Derived Hecke algebra for weight one forms

TL;DR

This paper investigates the action of the derived Hecke algebra on weight one modular forms, proposing a conjecture linking it to Stark units and verifying it through extensive numerical computations.

Contribution

It introduces a conjecture connecting derived Hecke algebra actions on weight one forms to Stark units and provides numerical evidence supporting this conjecture.

Findings

Numerical verification of the conjecture for levels 23 and 31.

Confirmation of the conjecture for many derived Hecke operators at primes less than 200.

Utilization of Merel's pairing evaluation between Shimura and cuspidal subgroups.

Abstract

We study the action of the derived Hecke algebra on the space of weight one forms. By analogy with the topological case, we formulate a conjecture relating this to a certain Stark unit. We verify the truth of the conjecture numerically, for the weight one forms of level $23$ and $31$, and many derived Hecke operators at primes less than $200$. Our computation depends in an essential way on Merel's evaluation of the pairing between the Shimura and cuspidal subgroups of $J_0(q)$.

Tables

Table 1. Table 5.1. Data for the weight one form associated to the cubic field with discriminant − 23 23 -23 ; in all cases the ratio is − 1 / 72 1 72 -1/72 modulo p 𝑝 p . All allowable p ≤ 100 𝑝 100 p\leq 100 and q ≤ 150 𝑞 150 q\leq 150 shown.

$p$	$q$	$\log \left(\overline{u}\right)/\log \left({\varpi }_{\mathrm{Merel}}\right)\in \mathbb{Z}/p$	$\eta \in \mathbb{Z}/p$	ratio
$5$	$11$	3 (5)	4(5)	2(5)
$5$	$61$	1(5)	3(5	2(5))
$5$	$81$	$\infty$	-	-
$7$	$43$	3(7)	1(7)	3(7)
$7$	$113$	1(7)	5(7)	3(7)
$11$	$67$	6(11)	8 (11)	-2(11)
$11$	$89$	1(11)	5(11)	-2(11)
$13$	$53$	6 (13)	10(13)	-2(13)
$13$	$79$	5 (13)	4 (13)	-2 (13)
$17$	$103$	$\infty$	-	-
$17$	$137$	5(17)	14 (17)	4 (17)
$37$	$149$	20(37)	3 (37)	19 (37)
$41$	$83$	12(41)	38(41)	-4(41)
$53$	$107$	30(53)	13(53)	39 (53)

Table 2. Table 5.2. Data for the weight one form associated to the cubic field with discriminant − 31 31 -31 ; in all cases the ratio is 72 72 72 modulo p 𝑝 p . All allowable p ≤ 100 𝑝 100 p\leq 100 and q ≤ 150 𝑞 150 q\leq 150 shown. ? means that we did not compute because it took too long; - means undefined.

$p$	$q$	$\log \left(\overline{u}/{\varpi }_{\mathrm{Merel}}\right)\in \mathbb{Z}/p$	$\eta \in \mathbb{Z}/p$	ratio
$5$	$11$	2 (5)	4(5)	2(5)
$5$	$61$	2(5)	4(5	2(5))
$7$	$29$	1(7)	2(7)	2(7))
$7$	$43$	4(7)	1(7)	2(7))
$7$	$127$	$\infty$	-	-
$11$	$23$	3 (11)	7 (11)	6(11)
$11$	$89$	7(11)	9(11)	6(11)
$13$	$53$	2(13)	1(13)	7(13)
$13$	$79$	3(13)	8 (13)	7 (13)
$17$	$137$	4(17)	16 (17)	4(17)
$23$	$139$	4(23)	12 (23)	3 (23)
$41$	$83$	28(41)	7(41)	31(41)

Equations

$$X_{01}(qN),X_1(qN)$$

$$\omega \otimes \omega (-D)\simeq {\Omega }_{X\to \mathbb{Z}[\frac{1}{N}]}^1,$$

$$\Gamma (\mathrm{Spec}(R),R^i{\pi }_{\ast }\omega ).$$

$${\pi }_{\ast }\omega \otimes {\mathrm{R}}^1{\pi }_{\ast }\omega (-D)⟶{\mathrm{R}}^1{\pi }_{\ast }{\Omega }_{X_R/R}^1$$

$$H^0(X_R,\omega )\times H^1(X_R,\omega (-D))\to R,$$

$$g\in H^0(X_{\mathcal{O}[\frac{1}{N}]},\omega (-D)).$$

$$\rho :\text{mbox}Gal(L/\mathbf{Q})⟶{\mathrm{GL}}_2(\mathcal{O})$$

$${\mathrm{Ad}}^{\ast }\rho =\mathrm{Hom}({\mathrm{Ad}}^0\rho ,\mathcal{O}),$$

$$g^{\prime }:=\sum \overline{a_n}{q}^n\in H^0(X_{\mathcal{O}[\frac{1}{N}]},\omega (-D))$$

$$H^0(X_{{\mathbb{Z}}_p},\omega )\to H^0(X_{{\mathbb{F}}_p},\omega )$$

$$k⟨1⟩=k\otimes (\mathbb{Z}/q{)}_p^{\ast },k⟨-1⟩=\mathrm{Hom}((\mathbb{Z}/q{)}_p^{\ast },k).$$

$$M⟨n⟩=M{\otimes }_{\mathbb{Z}}k⟨n⟩.$$

$${\mathrm{U}}_g:={\left({\mathrm{U}}_L{\otimes }_{\mathbb{Z}}{\mathrm{Ad}}^{\ast }\rho \right)}^{G_{L/\mathbf{Q}}}={\left({\mathrm{U}}_L{\otimes }_{\mathbb{Z}}{\mathrm{Hom}}_{\mathcal{O}}({\mathrm{Ad}}^0\rho ,\mathcal{O})\right)}^{G_{L/\mathbf{Q}}}$$

$$\stackrel{\sim }{\to }{\mathrm{Hom}}_{\mathcal{O}[G_{L/\mathbf{Q}}]}({\mathrm{Ad}}^0\rho ,{\mathrm{U}}_L\otimes \mathcal{O})$$

$${\mathrm{U}}_g^{\vee }:={\mathrm{Hom}}_{\mathcal{O}}({\mathrm{U}}_g,\mathcal{O}),$$

$$H_{\mathrm{mot}}^1(\mathbf{Q},{\mathrm{M}}_g(1){)}_{\mathrm{int}}$$

$$={\left({\mathrm{U}}_L\otimes {\mathrm{Ad}}^{\ast }\rho \otimes \mathbf{Q}\right)}^{G_{L/\mathbf{Q}}}={\mathrm{U}}_g\otimes \mathbf{Q}.$$

$${\theta }_q:{\mathrm{U}}_g⟶k⟨1⟩.$$

$${\mathsf{e}}_{\mathfrak{q}}=2\rho ({\mathrm{Frob}}_{\mathfrak{q}})-\mathrm{trace}\rho ({\mathrm{Frob}}_{\mathfrak{q}})\in {\mathrm{Ad}}^0\rho$$

$${\mathsf{e}}_{\mathfrak{q}}:{\mathrm{Ad}}^{\ast }\rho ⟶\mathcal{O}\to k,$$

$${\mathsf{e}}_{g\mathfrak{q}}=\mathrm{Ad}(\rho (g))e_{\mathfrak{q}}.$$

$$({\mathrm{U}}_g)\to (\prod _{\mathfrak{q}|q}{\mathbf{F}}_{\mathfrak{q}}^{\ast }\otimes {\mathrm{Ad}}^{\ast }\rho {)}^{G_{L/\mathbf{Q}}}\stackrel{\sim }{\to }{\left({\mathbf{F}}_{{\mathfrak{q}}_0}^{\ast }\otimes {\mathrm{Ad}}^{\ast }\rho \right)}^{D_{{\mathfrak{q}}_0}}\stackrel{{\mathsf{e}}_{{\mathfrak{q}}_0}}{⟶}({\mathbf{F}}_{{\mathfrak{q}}_0}^{\ast }\otimes k{)}^{D_{{\mathfrak{q}}_0}}\to k⟨1⟩.$$

$$\theta \text{mbox}or{\theta }_q:{\mathrm{U}}_g\to k⟨1⟩.$$

$$T_{q,z}:H^0(X_k,\omega )\to H^1(X_k,\omega ).$$

$$\mathfrak{S}\in H_{\mathrm{et}}^1(X_0(q{)}_k,k⟨1⟩)$$

$$H_{\mathrm{et}}^1(X_0(q{)}_k,{\mathbb{G}}_a⟨1⟩)\simeq H_{\mathrm{Zar}}^1(X_0(q{)}_k,\mathcal{O}⟨1⟩),$$

$$\mathfrak{S}\in H_{\mathrm{Zar}}^1(X_0(q{)}_k,\mathcal{O}⟨1⟩),$$

$$0\to {\mathbb{F}}_{\mathfrak{p}}\to {\mathbb{G}}_a\to {\mathbb{G}}_a\to 0$$

$$H^0(X_{01}(qN{)}_k,\omega )\stackrel{\cup {\mathfrak{S}}_X}{⟶}{H}^1(X_{01}(qN{)}_k,\omega ⟨1⟩).$$

$$H^0(X_k,\omega )\stackrel{{\pi }_1^{\ast }}{⟶}{H}^0(X_{01}(qN{)}_k,\omega )\stackrel{\cup {\mathfrak{S}}_X}{⟶}{H}^1(X_{01}(qN{)}_k,\omega )⟨1⟩\stackrel{{\pi }_{2\ast }}{⟶}{H}^1(X_k,\omega )⟨1⟩,$$

$$H^0(X_k,\omega )\to H^1(X_k,\omega )⟨1⟩,$$

Full text

We verify the truth of the conjecture numerically, for the weight one forms of level $23$ and $31$, and many derived Hecke operators at primes less than $200$. Our computation depends in an essential way on Merel’s evaluation of the pairing between the Shimura and cuspidal subgroups of $J_{0}(q)$.

M.H.’s research received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 290766 (AAMOT). M.H. was partially supported by NSF Grant DMS-1404769

A. V. research was partially supported by the NSF and by the Packard foundation

1. 1 Introduction
2. 2 Notation
3. 3 Derived Hecke operators and the main conjecture
4. 4 Relationship to Galois deformation theory
5. 5 Explication
6. 6 Flat cohomology and Merel’s computation
7. 7 Comparison with the theory of [17]
8. 8 Magma Code

1. Introduction

Let $G$ be an algebraic group over $\mathbf{Q}$. In [17] the second-named author studied the action of a derived version of the Hecke algebra on the singular cohomology of the locally symmetric space attached to $G$. One expects that this action transports Hecke eigenclasses between cohomological degrees and moreover (see again [17]) is related to a “hidden” action of a motivic cohomology group.

It is also possible for a Hecke eigensystem on coherent cohomology to occur in multiple degrees. The simplest situation is weight one forms for the modular curve. We study this case, explicating the action of the derived Hecke algebra and formulating a conjectural relationship with motivic cohomology.

The motivic cohomology is particularly concrete: a weight one eigenform $f$ is attached to a $2$-dimensional Artin representation $\rho_{f}$ of $\mathrm{Gal}(\bar{{\mathbf{Q}}}/{\mathbf{Q}})$ and the motivic cohomology group in question is generated by a certain unit in the splitting field of the adjoint of $\rho_{f}$. This makes the conjecture particularly amenable to numerical testing. We carry this out for the forms of conductor $23$ and $31$, and the first few derived Hecke operators; the numerics all support the conjecture.

We note that this story is also related to the Taylor–Wiles method for coherent cohomology and its obstructed version due to Calegari–Geraghty (see [11], and the detailed discussion of weight one forms in [4]). This is used implicitly in the discussion in §4 of the current paper.

Now we outline the contents of the paper. After giving notation in §2, we describe the derived Hecke algebra and the main conjecture in §3. Although the discussion to this point is self-contained, we postpone the comparison with the results of [17] until the final section. We translate the Conjecture to an explicitly computable form in §5; see in particular Proposition 5.4. Finally, in §5.6 we make the conjecture even more explicit in the case of a form associated to a cubic field $K$, and check it numerically in the case of $K$ with discriminant $-23$ and $-31$.

Our numerical computation depends, in a crucial way, on the evaluation of a certain pairing in coherent cohomology on the mod $p$ fiber of a modular curve; this evaluation is postponed to §6, where we do it by relating it to Merel’s remarkable computation [14]. It is worth emphasizing how important Merel’s computation is for us: it seemed almost impossible to carry through our computation until we learned about Merel’s results. Indeed, the role that Merel’s computation plays here suggests that it would be worthwhile to understand how it might generalize to Hilbert modular surfaces.

