebra $U{\mathfrak{u}}$. 2. (ii)

A statement equivalent to the $m=1$ case of Lemma 22.5 was communicated to me by Florian Naef in 2019. 3. (iii)

Not all irreducible representations of $R$ occur in ${\mathcal{O}}(R)$ when it is viewed as an $R$-module via conjugation. Only those representations that factor through the adjoint form $R/Z(R)$ of $R$ can appear. In particular, when $R={\mathrm{SL}}_{2}$, only the even symmetric powers of the defining representation occur. However, each occurs a countable number of times.

22.3. Constructing elements of ${\mathscr{C}\!\ell}_{m}({\mathcal{U}}\rtimes R)$

Here we give a few explicit constructions of class functions on ${\mathcal{U}}\rtimes R$. We continue with the notation and setup of the previous section. We will also denote ${\mathcal{O}}(R)$, regarded as a left $R$-module via right conjugation, by ${\mathcal{O}}(R)^{\mathrm{conj}}$.

22.3.1. Computing ${\mathscr{C}\!\ell}_{0}({\mathcal{U}}\rtimes R)$

This is just ${\mathscr{C}\!\ell}(R)$. Denote the representation ring of $R$ by ${\mathsf{Rep}}(R)$. Proposition 16.5 implies that the function

[TABLE]

that takes the isomorphism class of the $R$-module $A$ to the class function $R({\overline{{\Bbbk}}})\to{\Bbbk}$ defined by $r\mapsto\operatorname{tr}\rho_{A}(r)$ is an isomorphism, where $\rho_{A}:R\to\operatorname{Aut}V$ is the corresponding homomorphism.

When $G$ is finite, ${\mathscr{C}\!\ell}(G)$ is spanned by the irreducible characters of $G$. And when $G$ is the projective limit of $G_{\alpha}$, we have

[TABLE]

22.3.2. Computing ${\mathscr{C}\!\ell}_{1}({\mathcal{U}}\rtimes R)$

Elements of ${\mathscr{C}\!\ell}_{1}({\mathcal{G}})$ correspond to $R$-module homomorphisms

[TABLE]

This can be computed by decomposing

[TABLE]

and using Proposition 16.5.

Remark 22.8*.*

A function $F:V\to{\mathcal{O}}(R)^{\mathrm{conj}}$ extends to the function

[TABLE]

The $R$ invariance of $F$ is equivalent to saying that this function is a 1-cocycle on ${\mathcal{U}}\rtimes R$ with values in ${\mathcal{O}}(R)^{\mathrm{conj}}$:

[TABLE]

The class function corresponding to a coboundary is zero.

The pullback of a 1-cocycle $F$ along a homomorphism $\rho:{\Gamma}\to{\mathcal{U}}\rtimes R({\Bbbk})$ will define a class function on ${\Gamma}$. Conversely, each 1-cocycle $h:{\Gamma}\to{\mathcal{O}}(R)^{\mathrm{conj}}$ gives rise to the class function ${\Gamma}\to{\Bbbk}$ defined by

[TABLE]

Coboundaries give the trivial class function, so there is a well defined function

[TABLE]

This observation explains how and why modular forms give class functions on ${\mathrm{SL}}_{2}({\mathbb{Z}})$. See Section 23.3.1.

22.3.3. Constructing elements of ${\mathscr{C}\!\ell}_{m}({\mathcal{U}}\rtimes R)$ by averaging

Suppose that $m>1$ and that $F\in[\check{V}^{\otimes m}\otimes{\mathcal{O}}(R)^{\mathrm{conj}}]$. Define $\operatorname{rot}F\in[\check{V}^{\otimes m}\otimes{\mathcal{O}}(R)]$ by

[TABLE]

and, for each $j\in\{1,\dots,m\}$ define

[TABLE]

by

[TABLE]

In both cases $r\in R({\overline{{\Bbbk}}})$ and each $v_{k}\in V$.212121The second can be defined by combining the map $\nabla_{V}:\check{V}\to{\mathcal{O}}(R)\otimes\check{V}$ induced by the left action of $R$ on $V$ with multiplication in ${\mathcal{O}}(R)$.

Proposition 22.9**.**

If $F$ is $R$-invariant, then so are $\operatorname{rot}F$ and each $\nabla_{j}F$.

Proof.