The expression of the derived Hecke algebra action as a cohomological cup product (see (5.2)) strongly suggests a surprising relation with special values of the $p$-adic triple product $L$-function. We hope to explore this relation in forthcoming work with Henri Darmon.

1.1. Acknowledgements

We would like to thank Brian Conrad for helpful explanations about flat cohomology, Frank Calegari for pointing out Merel’s paper and for interesting discussions related to that paper, and Shekhar Khare for several helpful and inspiring discussions (in particular related to §4). We would also like to thank Nick Rozenblyum for help with Remark 7.1, and Henri Darmon for reading and providing comments on the paper. Finally, we thank Loic Merel and Emmanuel Lecouturier for exchanges concerning their work.

2. Notation

2.1. Modular curves

Fix an integer $N$. We will generically use the letter $R$ for a ${\mathbb{Z}}[1/N]$-algebra. For $R\rightarrow R^{\prime}$ a morphism of ${\mathbb{Z}}[1/N]$-algebras and $Y$ an $R$-scheme, we denote by $Y_{R^{\prime}}$ the base extension of $Y$ to $R^{\prime}$.

Let $X=X_{1}(N)$ be the compactification ([5]) of the modular curve parametrizing elliptic curves with with an $N$-torsion point. We may construct $X$ as a smooth proper relative curve over $\mathrm{Spec}\ {\mathbb{Z}}[\frac{1}{N}]$, and the cusps give rise to a relative divisor $D\subset X$. In particular, we obtain an $R$-scheme $X_{R}$ for each ${\mathbb{Z}}[1/N]$-algebra $R$. We denote the universal generalized elliptic curve over $X$ by $\mathcal{A}\longrightarrow X$.

We will denote by

[TABLE]

the modular curves that correspond to adding $X_{0}(q)$ and $X_{1}(q)$ level structure to $X$.

Let $\omega$ be the line bundle over $X$ whose sections are given by weight $1$ forms. More precisely, when $X$ is the modular curve, let $\Omega_{\mathcal{A}/X}$ denote the relative cotangent bundle of $\mathcal{A}/X$, pulled back to $X$ via the identity section. We define $\omega$ as the pullback of $\Omega_{\mathcal{A}/X}$ via the zero section $X\rightarrow\mathcal{A}$. Therefore the sections of $\omega$ correspond to weight $1$ forms, whereas sections of $\omega(-D)$ corresponds to weight $1$ cusp forms. Moreover, there is an isomorphism of line bundles ([12], (1.5), A (1.3.17)):

[TABLE]

which says that “the product of a weight $1$ form and a cuspidal weight $1$ form is a cusp form of weight $2$.”

Let $\pi:X_{R}\rightarrow\mathrm{Spec}R$ be the structure morphism and consider the space of weight $1$ forms over $R$, in cohomological degree $i$ – formally:

[TABLE]

We will denote this space, for short, by $H^{i}(X_{R},\omega)$, and use similar notation for $\omega(-D)$. Therefore $H^{0}(X_{R},\omega)$ (respectively $H^{0}(X_{R},\omega(-D))$) is the usual space of weight $1$ modular forms (respectively cusp forms) with coefficients in $R$.

2.2. The residue pairing

The pairing (2.1) induces

[TABLE]

Since $X_{R}$ is a projective smooth curve over $R$, there is a canonical identification of the last factor with the trivial line bundle; thus we get a pairing

[TABLE]

which we denote as $[-,-]_{\mathrm{res},R}$. (Here $\mathrm{res}$ stands for “residue.”) This pairing is compatible with change of ring, and if $R$ is a field it is a perfect pairing.

2.3. The fixed weight one form $g$

We want to localize our story throughout at a single weight one form $g$. Therefore, fix $g=\sum a_{n}q^{n}$ a Hecke newform of level $N$ and Nebentypus $\chi$, normalized so that $a_{1}=1$. Here $\chi$ is a Dirichlet character of level $N$.

We regard the $a_{n}$ as lying in some number field $E$, and indeed in the integer ring $\mathcal{O}$ of $E$. Thus $g$ extends to a section:

[TABLE]

We shall denote by $H^{*}(X_{\mathcal{O}[\frac{1}{N}],}\omega)[g]$, that part of the cohomology which transforms under the Hecke operators in the same way as $g$, i.e. the common kernel of all $(T_{\ell}-a_{\ell})$ over all primes $\ell$ not dividing $N$.

Extending $E$ if necessary, we may suppose that one can attach to $g$ a Galois representation, unramified away from $N$ [6]:

[TABLE]

where $L$ is a Galois extension of ${\mathbf{Q}}$. Here the Frobenius trace of $\rho$ at ${\ell}$ coincides with $a_{\ell}$, and the image of complex conjugation $c$ under $\rho$ is conjugate to \left(\begin{array}[]{cc}1&0\\ 0&-1\end{array}\right). (In the body of the text, we will primarily use the field cut out by the adjoint of $\rho$, and we could replace $L$ by this smaller field.)

We emphasize the distinction between $E$ and $L$:

$E$ is a coefficient field for the weight one form $g$, and $L$ is the splitting field for the Galois representation of $g$.

In our numerical examples, we will have $E={\mathbf{Q}}$ and $L$ the Galois closure of a cubic field.

It will be convenient to denote by $\mathrm{Ad}^{0}\rho$ the trace-free adjoint of $\rho$, i.e. the associated action of $\mbox{Gal}(L/{\mathbf{Q}})$ on $2\times 2$ matrices of trace zero and entries in $\mathcal{O}$. We denote by $\mathrm{Ad}^{*}\rho$ the $\mathcal{O}$-linear dual to $\mathrm{Ad}^{0}\rho$, i.e.

[TABLE]

a locally free $\mathcal{O}$-module endowed with an action of $\mbox{Gal}(L/{\mathbf{Q}})$. 111We apologize for the perhaps pedantic distinction between $\mathrm{Ad}^{*}\rho$ and $\mathrm{Ad}^{0}\rho$. Since we will shortly be localizing at a prime larger than $2$, one could identify them by means of the pairing $\mathrm{trace}(AB)$. However, when working in a general setting, one really needs to use $\mathrm{Ad}^{*}$, and following this convention makes it easier to compare with [17].

For later use, it is convenient to choose a dual form $g^{\prime}$ which will be paired with $g$ eventually. In order that a Hecke equivariant pairing between $g$ and $g^{\prime}$ be nonzero we should take $g^{\prime}$ to be the form corresponding to the contragredient automorphic representation, i.e.

[TABLE]

where $\alpha\mapsto\overline{\alpha}$ is the complex conjugation in the CM field $E$. (In our examples, $E={\mathbf{Q}}$, and therefore $g^{\prime}=g$).

2.4. The prime $\mathfrak{p}$

Let $\mathfrak{p}$ be a prime of $E$, above the rational prime $p$. We make the following assumptions:

• •

All weight one forms in characteristic $p$ lift to characteristic zero, i.e. the natural map

[TABLE]

is surjective.

• •

$p\geqslant 5$.

• •

$p$ is unramified inside $E$.

• •

There are no $p$th-roots of unity inside $L$.

• •

$p$ does not divide the order $[L:{\mathbf{Q}}]$.

The representation $\rho$ may be reduced modulo $\mathfrak{p}$, obtaining $\overline{\rho}:G_{{\mathbf{Q}}}\longrightarrow\mathrm{GL}_{2}(\mathbf{F}_{\mathfrak{p}})$, where $\mathbf{F}_{\mathfrak{p}}=\mathcal{O}/\mathfrak{p}$ is the residue field at $\mathfrak{p}$. As before we may define the trace-free adjoint $\mathrm{Ad}^{0}\overline{\rho}$ and its dual $\mathrm{Ad}^{*}\overline{\rho}$.

2.5. Taylor Wiles primes

A Taylor–Wiles prime $q$ of level $n\geq 1$ for $g$, or more precisely relative to the pair $(g,\mathfrak{p})$, will be, by definition:

• •

a rational prime $q\equiv 1$ modulo $p^{n}$, relatively prime to $N$;

• •

the data of $(\alpha,\beta)\in\mathbf{F}_{\mathfrak{p}}$ with $\alpha\neq\beta$, such that $\overline{\rho}(\mathrm{Frob}_{q})$ is conjugate to \left(\begin{array}[]{cc}\alpha&0\\ 0&\beta\end{array}\right).

Thus whenever we refer to Taylor–Wiles primes, we always regard the ordered pair $(\alpha,\beta)$ as part of the data: this amounts to an ordering of the eigenvalues of Frobenius.

If $\mathfrak{p},n,q$ have been fixed, where $q$ is a Taylor–Wiles prime of level $n$ and $\mathfrak{p}$ is a prime of $\mathcal{O}$ as above, it is convenient to use the following shorthand notation:

• •

Write $k=\mathcal{O}/\mathfrak{p}^{n}$.

• •

Write $({\mathbb{Z}}/q)^{*}_{p}$ for the quotient of $({\mathbb{Z}}/q)^{*}$ of size $p^{n}$, so that there is a noncanonical isomorphism $({\mathbb{Z}}/q)_{p}^{*}\cong{\mathbb{Z}}/p^{n}$.

• •

Write

[TABLE]

These are isomorphic as abelian groups to $k$, but not canonically so.

• •

Similarly for a ${\mathbb{Z}}$-module $M$ we shall write

[TABLE]

Thus, for example, $\mathbb{F}_{p}\langle 1\rangle$ is canonically identified with the quotient of $({\mathbb{Z}}/q)^{*}$ of size $p$.

These notations clearly depends on $\mathfrak{p},n,q$; however we do not explicitly indicate this dependence.

2.6. The Stark unit group

Let $\mathrm{U}_{L}$ be the group of units of the integer ring of $L$.

The key group of “Stark units” that we shall consider is the following $\mathcal{O}$-module:

[TABLE]

where $\mathrm{Ad}^{0}$ is the conjugation action of the Galois group on trace-free matrices in $M_{2}(\mathcal{O})$.

For instance, in the examples of modular forms attached to cubic fields, the group $\mathrm{U}_{L}$ will amount to (essentially) the unit group of that cubic field.

Lemma 2.7**.**

$\mathrm{U}_{g}\otimes_{{\mathbb{Z}}}{\mathbf{Q}}$* is an $E$-vector space of rank $1$ and $\mathrm{U}_{g}\otimes_{{\mathbb{Z}}}{\mathbb{Z}}_{p}$ is a free $\mathcal{O}\otimes_{{\mathbb{Z}}}{\mathbb{Z}}_{p}$-module of rank one.*

Proof.

Fix any embedding $\iota:E\hookrightarrow{\mathbb{C}}$; the $E$-dimension of $\mathrm{U}_{g}\otimes_{{\mathbb{Z}}}{\mathbf{Q}}$ then coincides with the complex dimension of Galois invariants on $\mathrm{U}_{L}\otimes_{{\mathbf{Q}}}(\mathrm{Ad}^{*}\rho)^{\iota}$.

In general, for any number field $L$, Galois over ${\mathbf{Q}}$, and any $\eta:G_{L/{\mathbf{Q}}}\rightarrow\mathrm{GL}_{m}({\mathbb{C}})$ without a trivial subrepresentation, the dimension of $(\mathrm{U}_{L}\otimes\eta)^{G_{L/{\mathbf{Q}}}}$ is known to be the dimension of invariants for complex conjugation in $\eta$. (In fact this is a straightforward consequence of the unit theorem.) It follows, therefore, that the $E$-dimension of $\mathrm{U}_{g}\otimes_{{\mathbb{Z}}}{\mathbf{Q}}$ is $1$ as claimed.

The final claim now follows since our assumption on $p$ means that $\mathrm{U}_{L}$ is free of $p$-torsion. ∎

Fix a nonzero element $u\in\mathrm{U}_{g}$ in such a way that $[\mathrm{U}_{g}:\mathcal{O}.u]$ is relatively prime to $p$ (e.g. a generator, if $\mathrm{U}_{g}$ is a free $\mathcal{O}$-module). Later we will also work with the $\mathcal{O}$-dual

[TABLE]

and denote by $u^{*}\in\mathrm{U}_{g}^{\vee}$ a nonzero element, chosen so that $\langle u,u^{*}\rangle\in\mathcal{O}$ is not divisible by any prime above $p$.

2.8. Aside: comparison of $\mathrm{U}_{g}$ with the motivic cohomology group from [17]

This section is not used in the remainder of this paper. It serves to connect the previous construction with the discussion in [17]:

We may construct a $3$ dimensional Chow motive $\mathrm{Ad}^{0}\mathrm{M}_{g}$, with coefficients in $E$, attached to the trace-free adjoint $\mathrm{Ad}^{*}\rho_{g}$ – in other words, the étale cohomology of $\mathrm{Ad}^{0}\mathrm{M}_{g}$ is concentrated in degree zero and identified, as a Galois representation, with $\mathrm{Ad}^{*}\rho_{g}$.

Now consider the motivic cohomology $H^{1}_{\mathrm{mot}}({\mathbf{Q}},\mathrm{M}_{g}(1))$, or more precisely the subspace of integral classes $(-)_{\mathrm{int}}$ described by Scholl [16].

The general conjectures of [17, 15], transposed to the current (coherent) situation, predict that the dual of $H^{1}_{\mathrm{mot}}({\mathbf{Q}},\mathrm{M}_{g}(1))_{\mathrm{int}}$ should act on $H^{*}(X_{E},\omega)[g]$.

There is a natural map

[TABLE]

Although we did not check it, this map is presumably an isomorphism. In the present paper we will never directly refer to the motivic cohomology group. Rather we work with the right-hand side (or its integral form $\mathrm{U}_{g}$) as a concrete substitute for the motivic cohomology group.

2.9. Reduction of a Stark unit at a Taylor–Wiles prime $q$

Let $q$ be a Taylor–Wiles prime (as in §2.5); we shall define a canonical reduction map

[TABLE]

For example, in the examples of modular forms attached to cubic fields, this will amount to the reduction of a unit in the cubic field at a degree one prime above $q$. Although explicit, the general definition is unfortunately opaque (the motivation comes from computations in [17]).

For any prime $\mathfrak{q}$ of $L$ above $q$, with associated Frobenius element $\mathrm{Frob}_{\mathfrak{q}}$, let $D_{\mathfrak{q}}=\langle\mathrm{Frob}_{\mathfrak{q}}\rangle\subset\mbox{Gal}(L/{\mathbf{Q}})$ be the associated decomposition group, the stabilizer of $\mathfrak{q}$. We may construct a $D_{\mathfrak{q}}$-invariant element

[TABLE]

where we regard the middle quantity as a $2\times 2$ matrix with coefficients in $\mathcal{O}$ and trace zero, thus belonging to $\mathrm{Ad}^{0}\rho.$ Pairing with $\mathsf{e}_{\mathfrak{q}}$ and reduction mod $\mathfrak{p}^{n}$ induces

[TABLE]

equivariantly for the Galois group of ${\mathbf{Q}}_{q}$. Also, for $g\in\mbox{Gal}(L/{\mathbf{Q}})$ we have

[TABLE]

Write $L_{q}=(L\otimes{\mathbf{Q}}_{q})$ and let $\mathcal{O}_{L_{q}}$ be the integer subring thereof. Thus $\mathcal{O}_{L_{q}}/q\simeq\prod_{\mathfrak{q}|q}\mathbf{F}_{\mathfrak{q}}$. Fix a prime $\mathfrak{q}_{0}$ of $L$ above $q$. The inclusion of units for the number field $L$ into local units $\mathcal{O}^{*}_{L_{q}}$ induces

[TABLE]

where the second map is projection onto the factor corresponding to $\mathfrak{q}_{0}$.

The resulting composite is independent of the choice of $\mathfrak{q}_{0}$, because of (2.9). We call it $\theta$, or $\theta_{q}$ when we want to emphasize the dependence on the Taylor-Wiles prime $q$:

[TABLE]

3. Derived Hecke operators and the main conjecture

We follow the notation of §2; in particular,

$g$ is a modular form with coefficients in the integer ring $\mathcal{O}$; we have associated to it a $\mathcal{O}$-module $\mathrm{U}_{g}$ of “Stark units” of rank $1$.

• -

Fixing a prime $\mathfrak{p}$ of $\mathcal{O}$, we will work with the coefficient ring $k=\mathcal{O}/\mathfrak{p}^{n}$ with residue field $\mathbb{F}_{\mathfrak{p}}$ of characteristic $p$.

In this section we define derived Hecke operators and formulate the main conjecture concerning their relationship to $\mathrm{U}_{g}$. This discussion is obtained by transcribing the theory of [17] to the present context; in this section, we just describe the conclusions of this process.

For each $q\equiv 1$ modulo $p^{n}$ and each $z\in k\langle-1\rangle$, we will produce an operator

[TABLE]

Note that $q$ need not be a Taylor–Wiles prime (in the sense of §2.4) for the definition of $T_{q,z}$ – in other words, we do not use the assumption on the Frobenius element. However, our conjecture pins down the action of $T_{q,z}$ only at Taylor–Wiles primes.

3.1. The Shimura class

Start with the Shimura covering $X_{1}(q)\rightarrow X_{0}(q)$, and pass to the unique subcovering with Galois group $({\mathbb{Z}}/q)^{*}_{p}$; call this $X_{1}(q)^{\Delta}\rightarrow X_{0}(q)$. By Corollary 2.3 of [13, Chapter 2], it extends to an étale covering of schemes over ${\mathbb{Z}}[\frac{1}{qN}]$, and in particular induces an étale cover $X_{1}(q)^{\Delta}_{k}\rightarrow X_{0}(q)_{k}$. It therefore gives rise to a class in the etale $H^{1}$, i.e.

[TABLE]

In the category of étale sheaves over $X_{0}(q)_{k}$ there is a natural map $k\rightarrow\mathbb{G}_{a}$. Then a class in $H^{1}_{\mathrm{et}}(X_{0}(q)_{k},k\langle 1\rangle)$ defines a class in

[TABLE]

because of the coincidence of the étale and Zariski cohomologies with coefficients in a quasi-coherent sheaf. This construction has thus given a class, associated to the Shimura cover, but now in Zariski cohomology:

[TABLE]

which we shall sometimes call the Shimura class.

It is reassuring to note that $\mathfrak{S}$ is in fact nonzero, even modulo the maximal ideal $\mathfrak{p}$, as one sees by computing with the Artin–Schreier sequence

[TABLE]

over $k/(\mathfrak{p})$.

3.2. Construction of the derived Hecke operator

The class $\mathfrak{S}$ just defined can be pulled back to $H^{1}_{\mathrm{Zar}}(X_{01}(qN)_{k},\mathcal{O}\langle 1\rangle)$. We denote this class by $\mathfrak{S}_{X}$, to distinguish it from $\mathfrak{S}$ at level $q$.

Thus cup product with $\mathfrak{S}_{X}$ gives a mapping

[TABLE]

Finally, to obtain the derived Hecke operator we add a push-pull as in the usual Hecke operator definition:

[TABLE]

where $\pi_{1},\pi_{2}:X_{01}(qN)\rightarrow X$ are the two natural degeneracy maps (at the level of the upper half-plane, we understand $\pi_{1}$ to be $z\mapsto z$, and $\pi_{2}$ to be $z\mapsto qz$.) Observe that without the middle $\cup\mathfrak{S}_{X}$ this would be the usual Hecke operator at $q$. In other words, we have constructed a map

[TABLE]

and correspondingly for $z\in k\langle-1\rangle$ we will denote by $T_{q,z}$ the corresponding “derived Hecke operator”

[TABLE]

obtained by multiplying (3.2) by $z$.

Although by presenting the bare definition the construction may seem a little ad hoc, this definition is really a specialization of the general theory of [17], and is indeed very natural. We explain this in more detail in §7.

3.3. The conjecture

We now formulate the main conjecture. It asserts that the various operators $T_{q,z}$ all fit together into a single action of $\mathrm{U}_{g}^{\vee}$ on the $g$-part of cohomology. As formulated in [17], the conjecture is ambiguous up to a rational factor, and we will not attempt to remove this ambiguity here (although our computations suggest that this factor might have a simple description).

Terminology:

• •

Suppose that $\alpha\in E$ and $V$ is a $k$-module. For $x,y\in V$, we will write

[TABLE]

if we may write $\alpha=A/B$, where $A,B\in\mathcal{O}$ are not both divisible by $\mathfrak{p}$, in such a way that $\bar{B}x=\bar{A}y$ (here $\bar{A},\bar{B}$ are the reductions of $A,B$ under $\mathcal{O}\rightarrow k$.)

In particular, if $V$ is a $k$-line this has the following meaning:

• –

if $x=y=0$, then (3.4) is understood to always be true.

• –

Otherwise, we can make sense of $[x:y]\in\mathbb{P}^{1}(k)$, and (3.4) means that the reduction of $\alpha\in\mathbb{P}^{1}(E)\rightarrow\mathbb{P}^{1}(k)$ equals $[x:y]$.

• •

For $h\in H^{*}(X_{\mathcal{O}[\frac{1}{N}]},\omega)$ we write $\bar{h}$ for the reduction of $h$ to $H^{*}(X_{k},\omega)$.

• •

Recall that we defined a reduction map $\theta_{q}:\mathrm{U}_{g}\rightarrow k\langle 1\rangle$. Also, the pairing between $\mathrm{U}_{g}$ and $\mathrm{U}_{g}^{\vee}$, which is perfect after localization at $p$, descends to a perfect pairing on $\mathrm{U}_{g}\otimes k$ and $\mathrm{U}_{g}^{\vee}\otimes k$. With respect to this pairing, the map $\theta_{q}$ has an adjoint:

[TABLE]

Explicitly for $z\in k\langle-1\rangle$,

[TABLE]

where $u\in\mathrm{U}_{g},u^{*}\in\mathrm{U}_{g}^{\vee}$ are as defined around (2.5).

Conjecture 3.1**.**

There is an action $\star$ of $\mathrm{U}_{g}^{\vee}$ on $H^{*}(X_{\mathcal{O}[\frac{1}{N}],}\omega)[g]$, and $\alpha\in E$ such that for every $(\mathfrak{p},n,q,z)$, with

$\mathfrak{p}$* a prime of $E$ satisfying the conditions of §2.4;*

• -

$n\geqslant 1$* an integer;*

• -

$q$* a Taylor–Wiles prime of level $n$, in particular $q\equiv 1(p^{n})$.*

• -

$z\in(\mathcal{O}/\mathfrak{p}^{n})\langle-1\rangle$,

we have the following equality:

[TABLE]

On the right hand side, $\theta^{\vee}_{q}(z)^{\sim}$ means that we choose an arbitrary lift of $\theta^{\vee}_{q}(z)\in\mathrm{U}_{g}^{\vee}\otimes k$ to $\mathrm{U}_{g}^{\vee}$, and the bar refers to reduction mod $\mathfrak{p}^{n}$.

In what follows, we will write (3.6) in the abridged form

[TABLE]

The meaning here is that equality holds, in the sense described above, for some fixed coefficient of proportionality $\alpha\in E$. (Note we have suppressed explicit mention of the lift $\theta^{\vee}_{q}(z)^{\sim}$ from the notation; in any case the right-hand side is independent of this choice of lift.)

4. Relationship to Galois deformation theory

In this section – which is not used in the rest of the paper – we shall sketch a proof that, in the case $n=1$,

[TABLE]

assuming an “$R=T$” theorem for weight one forms at the level of $g$, as well as further technical conditions. Such a theorem is known in some generality by the work of Calegari [3].

This result (and its proof) is in line with results and proofs from [17]. Indeed, our methods would show that (4.1) is an equivalence, if we knew an “$R=T$” theorem for weight one forms with (Taylor–Wiles) auxiliary level.

4.1. Setup

Let $q$ be a prime such that the eigenvalues of $\overline{\rho}$ on the Frobenius at $q$ are distinct elements of $\mathbf{F}_{\mathfrak{p}}$, say $\alpha$ and $\beta$. Let $\mathfrak{m}$ be the ideal of the Hecke algebra associated to the Galois representation $\overline{\rho}$.

In addition to the conditions from (2.4) we assume that:

• (i)

$n=1$ so that $k=\mathcal{O}/\mathfrak{p}$ is a field.

• (ii)

For each prime $\nu$ dividing $N$, the residual representation $\overline{\rho}$ is of the form $\chi_{1}\oplus\chi_{2}$, where $\chi_{1}$ is ramified and $\chi_{2}$ is unramified.

• (iii)

$p$ does not divide $\nu-1$, for each $\nu$ as above.

• (iv)

$p$ does not divide $[L:{\mathbf{Q}}]$, and does not divide the order of the class group of $L$.

• (v)

The $\mathfrak{m}$-completion of the space of modular forms at level $\Gamma_{1}(N)$, with coefficients in $\mathcal{O}$, is free rank one over $\mathcal{O}_{\mathfrak{p}}$. (In particular, there are no congruences modulo $\mathfrak{p}$ between $g$ and other weight one forms, either in characteristic zero or characteristic $p$.)