It is clear that $F$ invariant implies that $\operatorname{rot}F$ is invariant. The other assertions follow from that the fact that for all $t,r\in R({\overline{{\Bbbk}}})$ we have

[TABLE]

∎

The action of the cyclic group $C_{m}=\langle\sigma\rangle$ on an $R$-invariant function $F\in[\check{V}^{\otimes m}\otimes{\mathcal{O}}(R)^{\mathrm{conj}}]^{R}$ can be expressed in terms of these:

[TABLE]

so that the average $\overline{F}$ of $F$ over the $C_{m}$ action is

[TABLE]

It is $C_{m}$ invariant and therefore an element of ${\mathscr{C}\!\ell}_{m}({\mathcal{U}}\rtimes R)$.

It is not always clear when $\overline{F}$ is non-zero. The following somewhat technical result gives a criterion for the cyclic average to be non-zero. It will be used in Section  23.3.2 to construct class functions from modular forms of odd weight.

Proposition 22.10**.**

Suppose that, as an $R$-module, we can write $V=V_{1}\oplus\dots\oplus V_{m}\oplus E$. Let $q_{j}:\check{V}\to\check{V}_{j}$ be the projection dual to the inclusion $V_{j}\to V$. Suppose that

[TABLE]

is $R$-invariant. If $F$ is the composite

[TABLE]

where $q=q_{1}\otimes\dots\otimes q_{m}$, then $F\neq 0$ implies that its cyclic average $\overline{F}$ is also non-zero.

Proof.

Since $F\neq 0$, we have $v_{j}\in V_{j}$ such that $F(v_{1}\dots v_{m})\neq 0$. The definition of $F$ implies that for all non-identity permutations $\sigma$ of $\{1,\dots,m\}$,

[TABLE]

The definition of $\overline{F}$ implies that $\overline{F}(v_{1}\dots v_{m})=F(v_{1}\dots v_{m})\neq 0$. ∎

Example 22.11**.**

Suppose that $F_{1},\dots,F_{m}\in{\mathscr{C}\!\ell}_{1}({\mathcal{U}}\rtimes R)$. Define

[TABLE]

by

[TABLE]

This function is $R$-invariant. Its cyclic average is the $R$-invariant function

[TABLE]

defined by

[TABLE]

We will call this the cyclic average of $F_{1},\dots,F_{m}$. If $V$ contains $V_{1}\oplus\dots\oplus V_{m}$ and if $F_{j}\in\check{V}_{j}$, then $F_{1}\odot F_{2}\odot\cdots\odot F_{m}$ will be non-zero provided that $F\neq 0$.

Remark 22.12*.*

When $m=2$, $F_{1}\odot F_{2}$ is simply the extended shuffle product (53) of $F_{1}$ and $F_{2}$. When $m>2$ it appears that, in general, cyclic products of linearly independent elements $F_{j}$ of ${\mathscr{C}\!\ell}_{1}({\mathcal{U}}\rtimes R)$ will not, in general, be decomposable.

23. Examples of Conjugation Invariant Iterated Integrals

Our goal now is to give several constructions of interesting elements of ${\mathscr{C}\!\ell}({\mathcal{G}})$, where ${\mathcal{G}}$ is the relative completion of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ defined in Section 17. These class functions will be constructed from holomorphic modular forms. We begin with a few remarks about ${\mathscr{C}\!\ell}_{0}({\mathcal{G}})$ as ${\mathscr{C}\!\ell}({\mathcal{G}})$ is a module over it. The first step is the observation:

Lemma 23.1**.**

We have ${\mathscr{C}\!\ell}_{0}({\mathcal{G}})\cong{\mathscr{C}\!\ell}({\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}}))[\operatorname{tr}]$, where $\operatorname{tr}:{\mathrm{SL}}_{2}\to{\mathbb{Q}}$ is the trace.

Proof.

The discussion in Section 22.3.1 implies that ${\mathscr{C}\!\ell}({\mathrm{SL}}_{2}/{\Bbbk})={\Bbbk}[\operatorname{tr}]$, where $\operatorname{tr}$ is the trace function ${\mathrm{SL}}_{2}\to{\Bbbk}$. So

[TABLE]

∎

After tensoring with ${\mathbb{C}}$, the ring ${\mathscr{C}\!\ell}({\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}}))$ is spanned by the irreducible characters of ${\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$. Note that this ring has many zero divisors as each ${\mathrm{SL}}_{2}({\mathbb{Z}}/N)$ is totally disconnected.

23.1. Modular forms as cohomology classes

We begin by recalling how modular forms define cohomology classes. We continue with the setup and notation from Section 15.1.1. More detailed references for this discussion are Sections 9 and 11 of [13].