Let $\mathfrak{m}_{\alpha}$ be the maximal ideal of the Hecke algebra for $X_{01}(qN)$ obtained by adjoining $U_{q}-\alpha$ to the ideal $\mathfrak{m}$; similarly we define $\mathfrak{m}_{\beta}$. These ideals also have evident analogues where we add $\Gamma_{1}(q)$ level to $X$, rather than just $\Gamma_{0}(q)$ level, and we denote these analogues by the same letters.

Our assumption (v), and the assumption of torsion-freeness from (2.4), means that

[TABLE]

i.e. all three spaces above are $k$-lines; the same statement is true for $H^{1}(-)$.

Let $g_{\alpha}$ and $g_{\beta}$, respectively, span the second and third spaces in the line above. Therefore $U_{q}g_{\alpha}=\alpha g_{\alpha}$ and $U_{q}g_{\beta}=\beta g_{\beta}$. We normalize these so that $\pi_{1}^{*}g=g_{\alpha}+g_{\beta}$.

Since the pushforward $\pi_{1*}$ via the natural projection $\pi_{1}:X_{01}(qN)\rightarrow X$ induces an isomorphism on each of the $U_{q}$-eigenspaces, $g_{\alpha}\cup\mathfrak{S}_{X}$ vanishes if and only if $\pi_{1*}(g_{\alpha}\cup\mathfrak{S}_{X})$ vanishes. Observe $(U_{q}-\beta)\pi_{1}^{*}g=(\alpha-\beta)g_{\alpha}$. We are assuming $\alpha\neq\beta$ and therefore

[TABLE]

Now $\pi_{1*}(\pi_{1}^{*}g\cup\mathfrak{S}_{X})=g\cup(\pi_{1*}\mathfrak{S}_{X})=0$ and $\pi_{1*}\mathfrak{S}_{X}$ is trivial:

Lemma 4.2**.**

The pushforward of $\mathfrak{S}_{X}$ by the natural projection $\pi:X_{01}(qN)\rightarrow X$ is trivial.

Proof.

The existence of the trace map [1, Expose 17, Section 6.2] gives a map $\pi_{*}({\mathbb{Z}}/p)\rightarrow{\mathbb{Z}}/p$ of étale sheaves, compatible with the usual trace $\pi_{*}\mathbb{G}_{a}\rightarrow\mathbb{G}_{a}$. For this reason, it is sufficient to show that the (trace-induced) map

[TABLE]

pushes the Shimura class forward to the trivial class.

If $\iota$ is the inclusion of an open curve into a complete curve induces then $\iota^{*}$ is an injection on $H^{1}$. Therefore, it suffices to show a similar statement for the open modular curves; restricted to these, the map $\pi$ is étale.

Define finite groups

[TABLE]

Then $X_{01}(qN)$ and $X_{k}$ are quotients of a suitable modular curve by $B$ and $G$ respectively. This allows to reduce to verifying the triviality of the transfer in group cohomology, from $B$ to $G$, of $\alpha\in H^{1}(B,({\mathbb{Z}}/q)_{p}^{*})$, defined via

[TABLE]

This is a straightforward computation. ∎

Continuing from (4.3), we find

[TABLE]

The final expression can be verified to be an invertible multiple of $T_{q,z}g$ for some nonvanishing $z\in\mathcal{O}/\mathfrak{p}\langle-1\rangle$. Therefore:

[TABLE]

Write $\Delta=({\mathbb{Z}}/q)^{*}_{p}$; since we are assuming that $n=1$, the group $\Delta$ is cyclic of order $p$ and we have an isomorphism $k[\Delta]\simeq k[T]/T^{p}$, whose inverse sends $T$ to $\delta-1$, for any generator $\delta$ of $\Delta$. Let $X_{1}(Nq)^{\Delta}$ be the subcovering of $X_{1}(Nq)\rightarrow X_{01}(qN)$ which corresponds to the quotient $({\mathbb{Z}}/q)^{*}\rightarrow({\mathbb{Z}}/q)^{*}_{p}$ of deck transformation groups.

Lemma 4.3**.**

The cup product $\cup\mathfrak{S}_{X}$ is nonzero as a map on $H^{*}(X_{01}(qN)_{k},\omega)_{\mathfrak{m}_{\alpha}}$ if and only if

[TABLE]

The usual Taylor-Wiles method, for classical modular forms on $\mathrm{GL}_{2}$, relies crucially on producing “more” modular forms when adding “$\Gamma_{1}(q)^{\Delta}$ level” at auxiliary primes $q$. Thus the Lemma says: the derived Hecke operator is nontrivial precisely when this fails, a failure that is rectified in the Calegari–Geraghty approach [4].

Proof.

By the methods of [4], we may obtain a complex $C$ of free $k[\Delta]$-modules (with degree-decreasing differential) together with isomorphisms:

[TABLE]

With reference to the latter isomorphism, cup product with $\mathfrak{S}_{X}$ on the right is represented by the natural action of a nontrivial class in $\mathrm{Ext}^{1}_{k[\Delta]}(k,k)$ on the left hand side (note that $H^{i}\mathrm{Hom}_{k[\Delta]}(C,k)$ is identified with homomorphisms from $C$ to $k[i]$ in the derived category.)

Replacing $C$ by a minimal free resolution we may assume that $C$ is the complex given by

[TABLE]

where $A\in k[\Delta]$ belongs to the augmentation ideal. Under the identification of $k[\Delta]$ with $k[T]/T^{p}$, the element $A$ corresponds to an invertible multiple of $T^{i}$, for some $0\leq i\leq p-1$, and then (4.7) implies

[TABLE]

We shall show that cup product with $\mathfrak{S}_{X}$ is nontrivial if and only if $i=1$. To compute the action of $\mathrm{Ext}^{1}_{k[\Delta]}(k,k)$ we may consider the following diagram:

[TABLE]

The horizontal complexes are, respectively, $C$, a projective resolution of $k$, and a projective resolution of $k[1]$. Continuing to take $\mathrm{Hom}$ in the derived category of $k[\Delta]$-modules, the top vertical map of complexes represents a generator for $\mathrm{Hom}(C,k)$ and the bottom vertical map of complexes represents a nontrivial class in $\mathrm{Ext}^{1}_{k[\Delta]}(k,k)=\mathrm{Hom}(k,k[1])$. Therefore, the composite map in $\mathrm{Hom}(C,k[1])$ is represented by the diagram

[TABLE]

This is nullhomotopic exactly when $T^{i-1}$ is divisible by $T$, i.e. $i\geq 2$. ∎

Taking (4.5) together with the Lemma, we see

[TABLE]

Consider the map

[TABLE]

where $\mathrm{R}$ (resp. $\mathrm{R}^{\prime}$) are the weight 1, determinant $\chi$, deformation rings for $\overline{\rho}$ at level $\Gamma_{1}(N)$ and with level $\Gamma_{1}(Nq)$ respectively. The local conditions $\mathcal{L}$ for $R$ and $\mathcal{L}^{\prime}$ for $R^{\prime}$ are as follows:

• •

At $p$, we require that deformation remains unramified.

• •

At $q$ we impose unramified for $R$ and no condition for $R^{\prime}$.

• •

For primes $\nu$ dividing $N$, we do not need to impose any condition: We have assumed that $\overline{\rho}$ is a direct sum $\chi_{1}\oplus\chi_{2}$ of two characters, with $\chi_{1}$ ramified and $\chi_{2}$ unramified. In particular $H^{1}({\mathbf{Q}}_{\nu},\mathrm{Ad}^{0}\overline{\rho})$ is $1$-dimensional, corresponding to deforming $\chi_{1}\leftarrow\chi_{1}\psi,\chi_{2}\leftarrow\chi_{2}\psi^{-1}$ for a character $\psi$ with trivial reduction. In [3] the assumption is imposed that in fact $\chi_{2}\psi^{-1}$ remains unramified, but we do not need to explicitly impose this because we assumed that $p$ is relatively prime to $\nu-1$ – thus the character $\psi$ is automatically unramified at $\nu$. In particular, we have automatically

[TABLE]

where we recall that for a module $M$ under the Galois group of ${\mathbf{Q}}_{\ell}$, the “unramified” classes $H^{1}_{\mathrm{ur}}\subset H^{1}$ are defined to be those that arise from inflation from the Galois cohomology of ${\mathbf{F}}_{\ell}$ acting on inertial invariants on $M$.

Assuming an $R=T$ theorem for $g$, we have $R\otimes k=k$. The map on tangent spaces induced by $f$, call it $f^{*}$, fits into the following diagram, with reference to the usual identification of tangent spaces with Galois cohomology:

[TABLE]

$f^{*}$ is surjective exactly when $j$ is injective. Since the middle group in the exact sequence is one-dimensional, injectivity of $j$ is the same as nonvanishing of $j$. Under Tate global duality, the map $j$ is dual to

[TABLE]

where $\mathcal{L}^{\prime\vee}$ is the dual condition to $\mathcal{L}^{\prime}$: it refers to classes that are unramified at primes not dividing $pN$, unramified (equivalently: trivial) at primes dividing $N$, and at $p$ belong to the Bloch-Kato $f$-cohomology (a more concrete description is given below).

We will show in the next subsection that:

[TABLE]

Therefore, the nonvanishing of $\theta_{q}$ implies the injectivity of $j$, which implies the surjectivity of $f^{*}$, which implies $R^{\prime}\otimes k=k$, which implies (4.6) by a multiplicity one argument. Then (4.5) and Lemma 4.3 show that $T_{q,z}\bar{g}\neq 0$ as desired.

That concludes our proof for (4.1); note finally that if we had a theorem $R^{\prime}=T^{\prime}$ all this reasoning would be reversible and we get an equivalence in (4.1).

4.4. Relation of $\mathrm{U}_{g}$ to Galois cohomology

To conclude we must relate $j^{\vee}$ and $\theta_{q}$, and thereby prove (4.11).

As in (2.2), $\rho$ is a representation into $\mathrm{GL}_{2}(\mathcal{O})$; let $\rho_{\mathfrak{p}}$ be the same representation, but now considered as valued in $\mathrm{GL}_{2}(\mathcal{O}_{\mathfrak{p}})$; thus

[TABLE]

We write $(\mathrm{U}_{g})_{\mathfrak{p}}$ for $\mathrm{U}_{g}\otimes_{\mathcal{O}}\mathcal{O}_{\mathfrak{p}}$.

Consider

[TABLE]

where the subscript $\mathrm{ur}$ means that we consider classes that are unramified at primes away from $p$, and, at $p$, belong the Bloch-Kato $f$-space. (What this means is made explicit in the computation of $H^{1}_{\mathrm{ur}}(L,\mathcal{O}_{\mathfrak{p}}(1))$ in the diagram below.)

Restriction to $L$ gives horizontal maps in the following diagram

[TABLE]

The vertical columns are exact at top and middle, and, in the bottom row, the sum is taken over all places $v$ of ${\mathbf{Q}}$, and then over all places $w$ of $L$.

Lemma 4.5**.**

$i$* induces an isomorphism $H^{1}_{\mathrm{ur}}({\mathbf{Q}},\mathrm{Ad}^{*}\rho_{\mathfrak{p}}(1))\simeq(\mathrm{U}_{g})_{\mathfrak{p}}$. Also, as long as the class group of $L$ is prime to $p$, the reduction modulo $\mathfrak{p}$ map $H^{1}_{\mathrm{ur}}({\mathbf{Q}},\mathrm{Ad}^{*}\rho_{\mathfrak{p}}(1))\rightarrow H^{1}_{\mathrm{ur}}({\mathbf{Q}},\mathrm{Ad}^{*}\overline{\rho}(1))$ is surjective.*

Proof.

The map $j$ is an isomorphism by considering the inflation-restriction sequence: the group $G_{L/{\mathbf{Q}}}$ has order prime to $p$.

This means $i$ is injective. $i$ will be surjective if $k$ is injective. In fact, for a place $\mathfrak{q}$ of $L$ above $v$, the map

[TABLE]

is split, up to multiplication by $[L_{\mathfrak{q}}:{\mathbf{Q}}_{v}]$, by corestriction, and $[L_{\mathfrak{q}}:{\mathbf{Q}}_{v}]$ is invertible on $\mathcal{O}_{\mathfrak{p}}$.

This proves the first assertion, about $i$. For the second assertion, note that the assumption about class groups means that $H^{1}_{\mathrm{ur}}(L,\mathbb{F}_{\mathfrak{p}}(1))$ coincides with $\mathrm{U}_{L}\otimes\mathbb{F}_{\mathfrak{p}}$. The same analysis as above means that the rank of $H^{1}_{\mathrm{ur}}({\mathbf{Q}},\mathrm{Ad}^{*}\overline{\rho}(1))$ over $\mathbb{F}_{\mathfrak{p}}$ is bounded above by the dimension of

[TABLE]

and (again because $G_{L/{\mathbf{Q}}}$ has no Galois cohomology in characteristic $p$) this dimension coincides with the $\mathcal{O}_{\mathfrak{p}}$-rank of $\mathrm{U}_{g}\otimes\mathcal{O}_{\mathfrak{p}}$ (which is exactly $1$). The surjectivity now follows. ∎

Now, under the identification $i:H^{1}_{\mathrm{ur}}({\mathbf{Q}},\mathrm{Ad}^{*}\rho_{\mathfrak{p}}(1))\simeq(\mathrm{U}_{g})_{\mathfrak{p}}$, the composite

[TABLE]

[TABLE]

is identified with the map $\theta_{q}$ described in (2.10). Here we have made use of a map $\mathrm{Ad}^{*}\overline{\rho}\rightarrow\mathbb{F}_{\mathfrak{p}}$, which comes from pairing with the element defined in (2.8). In particular, $\theta_{q}$ vanishes if and only if $j^{\vee}$ does, as required.

5. Explication

Our main Conjecture 3.1, as formulated, involves a cup product in coherent cohomology on the special fiber of a modular curve. We want to translate it to a readily computable form, i.e. one that can be carried out just using manipulations with $q$-series. We will achieve this in this section, at least in the case $n=1$ and under modest assumptions on $q$, and then test the conjecture numerically.

5.1. Pairing with $g^{\prime}$

Recall (§2.3) that we have fixed another weight one modular form $g^{\prime}$ that is contragredient to $g$. To extract numbers from the Conjecture, we pair both sides of (3.6) with $g^{\prime}$, using the residue pairing (§2.2). Pairing (3.6) with $g^{\prime}$, and using $\theta_{q}^{\vee}(z)=u^{*}\otimes\frac{\langle z,\theta_{q}(u)\rangle}{\langle u^{*},u\rangle}$ from (3.5) we arrive at:

[TABLE]

where both sides lie in $k$; and we recall again that we have written $\bar{g}$ for the reduction of $g$ to a modular form with $k$ coefficients.

Now the square-bracketed quantity on the right hand side is an element of $E$, integral at $\mathfrak{p}$, and independent of choice of $(\mathfrak{p},n,q,z)$. We abridge (5.1) to

[TABLE]

This should hold true for any $(\mathfrak{p},n,q,z)$.

Unwinding the definition of the derived Hecke operator,

[TABLE]

where the residue pairing is now taken on $X_{01}(qN)_{k}$, $\pi_{1},\pi_{2}$ are the two projections $X_{01}(qN)\rightarrow X$, and

[TABLE]

(Recall that $\mathfrak{S}_{X}\in H^{1}(X_{01}(qN)_{k},\mathcal{O}\langle-1\rangle)$, so its product with $z\in k\langle 1\rangle$ lies in the right-hand group above.) To simplify notation, define the weight $2$ form

[TABLE]

In terms of classical modular forms, $G$ would be the form “$z\mapsto g(z)g^{\prime}(qz)$.” Then the right hand side of (5.2) is simply the (Serre duality) pairing of $G\in H^{0}(\Omega^{1})$ and $z\mathfrak{S}_{X}\in H^{1}(\mathcal{O})$ in the coherent cohomology of $X_{01}(qN)$. Therefore, the conjecture implies that $\langle z\mathfrak{S}_{X},G\rangle\propto\langle\theta_{q}(u),z\rangle$; and here we may as well cancel the $z$s from both sides:

[TABLE]

Here both sides lie in $k\langle 1\rangle$, that is to say, in $({\mathbb{Z}}/q)^{*}\otimes k$.

Now the class $\mathfrak{S}_{X}$ is pulled back from a class $\mathfrak{S}$ on $X_{0}(q)$, and correspondingly the pairing on the left-hand side can be pushed down to $X_{0}(q)$. Writing

[TABLE]

we have $\langle\mathfrak{S}_{X},G\rangle=\langle\mathfrak{S},G^{\mathrm{proj}}\rangle$.

Thus our conjecture implies that

[TABLE]

where we recall that:

• •

$\mathfrak{S}\in H^{1}(X_{0}(q)_{k},\mathcal{O}\otimes({\mathbb{Z}}/q)^{*})$ is constructed from the covering $X_{1}(q)\rightarrow X_{0}(q)$;

• •

$G^{\mathrm{proj}}\in H^{0}(X_{0}(q)_{k},\Omega^{1})$ is the pushforward of the form “$z\mapsto g(z)g^{\prime}(qz)$” from level $X_{01}(qN)$ to level $X_{0}(q)$; it is a weight $2$ cusp form.

• •

$\langle-,-\rangle$ is the pairing of Serre duality.

• •

The symbol $\propto$ is interpreted as in (3.7).

5.2. Localization at the Eisenstein ideal

To translate (5.5) to a computable form, we will use computations of Merel and Mazur. Let

[TABLE]

be the “Eisenstein” cusp form with $k$ coefficients, in other words, the unique element whose $q$-expansion coincides with the reduction modulo $p^{n}$ of the weight $2$ Eisenstein series; the condition that $q\equiv 1$ modulo $p^{n}$ means that this weight $2$ Eisenstein series indeed has cuspidal reduction in $k$. The pairing

[TABLE]

was considered by Mazur ([13], page 103, discussion of the element $u$) and was computed in a remarkable paper of Merel [14]. We will carefully translate Merel’s computation into our setting in the next section; unfortunately, in doing so, we will have to impose the restriction $n=1$, i.e. we can only compute things modulo $p$ and not higher powers of $p$.

Lemma 5.3**.**

(Merel; see §6 for details of the translation from Merel’s framework to this one).

[TABLE]

where mod $p$ means that the two sides have the same projection to $\mathbb{F}_{p}\langle 1\rangle$.222It seems likely that the two sides are actually equal in $({\mathbb{Z}}/q)^{*}_{p}$ but we do not prove this.

Here the Merel unit $\varpi_{\mathrm{Merel}}\in({\mathbb{Z}}/q)^{*}$ is the element

[TABLE]

In the remainder of this section, we will compute $\langle\mathfrak{S},G^{\mathrm{proj}}\rangle$ (the left-hand side of (5.5)) using Lemma 5.3.

Let $\mathbb{T}$ be the Hecke algebra for cusp froms on $X_{0}(q)_{{\mathbb{Z}}_{p}}$, i.e. the algebra of endomorphisms of $S_{2}(q):=H^{0}(X_{0}(q)_{{\mathbb{Z}}_{p}},\Omega^{1})$ generated by $T_{\ell}$ for all $\ell\neq q$. Let $\mathfrak{I}\leqslant\mathbb{T}$ be the Eisenstein ideal, i.e. the kernel of the character

[TABLE]

by which $\mathbb{T}$ acts on the modulo $p$ reduction of $\mathrm{E}$. In particular it’s a maximal ideal.

Let $\mathfrak{m}_{1},\dots,\mathfrak{m}_{r}$ be all other maximal ideals of $\mathbb{T}$. Then the natural map

[TABLE]

is an isomorphism (here $\mathbb{T}_{\mathfrak{I}}$ means the completion, and similarly for $\mathfrak{m}_{s}$). Let $e_{\mathfrak{I}}$ be the idempotent of $\mathbb{T}$ corresponding to the first factor; the splitting

[TABLE]

gives rise to a splitting

[TABLE]

where $S_{2}(q)_{\mathfrak{I}}$ is the image of the idempotent $e_{\mathfrak{I}}$, and the complementary subspace is the image of $1-e_{\mathfrak{I}}$. Therefore, if $T\in\mathfrak{I}$ is chosen so that $T\notin\bigcup_{i=1}^{s}\mathfrak{m}_{i}$, then $T$ acts invertibly on the second factor.

Decompose $G^{\mathrm{proj}}$ as

[TABLE]

according to the splitting above. The Shimura class $\mathfrak{S}$ is annihilated by $\mathfrak{I}$ (see for example [13], Lemma 18.7). Choose as above $T\in\mathfrak{I}$ that acts invertibly on the second factor of (5.8). We may write

[TABLE]

and so

[TABLE]

where as before the pairings come from Serre duality.

Next, Mazur proves [13, Proposition 19.2] that

[TABLE]

We will complete our computation only in this case.333Recently, Lecouturier has proposed a very interesting generalization of the conjectural equality (5.10) to the case when $\varpi_{\mathrm{Merel}}$ is zero modulo $p$ and has verified it numerically in some cases. Since $\mathrm{E}$ is annihilated by $\mathfrak{I}$, we have in fact $\mathrm{E}\in S_{2}(q)_{\mathfrak{I}}$, and since the first Fourier coefficient of $\mathrm{E}$ is $1$, we have (under the assumption of (5.9)) $S_{2}(q)_{\mathfrak{I}}={\mathbb{Z}}_{p}.E.$ Thus, after extending scalars to $\mathcal{O}$, we find

[TABLE]

where $a_{1}(G^{\mathrm{proj}}_{\mathfrak{I}})\in\mathcal{O}\otimes{\mathbb{Z}}_{p}$ denotes the first coefficient in the $q$-expansion. Putting this together with our prior discussion, we have shown

Proposition 5.4**.**

Conjecture 3.1 implies that there exists $\alpha\in E$ such that

[TABLE]

*for any $(\mathfrak{p},n,q)$ as in §2.4 with the additional property that $(\varpi_{\mathrm{Merel}})\in({\mathbb{Z}}/q)^{*}_{p}$ is nontrivial modulo $p$. 444To be absolutely clear, we write out the meaning of this statement. We understand

$L:=a_{1}(G^{\mathrm{proj}}_{\mathfrak{I}})\otimes\ (\varpi_{\mathrm{Merel}})_{p},R:=\theta_{q}(u)$

as elements of $\mathcal{O}\otimes({\mathbb{Z}}/q)^{*}_{p}$; and the statement above means that if we reduce $\bar{L},\bar{R}$ to $\mathcal{O}/p\otimes({\mathbb{Z}}/q)^{*}_{p}$, then $\bar{L}=\alpha\bar{R}$, in the sense of (3.4). Other conventions are as follows:*

• •

$\varpi_{\mathrm{Merel}}\in({\mathbb{Z}}/q)^{*}_{p}$* is the Merel unit, see (5.6);*

• •

$a_{1}(G^{\mathrm{proj}}_{\mathfrak{I}})\in\mathcal{O}\otimes{\mathbb{Z}}_{p}$* is the first Fourier coefficient of $G=(\pi_{1}^{*}g)(\pi_{2}^{*}g^{\prime})$, after taking projection $G^{\mathrm{proj}}$ to level $X_{0}(q)$ and then projection $G^{\mathrm{proj}}_{\mathfrak{I}}$ to the localization at the Eisenstein ideal.*

• •

$\theta_{q}(u)\in k\langle 1\rangle=\mathcal{O}\otimes({\mathbb{Z}}/q)^{*}_{p}$* is the reduction of the Stark unit*

5.5. Some philosophical worries

Let us take to examine some consequences of an inadequacy of our conjecture, namely, it is only formulated “up to $E^{*}$’.”

For each $(\mathfrak{p},n,q)$ as in §2.4, we can compute both $a_{1}(G^{\mathrm{proj}}_{\mathfrak{I}})\otimes\ (\varpi_{\mathrm{Merel}})_{p}$ and $\theta_{q}(u)$ and compare them. Let us also restrict to $(\mathfrak{p},n,q)$ for which $\theta_{q}(u)\neq 0$; there are infinitely many such $\mathfrak{p}$. Therefore (5.10) specifies the reduction $\alpha\in E$ to $\mathbb{P}^{1}(\mathbf{F}_{\mathfrak{p}})$, for an infinite collection of $\mathfrak{p}$. This uniquely specifies $\alpha$ if it exists.

The conjecture is numerically falsifiable to some extent. For example, if we find two different pairs $(\mathfrak{p},n,q)$ and $(\mathfrak{p},n^{\prime},q^{\prime})$ for which the predicted reductions of $\alpha$ mod $\mathfrak{p}$ differ, this clearly contradicts the conjecture. Indeed the fact that this did not occur in our numerical computations was very encouraging to us.

However, if this type of clash does not occur, no amount of computation can falsify the conjecture: we can, of course, produce an $\alpha\in E$ with any specified reduction at any number of places. Nonetheless this proves to be largely a theoretical worry. In our examples, we shall find an $\alpha$ of very low height for which (5.10) holds for many $(\mathfrak{p},n,q,z)$. Our sense is that this should be taken as satisfactory indication that the Conjecture, or something very close to it at least, is valid.

As a final excuse we may note that the conjectures about special values of $L$-functions were initially phrased with a ${\mathbf{Q}}^{*}$ ambiguity that is similarly unfalsifiable.