Suppose that $V_{\chi}$ is an irreducible ${\mathrm{SL}}_{2}({\mathbb{Z}}/N)$-module with character. Suppose that $f:{\mathfrak{h}}\to V_{\chi}$ is a vector valued modular form of weight $m$, level $N$ and character $\chi$. By this, we mean that

[TABLE]

where $\rho_{\chi}:{\mathrm{SL}}_{2}({\mathbb{Z}})\to{\mathrm{SL}}_{2}({\mathbb{Z}}/N)\to\operatorname{Aut}V_{\chi}$ is the associated representation.

We take the diagonal maximal torus in ${\mathrm{SL}}_{2}$. Set

[TABLE]

This is an element of the Lie algebra ${\mathfrak{sl}}_{2}$ of torus weight $-2$; and its transpose is the element of torus weight $2$. Denote the irreducible ${\mathrm{SL}}_{2}$-module with a highest weight vector $\mathbf{e}$ of (torus) weight $k$ by $S^{k}(\mathbf{e})$. The holomorphic 1-form

[TABLE]

on the upper half plane ${\mathfrak{h}}$ with values in $S^{m}(\mathbf{e})\otimes V_{\chi}$ is ${\mathrm{SL}}_{2}({\mathbb{Z}})$-invariant in the sense that

[TABLE]

for all $\gamma\in{\mathrm{SL}}_{2}({\mathbb{Z}})$. It defines a class in

[TABLE]

We need to know which characters of ${\mathrm{SL}}_{2}({\mathbb{Z}}/N)$ occur in $H^{1}({\Gamma}(N),S^{m}(\mathbf{e}))$. We have the following elementary necessary condition.

Lemma 23.2**.**

If the character $\chi$ of ${\mathrm{SL}}_{2}({\mathbb{Z}}/N)$ occurs in $H^{1}({\Gamma}(N),S^{m}H)$, then $\chi(-\operatorname{id})=(-1)^{m}$.

Proof.

If $\chi$ occurs in $H^{1}({\Gamma}(N),S^{m}H)$, there is a non-zero modular form $f$ satisfying (55). Taking $\gamma=-\operatorname{id}$ in this equation implies that $\rho(-\operatorname{id})=(-1)^{m}$. ∎

Jared Weinstein [38, Thm. 4.3] has shown that this is the only restriction.

Theorem 23.3** (Weinstein).**

An irreducible complex representation $\rho$ of ${\mathrm{SL}}_{2}({\mathbb{Z}}/N)$ occurs in $H^{1}({\Gamma}(N),S^{m}H_{\mathbb{C}})$ if and only if $\rho(-\operatorname{id})=(-1)^{m}$.

23.2. The conjugation action on ${\mathcal{O}}({\mathrm{SL}}_{2})$ and ${\mathcal{O}}({\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}}))$

To construct elements of ${\mathscr{C}\!\ell}({\mathcal{G}})$ from modular forms, we need to know which representations of ${\mathrm{SL}}_{2}\times{\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$ occur in ${\mathcal{O}}({\mathrm{SL}}_{2})^{\mathrm{conj}}\otimes{\mathcal{O}}({\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}}))^{\mathrm{conj}}$.

As an ${\mathrm{SL}}_{2}\times{\mathrm{SL}}_{2}$ module, ${\mathcal{O}}({\mathrm{SL}}_{2})$ is isomorphic to

[TABLE]

The conjugation action is obtained by restricting to the diagonal. Consequently, we have the isomorphism

[TABLE]

Proposition 23.4**.**

Only even symmetric powers of $H$ occur in the conjugation representation of ${\mathrm{SL}}_{2}$ on its coordinate ring and each occurs with infinite multiplicity.

The Chinese Remainder Theorem implies that

[TABLE]

So, to understand ${\mathcal{O}}({\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}}))^{\mathrm{conj}}$, it suffices to understand ${\mathcal{O}}({\mathrm{SL}}_{2}({\mathbb{Z}}/p^{n}))^{\mathrm{conj}}$ for all $p$ and $n>0$.

Since $-\operatorname{id}$ is central in ${\mathrm{SL}}_{2}({\mathbb{Z}}/p^{n})$, it will act trivially on ${\mathcal{O}}({\mathrm{SL}}_{2}({\mathbb{Z}}/p^{n}))^{\mathrm{conj}}$. The converse is true in most cases.

Theorem 23.5** (Tiep).**

Suppose that $p>3$ and $n>0$. An irreducible complex representation $\rho$ of ${\mathrm{SL}}_{2}({\mathbb{Z}}/p^{n})$ occurs in ${\mathcal{O}}({\mathrm{SL}}_{2}({\mathbb{Z}}/p^{n}))^{\mathrm{conj}}\otimes{\mathbb{C}}$ if and only if $\rho(-\operatorname{id})=1$.