Eventually, we hope that these issues will be solved by formulating an integral form of the conjecture; this could perhaps be done using the theory of derived deformation rings.

5.6. Forms associated to cubic fields

We now make the foregoing discussion even more explicit for the form $g$ associated to a cubic field $K$; write $L$ for the Galois closure of $K$. (This will coincide with our previously defined $L$ in a moment.)

Such a field $K$ defines a representation $\mbox{Gal}(L/{\mathbf{Q}})\rightarrow S_{3}$; if we regard $S_{3}$ as acting on $M=\{(x_{1},x_{2},x_{3})\in{\mathbb{Z}}^{3}:\sum x_{i}=0\}$ by permuting the coordinate axes, we may regard $\rho$ as a rank $2$ Galois representation:

[TABLE]

Under the representation (5.11), there is a basis for $M$ such that the transposition $\sigma=(12)\in S_{3}$ is sent to S:=\left(\begin{array}[]{cc}0&1\\ 1&0\end{array}\right), whereas a $3$-cycle $\tau=(123)\in S_{3}$ is sent to T:=\left(\begin{array}[]{cc}-1&1\\ -1&0\end{array}\right). We may set things up so that the fixed field of $(12)\in S_{3}$ is equal to $K$.

In our previous notation, take

• •

$L$ as above, namely, the Galois closure of the cubic field $K$;

• •

$E={\mathbf{Q}}$ and $\mathcal{O}={\mathbb{Z}}$.

• •

$\mathfrak{p}=p\geqslant 5$ to be a rational prime of ${\mathbf{Q}}$.

• •

$n=1$ (thus we work only modulo $p$ rather than $p^{n}$).

• •

$q\equiv 1(p)$ to be a prime such that the $q$th Hecke eigenvalue $a_{q}(g)=0$. In this case, the Frobenius is a transposition555The primes $q$ for which $\rho(\mathrm{Frob}_{q})$ is a $3$-cycle also are Taylor–Wiles primes, but it is then easy to see that $T_{q,z}g=0$ for such $q$. To verify this, one can use the fact – notation as in (5.3) – that the Atkin-Lehner involution at $q$ for $X_{01}(qN)$ acts by $-1$ on $\mathfrak{S}_{X}$, but it acts by $\chi(q)$ on $G$, where $\chi$ is the quadratic Nebentypus character for $g$. in $S_{3}$. Thus $q$ is a Taylor–Wiles prime with eigenvalues $(1,-1)$.

• •

Therefore in this case $k\langle 1\rangle=\mathbb{F}_{p}\langle 1\rangle$ is just the unique quotient of $({\mathbb{Z}}/q)^{*}$ of order $p$.

• •

We also fix a prime $\mathfrak{q}_{0}$ of $L$ over $q$ such that the image of the Frobenius for $\mathfrak{q}_{0}$ is equal to $S$. In particular, this fixes $K$, so the prime $\tilde{q}$ of $K$ below $\mathfrak{q}_{0}$ is of degree $1$ over $q$.

Lemma 5.7**.**

Consider the isomorphism $\mathrm{U}_{g}\simeq\mathrm{Hom}_{G_{L/{\mathbf{Q}}}}(\mathrm{Ad}^{0}\rho,\mathrm{U}_{L})$ of (2.4). (Recall that $\mathrm{U}_{L}$ is the unit group of $L$.) Then computing the image of $S\in\mathrm{Ad}^{0}\rho$ gives rise to an isomorphism

[TABLE]

where $\mathcal{O}_{K}^{(1)}$ is the group of norm one units of $K$.

Moreover, for $p\geq 5$ the reduction map $\theta_{q}:\mathrm{U}_{g}\rightarrow\mathbb{F}_{p}\langle 1\rangle$ described in (2.10) becomes identified with the composite

[TABLE]

where $\tilde{q}$ is the unique degree one prime of $K$ above $p$.

Proof.

Indeed we may split

[TABLE]

where $e$ is the projection of $T\in\mathrm{Hom}(M,M)$ to the trace zero subspace $\mathrm{Hom}^{0}$, and $\mathsf{W}$ is the ${\mathbb{Z}}[\frac{1}{6}]$-submodule of $\mathrm{Hom}(M,M)\otimes{\mathbb{Z}}[\frac{1}{6}]$ spanned by the images of $(12),(13),(23)$ under $\rho$.

Therefore $S_{3}$ acts on $e$ by the sign character, whereas for any $S_{3}$-module $V$, the space of homomorphisms $\mathrm{Hom}_{S_{3}}(\mathsf{W},V)$ is identified with the subspace of

[TABLE]

such that $v+(123)v+(132)v=0$.

Using the definition of $\mathrm{U}_{g}$ and the splitting above, we find that evaluation at $S$ induces an isomorphism

[TABLE]

The first factor corresponds to units in the imaginary quadratic field ${\mathbf{Q}}(\sqrt{\mbox{disc}(L)})$, and is thus trivial upon inverting $6$. This proves (5.12).

Now let $u$ be a norm one unit in $K$; we may now identify it with an element of $\mathrm{U}_{g}\otimes{\mathbb{Z}}[1/6]$. We will compute its image under the reduction map. Let $\mathbf{u}\in\mathrm{Hom}(\mathrm{Ad}^{0}\overline{\rho},\mathrm{U}_{L})$ be the element associated to $u$. By definition $\mathbf{u}(S)=u$. Let $\mathfrak{q}_{0}$ be the prime of $L$ above $\tilde{q}$, as before; to compute $\theta_{q}(u)$ we must, by definition, compute the image of $\mathbf{u}$ under the sequence (2.10):

[TABLE]

where we phrased the previous definition dually. The element $\mathsf{e}_{q}$ from (2.8) is identified here with $S$, so that the last map is evaluation at $S$. It follows that this map is simply the reduction of $u$ at $\tilde{q}$.

∎

It follows from this discussion and Proposition 5.4 that we can rephrase our conjecture in the following way:

Conjecture 5.1**.**

Let $K$ be a cubic extension with negative discriminant $-D$, with sextic Galois closure $L$. Let $g$ be the associated weight one form of level $D$. Let $u\in\mathcal{O}_{K}^{*}$ be a unit. Let $q\equiv 1$ modulo $p$ be as above; suppose that $\left(\frac{-D}{q}\right)=-1$, and $p\geq 5$, and finally $\varpi_{\mathrm{Merel}}\in({\mathbb{Z}}/q)^{*}$ (see (5.7) for definition) is nonzero modulo $p$, i.e. upon projection to the quotient $\mathbb{F}_{p}\langle 1\rangle$.

Then there exist $A,B\in\mathbf{Z}$ such that, for all such $q$ we have

[TABLE]

where:

• •

$\eta\in{\mathbb{Z}}$* is the first Fourier coefficient of the Eisenstein component of $G^{\mathrm{proj}}_{\mathfrak{I}}$, the projection of $g(z)g(qz)$ to the Eisenstein component at level $q$. (This is well defined modulo the numerator of $\frac{q-1}{12}$, which is sufficient to make sense of the above definition.)*

• •

$\bar{u}\in({\mathbb{Z}}/q)^{*}$* is the reduction of $u$ modulo the unique degree one prime of $K$, above $q$.*

We have tested this conjecture numerically (see data tables) for the fields $K$ of discriminant $-23$ and $-31$. In all the cases for discriminant $-23$ we find $\frac{A}{B}=\frac{-1}{72}$; in all the cases for discriminant $-31$ we find $\frac{A}{B}=72$. The fact that $72$ is divisible only by $2$ and $3$ is striking.

6. Flat cohomology and Merel’s computation

We now explain why Merel’s computation implies Lemma 5.3. The issue is that Merel’s computation is in characteristic zero. To relate it to $\langle\mathrm{E},\mathfrak{S}\rangle$, which is defined in characteristic $p$, we will need to do a little setup in flat cohomology.

Let $X=X_{0}(q)$ regarded now as a a proper smooth curve over ${\mathbb{Z}}_{p}$; here $q\equiv 1$ modulo $p$. Let $J_{p}$ be the $p$-torsion of the Jacobian of $X_{\overline{{\mathbf{Q}}_{p}}}$. We shall define several incarnations of both the Shimura class and the Eisenstein class.

6.0.1. The (Shimura) class $\alpha$

The Shimura cover $X_{1}(q)^{\Delta}\rightarrow X_{0}(q)$ (from §3.1) is a $({\mathbb{Z}}/q)^{*}_{p}$ torsor for the etale topology. As before it define a class $\mathfrak{S}\in H^{1}_{\mathrm{et}}(X,\mathbb{F}_{p}\langle 1\rangle)$, which can be pulled back to flat cohomology:

[TABLE]

Restricting $\mathfrak{S}$ to the geometric generic fiber $X_{\overline{{\mathbf{Q}}_{p}}}$ we get a class in étale cohomology

[TABLE]

The inclusion $\mu_{p}\hookrightarrow\mathbb{G}_{m}$ induces $H^{1}_{\mathrm{et}}(X_{\overline{{\mathbf{Q}}_{p}}},\mu_{p})\rightarrow J_{p}$, and thus $\alpha_{\mathrm{et}}$ gives

[TABLE]

(we use the notation $P_{\alpha}$ to suggest that this is a point on the Jacobian).

Finally, we also obtain a Zariski class on the geometric special fiber, using the inclusion $\mathbb{F}_{p}\hookrightarrow\mathcal{O}$ and the identification of Zariski and etale cohomology for $\mathcal{O}$:

[TABLE]

6.0.2. The (Eisenstein) class $\beta$

Let $\Delta$ be the weight $12$ cusp form $q\prod(1-q^{n})^{24}$, and consider the function $f:=\Delta(qz)/\Delta(z)$ on $X$. Extracting its $p$th root gives a $\mu_{p}$-torsor (in the flat topology) on $X$. Indeed, $f$ is invertible except for the divisors corresponding to [math] and $\infty$, and along those divisors its valuation is divisible by $p$. Thus, we get a class

[TABLE]

The $\mu_{p}$-torsor is étale over the geometric generic fiber $X_{\overline{{\mathbf{Q}}_{p}}}$ and we we get a corresponding class in étale cohomology

[TABLE]

There is a corresponding class in the $p$-torsion of the Jacobian, namely, writing [math] and $\infty$ for the two cusps of $X$ we may form

[TABLE]

– this is related to our prior discussion because $p.Q_{\beta}$ is the divisor of $f$.

Finally, there is also a Zariski class “corresponding” to $\beta$ on the special fiber. Namely, the logarithmic derivative $\frac{df}{f}$ in fact extends to a global section of $\Omega^{1}$, i.e. a class

[TABLE]

Observe that $\frac{df}{f}$ is the differential form associated to the “Eisenstein cusp form” $G$ of weight $2$.

With these preliminaries, the main point is to check the following

Proposition 6.1**.**

We have an equality in $\mathbb{F}_{p}\langle 1\rangle$:

[TABLE]

Here $\langle-,-\rangle_{\mathrm{Weil}}$ is the Weil pairing, $\langle-,-\rangle_{\mathrm{et}}$ is the pairing given by Poincaré duality in étale cohomology on the geometric fiber, and $\langle-,-\rangle_{\mathrm{Zar}}$ is the pairing given by Serre duality in coherent cohomology on the special fiber. Keeping track of twists we see that these all take values in $\mathbb{F}_{p}\langle 1\rangle$.

Now $\langle\mathrm{E},\mathfrak{S}\rangle$ is given by $\langle\alpha_{\mathrm{Zar}},\beta_{\mathrm{Zar}}\rangle_{\mathrm{Zar}}$; the Proposition shows this coincides (in $\mathbb{F}_{p}\langle 1\rangle$) with $\langle P_{\alpha},Q_{\beta}\rangle_{\mathrm{Weil}}$. The Weil pairing on the right is computed by Merel; we pin down the relation to Merel’s computation in §6.2. Taken together, the Proposition and this computation prove Lemma 5.3.

Proof.

The first equality is straightforward: an explicit representative for $Q_{\beta}\in J_{p}\simeq H^{1}(X,\mu_{p})$ is given by the $\mu_{p}$-torsor associated to $f:=\Delta(qz)/\Delta(z)$, because the divisor of $f$ is $pQ_{\beta}$.

We now discuss the second equality. We will compare everything to the cup product in flat cohomology, i.e.

[TABLE]

There is a degree map $H^{2}_{\mathrm{fl}}(X,\mu_{p})\rightarrow\mathbb{F}_{p}$; let us explicate it. On any scheme, the sequence $\mu_{p}\rightarrow\mathbb{G}_{m}\rightarrow\mathbb{G}_{m}$ induces an exact sequence of represented sheaves for the flat topology. This identifies the flat cohomology of $\mu_{p}$ with the hypercohomology of $[\mathbb{G}_{m}\stackrel{{\scriptstyle x\mapsto x^{p}}}{{\longrightarrow}}\mathbb{G}_{m}]$.