The proof appears in Appendix A.

Remark 23.6*.*

The even symmetric powers $S^{2n}H$ of ${\mathrm{SL}}_{2}$ are precisely the representations on which $-\operatorname{id}$ acts trivially. So, in both the case of ${\mathrm{SL}}_{2}$ and ${\mathrm{SL}}_{2}({\mathbb{Z}}/p^{n})$, the only restriction on the representations $\rho$ that occur in the conjugation representation is that $\rho(-\operatorname{id})=1$, provided of course that $p>3$.

Remark 23.7*.*

This remark, I hope, will become obsolete and will be removed. There is now significant computational evidence to suggest that Tiep’s result above extends to the cases $p=2,3$. Tiep checked some cases by hand and recruited Eamonn O’Brien who has, using a computer, verified it for $2^{n}$ with $n\leq 11$ and $3^{n}$ with $n\leq 7$.

23.3. Class functions from modular forms

Here we show all modular forms of all levels give rise to class functions. These constructions imply that their motives occur in the weight graded quotients of ${\mathscr{C}\!\ell}({\mathcal{G}})$. In view of the result in the previous 2 sections, we will need to separate the cases of odd and even weight.

23.3.1. The even weight case

Here we elaborate on Remark 22.8. Suppose that $N\geq 1$, that $m\geq 2$ is even and that that $\chi$ is a character of ${\mathrm{SL}}_{2}({\mathbb{Z}}/N)$ that appears in its conjugation representation. Theorem 23.3 implies that there is a non-zero vector valued modular form $f:{\mathfrak{h}}\to V_{\chi}$ of weight $m$ and level $N$. The corresponding form ${\omega}_{f}(\mathbf{e})$ takes values in $S^{m}(\mathbf{e})\otimes V_{\chi}$.

The following result implies that the Hodge structure $V_{f}$ associated to a Hecke eigen cusp form $f$ of even weight will appear in $\operatorname{Gr}^{W}_{1}{\mathscr{C}\!\ell}({\mathcal{G}})$.

Proposition 23.8**.**

For each ${\mathrm{SL}}_{2}\times{\mathrm{SL}}_{2}({\mathbb{Z}}/N)$ invariant function

[TABLE]

(where the groups act on the target via conjugation), the function

[TABLE]

is a class function on ${\mathrm{SL}}_{2}({\mathbb{Z}})$ that is the restriction of an element of ${\mathscr{C}\!\ell}({\mathcal{G}})$.

Such class functions correspond to elements of

[TABLE]

This construction does not work when $m$ is odd or when $\chi$ does not appear in the conjugation action ${\mathcal{O}}({\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}}))$, such as when $\chi(-\operatorname{id})=1$.

Proof.

We first explain why the value of $F_{f,\varphi}$ on $\alpha$ is well defined. To compute the integral, we first choose a base point $\tau_{0}\in{\mathfrak{h}}$ and a path $c_{\alpha}$ from $\tau_{0}$ to $\alpha\tau_{0}$. It is unique up to homotopy. Define

[TABLE]

We need to explain why it does not depend on $\tau_{0}$. Suppose that $\tau_{1}$ is another base point. Choose a path $c_{1}$ in ${\mathfrak{h}}$ from $\tau_{1}$ to $\tau_{0}$. Then $c_{\alpha}^{\prime}=c_{1}c_{\alpha}(\alpha\cdot c_{1}^{-1})$ is a path from $\tau_{1}$ to $\alpha\tau_{1}$. Since ${\omega}_{f}(\varphi)$ is invariant in the sense that

[TABLE]

we have

[TABLE]

so that

[TABLE]

as ${\omega}_{f}(\varphi)$ takes values in ${\mathcal{O}}({\mathrm{SL}}_{2})^{\mathrm{conj}}\boxtimes{\mathcal{O}}({\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})^{\mathrm{conj}})$. Thus the definition of $F_{f,\varphi}$ does not depend on the choice of the base point.

Conjugation invariance can be proved similarly. Alternatively, Proposition 17.3 implies that ${\omega}_{f}(\varphi)$ represents an element of

[TABLE]

and thus to a continuous ${\mathrm{SL}}_{2}\times{\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$ invariant function

[TABLE]

This gives a class function on ${\mathcal{G}}$ via the construction in Section 22.3.2. ∎

For future use, we compute the action of the Adams operator $\psi^{m}$ on $F_{f,\varphi}$.

Lemma 23.9**.**

With notation as above, we have, for all $m\geq 0$,

[TABLE]

Proof.