Let $X_{\overline{{\mathbb{Z}}_{p}}}$ be the base change of $X$ to $\overline{{\mathbb{Z}}_{p}}$ (the Witt vectors of $\overline{\mathbb{F}_{p}}$). We obtain an exact sequence

[TABLE]

Flat and étale cohomology of $\mathbb{G}_{m}$ coincide (see [***]), and the right-hand side is a subgroup of the Brauer group of $X_{\overline{{\mathbb{Z}}_{p}}}$, which vanishes (Théorème 3.1 of [10]). Accordingly, any class in $H^{2}_{\mathrm{fl}}(X_{\overline{{\mathbb{Z}}_{p}}},\mu_{p})$ is the coboundary of a line bundle, and computing degree gives the desired homomorphism

[TABLE]

We see that $\mathrm{deg}(\alpha\cup\beta)=\langle\alpha_{\mathrm{et}},\beta_{\mathrm{et}}\rangle_{\mathrm{et}}$ and so it remains to see

[TABLE]

Let $\pi$ be the morphism from the flat site on $X_{\overline{\mathbb{F}_{p}}}$ to the étale site. As a reference for what follows, we refer to the paper of Artin and Milne [2]. We have isomorphisms:

[TABLE]

[TABLE]

where $\mathrm{F}$ and $\mathrm{C}$ are, respectively, the Frobenius and Cartier maps. and Artin–Milne show that the pairing ${\mathbb{Z}}/p{\mathbb{Z}}\times\mu_{p}\rightarrow\mu_{p}$ induces, after push-forward, the “obvious” pairing on the complexes on the right, which can be computed in the Zariski topology, because flat and Zariski cohomology coincide for quasi-coherent sheaves.

For the same reason, the second identification induces an isomorphism

[TABLE]

where the map $H^{1}(\Omega^{1})\rightarrow\overline{\mathbb{F}_{p}}$ comes from Serre duality. Moreover, the resulting identification is simply the degree map, alluded to above; this comes down to the fact that the map

[TABLE]

again computes the degree of a line bundle modulo $p$.

With respect to the resulting identification of $H^{1}_{\mathrm{fl}}(X,\mu_{p})\simeq\mathbb{H}^{0}(\Omega^{1}\stackrel{{\scriptstyle 1-\mathrm{C}}}{{\rightarrow}}\Omega^{1})$, and the Čech representation of this last hypercohomology, the class $\beta$ is represented by $\frac{df}{f}\in\check{C}^{0}(\Omega^{1})$, which has zero boundary and which is annihilated on the nose by $1-\mathrm{C}$. Similarly the class $\alpha$ in étale cohomology is represented by a Cech cocycle $c^{1}\in\check{C}^{1}(\mathcal{O})$ together with a class $c^{0}\in\check{C}^{0}(\mathcal{O})$ satisfying $(1-\mathrm{F})c^{1}=dc^{0}$. The image of the pairing $\alpha\cup\beta\in H^{2}_{\mathrm{fl}}(\mu_{p})$, under the map $H^{2}_{\mathrm{fl}}(\mu_{p})\rightarrow H^{1}(\Omega^{1})^{\mathrm{C}}$, is represented by $c^{1}\cdot\frac{df}{f}\in\check{C}^{1}(\Omega^{1})$; its image by the trace pairing is the usual Serre duality pairing between the cohomology classes of $c^{1}$ and $\frac{df}{f}$. This concludes the proof. ∎

6.2. Merel’s computation

Although routine, we write out the details involving $\langle P_{\alpha},Q_{\beta}\rangle$ to be sure of factors involving $\gcd(q-1,12)$. In what follows, we understand our modular curves to be considered over an algebraically closed field of characteristic zero.

Recall that $P_{\alpha}$ is an element of $\mathrm{Hom}(\mu_{p}\langle-1\rangle,J_{p})$. Thus the Weil pairing $\langle P_{\alpha},Q_{\beta}\rangle\in\mathbb{F}_{p}\langle 1\rangle$ has the property that

[TABLE]

where, on the left hand side we have the “usual” Weil pairing of two torsion points in $J_{p}$.

Following Merel, let $\nu$ be the gcd of $q-1$ and $12$; let $n=\frac{q-1}{\nu}$. Let $U\subset({\mathbb{Z}}/q)^{*}$ be the subgroup of $\nu$th powers; the map $({\mathbb{Z}}/q)^{*}\rightarrow\mathbb{F}_{p}\langle 1\rangle$ factors through the $\nu$th power map, and we get a sequence

[TABLE]

The Galois group of the covering $X_{1}(q)\rightarrow X_{0}(q)$ can be identified with $U$ (as in §3.3 [14]). This gives rise to a map

[TABLE]

Also $Q^{\prime}=(\infty)-(0)$ is $n$-torsion in the divisor class group, thus defining another class in $J_{n}$. Then Merel shows that

[TABLE]

where the equality is in $\mu_{n}$ and the subscript $n$ means we are using the Weil pairing at the $n$-torsion level.

We want to compare $\alpha^{\prime}$ to $P_{\alpha}$. Note that if $t\in\mathrm{Hom}(U,\mu_{n})$ the power $t^{n/p}$ defines an element of $\mathrm{Hom}(U,\mu_{p})$ which, considered as an element of $\mathrm{Hom}(({\mathbb{Z}}/q)^{*},\mu_{p})$, factors through $\mathbb{F}_{p}\langle 1\rangle$. We refer to the resulting element as $\bar{t}\in\mathrm{Hom}(\mathbb{F}_{p}\langle 1\rangle,\mu_{p})$. Explicitly, if $\mu\in({\mathbb{Z}}/q)^{*}$, we have

[TABLE]

Now consider the commutative diagram (where we write $X=X_{0}(q)$ for short)

[TABLE]

When we evaluate at the element of $H^{1}(X,U)$ corresponding to the cover $X_{1}(q)\rightarrow X_{0}(q)$, the top horizontal map becomes $\alpha^{\prime}$ and the bottom map becomes $P_{\alpha}$ from (6.1). Thus we have

[TABLE]

Pairing with $Q_{\beta}=\frac{q-1}{p}Q^{\prime}\in J_{p}$ and comparing with (6.3):

[TABLE]

and so

[TABLE]

where the equality is once again in $\mu_{p}$. We conclude that $\langle P_{\alpha},Q_{\beta}\rangle$ is indeed the image of $\varpi_{\mathrm{Merel}}$ inside $\mathbb{F}_{p}\langle 1\rangle$.

7. Comparison with the theory of [17]

Derived Hecke operators at Taylor-Wiles primes have been defined abstractly for general $q$-adic groups in [17]. The purpose of the present section is to identify the operators introduced in 3.1 with those defined in [17]. (The results of this section are, strictly speaking, not used elsewhere in the paper; however they show that all the constructions we have made are inevitable.)

Write $G=\mathrm{GL}_{2}({\mathbf{Q}}_{q})$ where $q\equiv 1\pmod{p}$, and $K=\mathrm{GL}_{2}({\mathbb{Z}}_{q})$. Fix a base ring $S$ that is a ${\mathbb{Z}}_{p}$-algebra.

What we will need to do, in order to study the derived Hecke operator at $q$, is to identify the cohomology of the modular curve with the cohomology of the $K$-invariants of a complex of $G$-representations. Unsurprisingly, this is done by adding infinite level at $q$; we just pin down the details. We need to take a little care because the tower of coverings that one gets by adding infinite $q$-level is not étale; however, its ramification is prime to $p$, which will be enough for our purposes.

In particular, we will use666with an apology to 21st century readers, see below… Lemma A.10 of Appendix A of [17], which explicates the action of the abstract derived Hecke algebra in terms of restrictions, corestrictions, and cup products.

7.1. Construction of complexes with an action of $\mathrm{GL}_{2}({\mathbf{Q}}_{q})$.

Let us fix a level structure away from $q$ for the usual modular curve, i.e., an open compact subgroup $K^{(q)}\subset\mathrm{GL}_{2}(\mathbb{A}^{(\infty,q)})$. We require that $K^{(q)}=\prod_{v\neq q}K_{v}$, where $K_{v}$ is hyperspecial maximal for almost all $v$.

For $U\subset\mathrm{GL}_{2}({\mathbf{Q}}_{q})$ an open compact subgroup, let $X(U)$ be the Deligne–Rapoport compactification of the modular curve with level structure $K^{(q)}\times U$. This again has (Deligne–Rapoport) a smooth proper model over $\mathrm{Spec}\ S$, denoted $X(U)_{S}$. We denote again by $\omega_{U}\rightarrow X(U)$ the relative cotangent bundle of the universal elliptic curve; this defines a locally free sheaf over $X(U)_{S}$.

Let us consider the pro-system of schemes

[TABLE]

indexed by the collection of all open compact subgroups of $\mathrm{GL}_{2}({\mathbf{Q}}_{q})$; the maps are inclusions $V\subset U$ of open compact subgroups.

The isomorphisms $X(g^{-1}Ug)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}X(U)$ induce an action of $G=\mathrm{GL}_{2}({\mathbf{Q}}_{q})$ on $X_{\infty}$ (considered as a pro-object in the category of schemes). Let $\omega_{\infty}$ be the “vector bundle” over $X_{\infty}$ defined by $\omega$: by this we mean that $\omega_{\infty}$ is a pro-scheme over $X_{\infty}$, which is level-wise a vector bundle.

We will need the following properties:

• (i)

The action of $G$ on $X_{\infty}$ lifts to an action on $\omega_{\infty}$.

• (ii)

Suppose that $V$ is a normal subgroup of $U$. Then the natural map

[TABLE]

is finite, and identifies $X(U)_{S}$ with the quotient of $X(V)_{S}$ by $U/V$ in the category of schemes. (See [5, 3.10]).

Moreover, there is a natural (in $S$) identification $f_{UV}^{*}\omega_{U}\simeq\omega_{V}$.

• (iii)

With notation as in (ii), if the order of $U/V$ is a power of $p$, then the map $X(V)_{S}\rightarrow X(U)_{S}$ is étale.

Proof.

(of (iii) only:) We may suppose that $S={\mathbb{Z}}_{p}$. The map is étale over the interior of the modular curve, so, by purity of the branch locus, it is enough to check that it is étale at the cusps in characteristic zero. The cusps of a modular curve are parameterized by an adelic quotient, but replacing the role of an upper half-plane by $\mathbb{P}^{1}({\mathbf{Q}})$; so we must verify that the map

[TABLE]

considered as a morphism of groupoids, induces isomorphisms on each isotropy group.

Let $\mathbf{B}$ be a Borel subgroup in $\mathrm{GL}_{2,/{\mathbf{Q}}}$ and $\mathbf{N}$ its unipotent radical. We can identify $\mathbb{P}^{1}({\mathbf{Q}})$ with $\mathrm{GL}_{2}({\mathbf{Q}})/\mathbf{B}({\mathbf{Q}})$. The desired result follows, then, if for each $g\in\mathrm{GL}_{2}(\mathbb{A}_{f})$ we have

[TABLE]

However the projection of $\mathbf{B}({\mathbf{Q}})\cap gUg^{-1}$ to the toral ${\mathbf{Q}}^{*}$ is a finite subgroup of ${\mathbf{Q}}^{*}$, thus contained in $\{\pm 1\}$. It follows that an index $2$ subgroup of the left-hand side is contained in $\mathbf{N}({\mathbf{Q}})\cap gUg^{-1}$, which is certainly contained in $gVg^{-1}$ because any open compact of $\mathbf{N}({\mathbf{Q}}_{q})$ is pro-$q$. ∎

Lemma 7.2**.**

Suppose that, as above, $V$ is a normal subgroup of $U$. Let $f=f_{UV}$ be as in (ii) above. Let $\mathcal{F}$ be any sheaf of $\mathcal{O}_{X(V)}$-modules on $X(V)$, equipped with a compatible action of $U/V$.

Then

• (i)

For each $x\in X(V)$, the higher cohomology of the stabilizer $(U/V)_{x}$ on $\mathcal{F}_{x}$ is trivial.

• (ii)

For each $y\in X(U)$, the higher cohomology of $(U/V)$ acting on $(\pi_{*}\mathcal{F})_{y}$ is trivial.

Proof.

Note that we can reduce (i) to the case when $(U/V)_{x}=(U/V)$ by shrinking $U$. Both (i) and (ii) will follow, then, if we prove that for any $U/V$-stable affine set $\mathrm{Spec}(A)\subset X(V)$,

[TABLE]

since the stalks appearing in (i) and (ii) are direct limits of such spaces.