Since integrating elements of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ over ${\omega}_{f}(\varphi)$ is a 1-cocycle, we have

[TABLE]

Since ${\omega}_{f}(\varphi)$ takes values in $[{\mathcal{O}}({\mathrm{SL}}_{2})\otimes{\mathcal{O}}({\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}}))]^{\mathrm{conj}}$, we have

[TABLE]

∎

23.3.2. Odd weight

The construction given in the previous section does not work for forms of odd weight. This is because odd symmetric powers of ${\mathrm{SL}}_{2}$ do not occur in ${\mathcal{O}}({\mathrm{SL}}_{2})^{\mathrm{conj}}$, nor do representations of ${\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$ that occur in spaces of modular forms of odd weight as they take the value $-1$ on $-\operatorname{id}$. To construct class functions from modular forms of odd weight, we need to consider iterated integrals of length 2 or more and use Proposition 22.10.

Suppose that $f$ and $g$ are vector valued modular forms of level $\leq N$ of weights $m$ and $n$, both odd. Suppose that the corresponding characters of ${\mathrm{SL}}_{2}({\mathbb{Z}}/N)$ are $\chi$ and $\psi$. Denote the corresponding cohomology classes by

[TABLE]

These generate ${\mathrm{SL}}_{2}\times{\mathrm{SL}}_{2}({\mathbb{Z}}/N)$ submodules $U$ and $V$ of $H^{1}({\mathfrak{u}})$. Suppose $U\cap V=0$. Assume that $\chi\psi$ occurs in ${\mathcal{O}}({\mathrm{SL}}_{2}({\mathbb{Z}}/N))^{\mathrm{conj}}$. Since $m+n$ is even, there is a non-zero ${\mathrm{SL}}_{2}\times{\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$ invariant map

[TABLE]

We therefore have the twice iterated integral

[TABLE]

which takes values in ${\mathcal{O}}({\mathrm{SL}}_{2})^{\mathrm{conj}}\otimes{\mathcal{O}}({\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}}))^{\mathrm{conj}}$. Its cyclic average $\overline{I}_{\varphi}$ gives the element

[TABLE]

of ${\mathscr{C}\!\ell}_{2}({\mathcal{G}})$. Proposition 22.10 implies that it is non-zero.

Proposition 23.10**.**

With these assumptions, the function $F_{f\times g,\varphi}$ is a non-zero class function on ${\mathcal{G}}$.

23.4. Weight graded quotients

Denote the “motive” of a Hecke eigen cusp form $f$ of weight $m$ by $V_{f}$. By this, we mean the ${\mathbb{Q}}$ Hodge structure of weight $m+1$ associated to $f$, as well as its $\ell$-adic Galois realizations. To say that a tensor product of such motives appears in ${\mathcal{O}}({\mathcal{G}})$ we mean that the corresponding Hodge structure appears in ${\mathscr{C}\!\ell}({\mathcal{G}}^{B})$ and the $\ell$-adic Galois representation appears compatibly in ${\mathscr{C}\!\ell}({\mathcal{G}}^{{\mathrm{\acute{e}t}}_{\ell}}_{{\mathbb{Q}}_{\ell}})$. The following follows by taking products (or cyclic products) of the class functions $F_{f,\varphi}$ constructed in Section 23.3.1 and using the fact that the motive $V_{f}$ associated to a cusp form $f$ is a simple Hodge structure.

Theorem 23.11**.**

If $f_{1},\dots,f_{n}$ are linearly independent Hecke eigenforms of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ of level 1, then for all positive integers $r_{1},\dots,r_{n}$, a Tate twist of the motive

[TABLE]

appears in $\operatorname{Gr}^{W}_{\bullet}{\mathscr{C}\!\ell}({\mathcal{G}})$. In particular, the ring ${\mathscr{C}\!\ell}({\mathcal{G}}_{1})$ is not finitely generated.

One can obtain Tate twists of these Hodge structures by extending the list by Eisenstein series.

Similarly, subject to finding the appropriate characters in the conjugation representation ${\mathscr{C}\!\ell}({\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}}))$, one can find the motives of all tensor products of symmetric powers of cusp forms of all levels in ${\mathscr{C}\!\ell}({\mathcal{G}})$.

23.5. The Hecke action on ${\mathscr{C}\!\ell}_{1}({\mathcal{G}})$ and its periods

Suppose that $f$ is a modular form of even weight $m$ and level 1. Fix an ${\mathrm{SL}}_{2}$ invariant function $\varphi:S^{m-2}H\to{\mathcal{O}}({\mathrm{SL}}_{2})^{\mathrm{conj}}$ and a prime number $p$. Our goal in this section is to compute the action of $T_{p}$ on selected periods of the class function $F_{f,\varphi}$ that was constructed in Section 23.3.1.

As in Proposition 9.3, we write the Hecke correspondence $T_{p}$ as

[TABLE]

where $\pi$ corresponds to the inclusion ${\Gamma}_{0}(p)\hookrightarrow{\mathrm{SL}}_{2}({\mathbb{Z}})$ and $\pi_{\mathrm{op}}$ to the inclusion of ${\Gamma}_{0}(p)$ into ${\mathrm{SL}}_{2}({\mathbb{Z}})$ defined by $\gamma\mapsto g_{p}^{-1}\gamma^{-T}g_{p}$. For each $\alpha\in\boldsymbol{\lambda}({\mathrm{SL}}_{2}({\mathbb{Z}}))$ we have $\langle T_{p}F_{f,\varphi},\alpha\rangle=\langle\pi_{\mathrm{op}}^{\ast}F_{f,\varphi},\pi^{\ast}\alpha\rangle$.

To compute $\pi^{\ast}\alpha$ we lift $\alpha$ to ${\mathrm{SL}}_{2}({\mathbb{Z}})$. According to Proposition 10.2

[TABLE]

where $\gamma_{0},\dots,\gamma_{p}$ are the coset representatives given in Proposition 10.1 and $d_{j}(\alpha)$ is the length of the orbit of the orbit of ${\Gamma}_{0}(p)\gamma_{j}$ under the right action by $\langle\alpha\rangle$. So222222Note that these integrals are well defined by the discussion in the proof of Proposition 23.8.

[TABLE]

where we have abbreviated $d_{j}(\alpha)$ to $d_{j}$. Unfortunately, we cannot use conjugation invariance to replace

[TABLE]

by

[TABLE]

in this formula as $\gamma_{j}\notin{\Gamma}_{0}(p)$.

At this stage, it is useful to relate $T_{p}F_{f,\varphi}$ to the standard action of $T_{p}$ on modular forms. As defined in [34, VII§5.3], the image of $f$ under this action is

[TABLE]

The Hecke operator $T_{p}$ also acts on ${\omega}_{f}(\varphi)$ as an element of $H^{1}({\mathrm{SL}}_{2}({\mathbb{Z}}),{\mathcal{O}}({\mathrm{SL}}_{2}))$. These actions are related:

Lemma 23.12**.**

If $f$ is a modular form of even weight $m$ and level 1, then

[TABLE]

Note that, in general, this is not the same as the action of $T_{p}$ on $F_{f,\varphi}$, as we shall see in the example below.

Proof.

We use the notation of Sections 15.1.1 and 23.1. Additional details can be found in [13, §11]. The expression

[TABLE]

is the cohomology class of the holomorphic differential on ${\mathbb{C}}/({\mathbb{Z}}\oplus{\mathbb{Z}}\tau)$ that takes the value 1 on the loop corresponding to the interval $[0,1]$. It should be regarded as a section of the Hodge bundle over the upper half plane. The group ${\mathrm{GL}}_{2}({\mathbb{R}})^{+}$ acts on this line bundle and

[TABLE]

This implies that

[TABLE]

In the notation of Section 23.1

[TABLE]

Denote it by ${\omega}_{f}$. Let $\gamma_{0},\dots,\gamma_{p}$ be the coset representatives from Proposition 10.2. The pullback of ${\omega}_{f}$ to ${\mathcal{M}}_{1,1}^{\mathrm{an}}$ along $T_{p}$ is the sum of the pullbacks along

[TABLE]

The formulas (58) and (59) imply that

[TABLE]

We have to adjust this as $(\mathbf{a}^{\vee})^{m-2}\in S^{m}H^{\vee}$, while $\varphi$ takes values in the highest weight part of $S^{k}H\otimes S^{k}H^{\vee}\cong\operatorname{End}(S^{m-2}H)^{\vee}$, where $k=(m-2)/2$. This has highest weight vector $(\mathbf{a}^{\vee})^{k}\mathbf{b}^{k}$. Since $g_{p}^{\ast}\mathbf{a}^{\vee}=\mathbf{a}^{\vee}$ and $g_{p}^{\ast}\mathbf{b}=p\mathbf{b}$, we see that

[TABLE]

∎

23.6. The Hecke action on periods: an example

The dual Hecke operators act on motivic periods of ${\mathscr{C}\!\ell}({\mathcal{G}})$. The action is defined by

[TABLE]

where $\alpha\in{\mathrm{SL}}_{2}({\mathbb{Z}})$ and $F\in{\mathscr{C}\!\ell}({\mathcal{G}})$. To illustrate how this works, we compute one example. Suppose that $\alpha$ is an element of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ whose image in ${\mathrm{PSL}}_{2}({{\mathbb{F}}_{p}})$ has order $p+1$. (Such elements exist by the discussion in Section 10.2.2.)

Proposition 23.13**.**

If $\alpha\in{\mathrm{SL}}_{2}({\mathbb{Z}})$ is as above and if $f$ is a modular form of level 1 and weight $m$, then

[TABLE]

Proof.

We will abuse notation and write, for example,

[TABLE]

in place of $[{\mathscr{C}\!\ell}({\mathcal{G}});\alpha,F_{f,\varphi}]$. The proof holds for motivic periods as we appeal only to formal properties of integrals and not to their specific values.

The assumption implies that $\alpha^{p+1}\in{\Gamma}_{0}(p)$. This implies that $\pi^{\ast}\alpha^{p+1}=(p+1)\alpha^{p+1}$. This, Lemma 23.12 and the fact that integrating $T_{p}{\omega}_{f,\varphi}$ defines a 1-cocycle on ${\mathrm{SL}}_{2}({\mathbb{Z}})$ implies that

[TABLE]

Plugging into the formula (56) and applying Lemma 23.9, we get

[TABLE]

∎

24. Filtrations

The ring ${\mathcal{O}}({\mathcal{G}})$ and its subring ${\mathscr{C}\!\ell}({\mathcal{G}})$ have several natural filtrations which are defined below, where we also describe their behaviour under Hecke correspondences.

24.1. The (relative) coradical filtration

This is also called the length filtration as it corresponds to the filtration of iterated integrals by length. It is an increasing filtration

[TABLE]

It is defined by setting $C_{0}{\mathcal{O}}({\mathcal{G}})={\mathcal{O}}(R)$, where $R={\mathrm{SL}}_{2}\times{\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}})$. When $n>0$, $C_{r}{\mathcal{O}}({\mathcal{G}})$ is defined to be the kernel of the $r$th “reduced diagonal”

[TABLE]

It restricts to the filtration

[TABLE]

This agrees with the definition (54).

This filtration is preserved by the Hecke operators:

[TABLE]

The shuffle product formula (53) implies that multiplication induces a map

[TABLE]

Note, however, that its graded quotients are infinite dimensional.

24.2. Hodge and weight filtrations

These are self explanatory. Since the Hecke operators act as morphism of MHS, they preserve both. The weight filtration satisfies

[TABLE]

This is finer than the length filtration:

[TABLE]

This inclusion is strict, except when $r=0$ when we have

[TABLE]

The Hodge filtration

[TABLE]

extends infinitely in both directions. To see why, consider the class functions $F_{f,\varphi}$ constructed from cusp forms $f$ of weight $2n$ and level 1 in Section 23.3.1. The Hodge structures they generate are all isomorphic to that on the cuspidal cohomology group

[TABLE]

which has weight 1 as the coefficient module $S^{2n-2}H(n-1)$ has weight 0. Zucker’s work [40] implies that it has Hodge numbers $(1-n,n)$ and $(n,1-n)$. (See also [13, Thm. 11.4].) This implies that the dimension of $F^{1-n}{\mathscr{C}\!\ell}_{1}({\mathcal{G}})$ becomes infinite as $n\to\infty$.

24.3. Filtration by level

Recall that ${\mathcal{G}}_{N}$ is the relative completion of ${\mathrm{SL}}_{2}({\mathbb{Z}})$ with respect to ${\mathrm{SL}}_{2}\times{\mathrm{SL}}_{2}({\mathbb{Z}}/N)$ and that ${\mathcal{G}}$ is the inverse limit of the ${\mathcal{G}}_{N}$. This implies that

[TABLE]

The level filtration (which is actually a net) of ${\mathscr{C}\!\ell}({\mathcal{G}})$ is the net defined by $L_{N}{\mathscr{C}\!\ell}({\mathcal{G}})={\mathscr{C}\!\ell}({\mathcal{G}}_{N})$. Each $L_{N}{\mathscr{C}\!\ell}({\mathcal{G}})$ is a subring of ${\mathscr{C}\!\ell}({\mathcal{G}})$. Its elements are closed iterated integrals of (not necessarily holomorphic) modular forms of level dividing $N$. They are ordered by divisibility:

[TABLE]

when $N|M$. This “filtration” is not preserved by the Hecke operators. Rather, we have

[TABLE]

24.4. The modular filtration

Roughly this is the filtration by modular weight. To make this precise, we use the construction and notation from Section 16.3. We will construct it as a filtration of ${\mathscr{C}\!\ell}({\mathcal{G}}^{B})$. We take ${\mathsf{C}}$ and ${\mathsf{S}}$ as in Section 17.1. For each positive integer $m$, we consider all simple $V_{\alpha}$ in ${\mathsf{Rep}}({\mathrm{SL}}_{2}\times{\mathrm{SL}}_{2}({\widehat{{\mathbb{Z}}}}))$ of the form $S^{n}{\mathbb{H}}\boxtimes V$, where $n\leq m$ and $V$ is arbitrary. We take

[TABLE]

In other words, we allow arbitrary extensions that are sums of extensions of the form

[TABLE]

where $n\leq m$ and $A$ is simple. There is a natural homomorphism

[TABLE]

Set

[TABLE]

This is an increasing filtration satisfying

[TABLE]

There are similar compatible constructions for ${\mathcal{O}}({\mathcal{G}}^{\mathrm{DR}})$ and ${\mathcal{O}}({\mathcal{G}}^{{\mathrm{\acute{e}t}}_{\ell}}_{{\mathbb{Q}}_{\ell}})$, so we will consider it to be a filtration of ${\mathcal{O}}({\mathcal{G}})$. In the de Rham realization, $M_{m}{\mathcal{O}}({\mathcal{G}})$ consists of all closed iterated integrals of (not necessarily holomorphic) modular forms of all levels of weight $\leq m+2$.

This filtration restricts to the modular filtration

[TABLE]

of ${\mathscr{C}\!\ell}({\mathcal{G}})$. Each $M_{m}{\mathscr{C}\!\ell}({\mathcal{G}})$ is a subring and is preserved by the Hecke operators. Its graded quotients are not finite dimensional.

24.5. Finiteness properties

The coradical, weight, level and modular filtrations are all filtrations by mixed Hodge structures and (after tensoring with ${{\mathbb{Q}}_{\ell}}$), Galois representations. Even though the individual terms of these filtrations are infinite dimensional, certain of their intersections are finite dimensional.

Proposition 24.1**.**

For each $N,m,r\geq 0$, the subspaces

[TABLE]

of ${\mathscr{C}\!\ell}({\mathcal{G}})$ are finite dimensional. Each is a mixed Hodge structure and, after tensoring with ${{\mathbb{Q}}_{\ell}}$, a $G_{\mathbb{Q}}$-module.

Example 24.2**.**

If $f$ is a $V_{\chi}$-valued modular form of even weight $m$ and level $N$. If $\varphi:S^{m-2}H\otimes V_{\chi}\to{\mathcal{O}}({\mathrm{SL}}_{2})^{\mathrm{conj}}$ is ${\mathrm{SL}}_{2}({\mathbb{Z}})$-invariant, then the class function $F_{f,\varphi}$ defined in Section 23.3.1 satisfies:

[TABLE]

This will lie in $F^{m/2}\cap W_{1}{\mathscr{C}\!\ell}({\mathcal{G}})$ if $f$ is a cusp form and in $F^{m/2}W_{m/2}{\mathscr{C}\!\ell}({\mathcal{G}})$ if $f$ is an Eisenstein series.

More generally, if $f_{1},\dots,f_{r}$ are vector valued modular forms of even weights $m_{1},\dots,m_{r}$, all of level dividing $N$, and if $\varphi_{j}:S^{m_{j}-2}H\otimes V_{\chi_{j}}\to{\mathcal{O}}({\mathrm{SL}}_{2})^{\mathrm{conj}}$ is ${\mathrm{SL}}_{2}$ for each $j$, then

[TABLE]

where $m=\max\{m_{1},\dots,m_{r}\}$.

Appendix A The Conjugation Representation of ${\mathrm{SL}}_{2}({\mathbb{Z}}/p^{n})$

by Pham Huu Tiep222Rutgers University, Piscataway, NJ 08854; tiep@math.rutgers.edu 333The author gratefully acknowledges the support of the NSF (grant DMS-2200850), the Simons Foundation, and the Joshua Barlaz Chair in Mathematics. Part of this work was done while the author was visiting Princeton University and MIT. It is a pleasure to thank both institutions for their generous hospitality and stimulating environment. Finally, the author thanks Gabriel Navarro and Eamonn O’Brien for helpful conversations on the problem and for several computer calculations in the cases $p=2,3$.

The body of the appendix is [arXiv:2303.02807]