Let $\Delta=U_{1}/V$ be a Sylow $p$-subgroup of $U/V$; it is sufficient to make the same verification for the higher cohomology of $\Delta$. Write $B=A^{\Delta}$. The map $\mathrm{Spec}(A)\rightarrow\mathrm{Spec}(B)$ is finite étale with Galois group $\Delta$, by (iii) above. It is now sufficient to show:

If $M$ is an $A$-module, equipped with a $\Delta$-action compatible with its module structure, then $H^{q}(\Delta,M)=0$ for $q>0$.

Let $M^{\prime}=M\otimes_{B}A$; define a $\Delta$-action on $M^{\prime}$ using $g(m\otimes a)=gm\otimes a$ for $g\in\Delta$. Since $A$ is a flat $B$-module, the natural map $H^{q}(\Delta,M)\otimes_{B}A\rightarrow H^{q}(\Delta,M^{\prime})$ is an isomorphism. We shall show $H^{q}(\Delta,M^{\prime})=0$; the vanishing of $H^{q}(\Delta,M)$ follows from faithful flatness.

Now $M^{\prime}$ is a module over $A\otimes_{B}A\simeq\prod_{\delta\in\Delta}A$, and this module structure is compatible with the $\Delta$-action on $\prod_{\delta\in\Delta}A$ which permutes the factors. Therefore, $M^{\prime}$ is induced (as a $\Delta$-module) from a representation of the trivial group, and thus has vanishing higher $\Delta$-cohomology by Shapiro’s lemma. ∎

7.3. Godement resolution

Let $T$ be the “Godement functor”, which assigns to a sheaf $\mathcal{F}$ the sheaf $U\mapsto\prod_{x\in U}\mathcal{F}_{x}$ of discontinuous sections. It carries a sheaf of $\mathcal{O}$-modules to another sheaf of $\mathcal{O}$-modules.

We will need to discuss the behavior under images. Suppose given a map $f:X^{\prime}\rightarrow X$ of schemes. There is a map of functors

[TABLE]

For a sheaf $\mathcal{F}$ on $X$ and an open set $V\subset X$, this is given by the natural pullback of discontinuous sections

[TABLE]

If we are working with sheaves of $\mathcal{O}$-modules, then, composing with the natural $f^{-1}\rightarrow f^{*}$, we get $T\rightarrow f_{*}Tf^{*}$, or, what is the same by adjointness, a natural transformation

[TABLE]

In particular, for a sheaf $\mathcal{F}$ on $X$, there is a map

[TABLE]

This gives rise to the pullback map in cohomology $H^{*}(X,\mathcal{F})\rightarrow H^{*}(X^{\prime},f^{*}\mathcal{F})$.

7.4.

It follows from Lemma 7.2 that (with notations as in that Lemma and) for any sheaf $\mathcal{F}$ of $\mathcal{O}_{X(V)}$-modules,

[TABLE]

Indeed group cohomology commutes with products (even infinite ones).

Now let $\mathcal{G}^{\bullet}(U)$ be the Godement resolution of $\omega_{U}$. It is a complex of sheaves of $\mathcal{O}_{X(U)}$-modules on $X(U)_{S}$. Let $M^{\bullet}(U)$ be the global sections of $\mathcal{G}^{\bullet}(U)$: this is a complex of $S$-modules. If $V\subset U$, there is a natural action of $U/V$ on $M^{\bullet}(V)$. It follows from (7.3) that

Lemma 7.5**.**

For each degree $i$, the $U/V$-cohomology of $M^{i}(V)$ vanishes, i.e. $H^{p}(U/V,M^{i}(V))=0$ for $p>0$.

The following result is the crucial one for us.

Lemma 7.6**.**

The map arising from (7.2)

[TABLE]

(where $U/V$ denotes invariants) induces on global sections a quasi-isomorphism

[TABLE]

Proof.

It is enough to verify that (7.4) is a quasi-isomorphism: the sheaves $\mathcal{G}^{\bullet}(U)$ and $f_{*}\mathcal{G}^{\bullet}(V)^{U/V}$ are flasque – the latter follows just by examining the definition of the Godement functor $T$ – and so taking global sections will preserve the quasi-isomorphism.

Consider the following diagram:

[TABLE]

The left vertical arrow is a quasi-isomorphism: we have an isomorphism $f_{*}\omega_{V}\simeq\omega_{U}\otimes f_{*}\mathcal{O}_{V}$, and $(f_{*}\mathcal{O}_{V})^{U/V}=\mathcal{O}_{U}$. The top horizontal arrow is also a quasi-isomorphism. It then suffices to show that the arrow $j$ is also a quasi-isomorphism.

The complex $f_{*}\mathcal{G}^{\bullet}(V)$ is a resolution of $f_{*}\omega_{V}$ because $f_{*}$ has no higher cohomology on the quasi-coherent sheaf $\omega_{V}$. Next the stalks of $f_{*}\omega_{V}$ and $f_{*}\mathcal{G}^{\bullet}(V)$ have vanishing $U/V$-cohomology by Lemma 7.2. Given an acyclic complex of $U/V$-modules supported in degrees $\geq 0$, each of which have no higher $U/V$-cohomology, the $U/V$-invariants remain acyclic. This implies that $f_{*}\mathcal{G}^{\bullet}(V)^{U/V}$ is a resolution of $(f_{*}\omega_{V})^{U/V}$ as desired. ∎

7.7. Compatibility with traces

We must also mention the compatibility with trace maps. Suppose we are given a subgroup $U^{\prime}$ intermediate between $U$ and $V$:

[TABLE]

We don’t require that $U^{\prime}$ be normal.

There is a natural trace map

[TABLE]

Explicitly the trace $f_{*}\mathcal{O}_{X(U^{\prime})}\rightarrow\mathcal{\mathcal{missing}}{O}_{X(U)}$ induces

[TABLE]

With reference to the identifications of the previous lemma, this trace map is induced at the level of cohomology by

[TABLE]

where $\mathrm{T}\in S[U/V]$ is the sum of a set of coset representatives for $U/U^{\prime}$.

7.8. Derived invariants and the derived Hecke algebra

As in §7.4, $M^{\bullet}(U)$ is a Godement complex computing the complex of $\omega_{U}$ Now set

[TABLE]

which is now a complex of $S$-modules equipped with an action of $G=\mathrm{GL}_{2}({\mathbf{Q}}_{q})$.

We will argue that the “derived invariants” of $U$ on $M_{\infty}^{\bullet}$ gives a complex that computes the cohomology of $X(U)_{S}$. We first recall the notion of derived invariants, and its relationship with the derived Hecke algebra.

Let $U$ be an open compact subgroup of $G$. Let $U_{1}\subset U$ be a normal subgroup with the property that the pro-order of $U_{1}$ is relatively prime to $p$. Let $\mathbf{Q}$ be a projective resolution of $S$ in the category of $S[U/U_{1}]$-modules; we regard this as a complex with degree-increasing differential concentrated in degrees $\leq 0$:

[TABLE]

We may of course regard $\mathbf{Q}$ as a complex of $S[U]$-modules.

Let $\mathbf{P}=\mathrm{ind}_{U}^{G}\mathbf{Q}$. This is a projective resolution of the smooth $S[G]$ module $S[G/U]$ (in the category of smooth $S[G]$ modules). For any complex $R^{\bullet}$ of $G$-modules, we define the derived $U$-invariants to be the complex

[TABLE]

Explicitly, this is a complex whose cohomology computes the hypercohomology $\mathbb{H}^{*}(U,R^{\bullet})$.

In the case above, the derived invariants of $U$ on $M^{\bullet}_{\infty}$ compute the cohomology of $X(U)$, in the following sense:

Lemma 7.9**.**

The natural inclusion of $M^{\bullet}(U)\hookrightarrow M^{\bullet}_{\infty}$ and the augmentation $\mathbf{Q}\rightarrow S$ induce a quasi-isomorphism:

[TABLE]

Proof.

Using the remarks after (7.5), we see that

[TABLE]

(for the second arrow: since $U_{1}$ is prime to $p$ the functor of taking $U_{1}$ invariants commutes with taking a direct limit of smooth $S[U_{1}]$-modules).

The inclusion $M^{\bullet}(U)\hookrightarrow M^{\bullet}(U_{1})$ and the homomorphism $\mathbf{Q}\rightarrow S$ induce

[TABLE]

and it remains to show that this composite is a quasi-isomorphism.

The first map is a quasi-isomorphism by Lemma 7.6. To show that the second map is a quasi-isomorphism, it is enough (by a devissage) to show that for each fixed degree $j$

[TABLE]

induces a quasi-isomorphism. But the right hand side computes the $U/U_{1}$ cohomology of $M^{j}(U_{1})$, and we have seen (Lemma 7.5) that this is concentrated in degree zero, where it is just the $U/U_{1}$ invariants, as needed. ∎

Now we may imitate all the reasoning above, with the role of $\omega$ replaced by $\mathcal{O}$. Let $N^{\bullet}$ be the corresponding complex. Reasoning as in Lemma 7.9, we get a quasi-isomorphism

[TABLE]

The identification of $S$ with global sections of $\mathcal{O}_{X(V)}$ induce compatible maps $S\rightarrow N^{\bullet}(V)$ for each level $V$, and so by passage to the limit a map $S\hookrightarrow N^{\bullet}_{\infty}$. This induces

[TABLE]

For $\alpha\in H^{j}(U,S)$ write $\langle\alpha\rangle\in H^{j}(X(U),\mathcal{O})$ for its image under this map. Then we have:

Lemma 7.10**.**

Under the identification $H^{*}(X(U),\omega_{U})$ with the hypercohomology $\mathbb{H}^{*}(U,M^{\bullet}_{\infty})$, (as in the prior Lemma), cup product with $\langle\alpha\rangle$ in Zariski cohomology is carried to cup product with $\alpha$ in hypercohomology.

Proof.

The product $\mathcal{O}\otimes\omega_{U}\rightarrow\omega_{U}$ extends to a map $N^{\bullet}(U)\otimes M^{\bullet}(U)\rightarrow M^{\bullet}(U)$ (see [8, Chapter 6]), which computes on cohomology the cup product. This exists compatibly at every level, and by passage to the direct limit, we arrive at a map $N^{\bullet}_{\infty}\otimes M^{\bullet}_{\infty}\rightarrow M^{\bullet}_{\infty}$ (the tensor product can be passed through the direct limit, by [BourbakiLinearAlgebra, Chapter 2, Prop. 7, §6.3]).

Fix a quasi-isomorphism of $S[U]$-modules:

[TABLE]

Consider the following diagram, with commutative squares:

[TABLE]

where:

• •

$\mathrm{Hom}$ means in every case homomorphisms of chain complexes of $S[U]$ modules, taken modulo chain homotopy;

• •

$\otimes$ comes from the tensor product, which induces a bifunctor on the homotopy category of chain complexes.

• •

We fix $m\in\mathrm{Hom}^{i}(\mathbf{Q},M^{\bullet}_{\infty})$, amd $m^{\prime}$ is the cohomology class corresponding to $m$ under the quasi-isomorphism (7.6).

• •

We identify $\alpha$ with a class in $\mathrm{Hom}^{j}(\mathbf{Q},S)$ and $\alpha^{\prime}$ is the image of this class, under $S\rightarrow N^{\bullet}_{\infty}$. Also $\alpha^{\prime\prime}$ is a cohomology class in $H^{j}(N^{\bullet}(U))$ that matches with $\alpha^{\prime}$ under the quasi-isomorphism (7.8).

The image of $m\otimes\alpha$, under the bottom horizontal arrows, computes the cup product of $m$ and $\alpha$ in $U$-hypercohomology. This corresponds to the image of $m\otimes\alpha^{\prime}$ in the middle horizontal row. Finally, this corresponds to the image of $m^{\prime}\otimes\alpha^{\prime\prime}$ in the top row, which gives the Zariski product. ∎

7.11. Derived Hecke algebra

Let notation be as above, but specialized to the case $U=K$, a maximal compact of $\mathrm{GL}_{2}({\mathbf{Q}}_{q})$. We may form the differential graded algebra $\mathrm{End}_{S[G]}(\mathbf{P})$ whose cohomology we understand to be the (graded) derived Hecke algebra for the pair $(G,K)$. There is an isomorphism ([17], (148))

[TABLE]

where $\mathbf{Q}_{x}$ is the complex $\mathbf{Q}$ but with the twisted action of $K_{x}=K\cap\mathrm{Ad}(g_{x})K$ defined by $\kappa\ast q=(Ad(g_{x}^{-1})\kappa)q$; here we have implicitly chosen coset representatives $g_{x}K$ for each $x\in K\backslash G/K$. Taking cohomology, one finds that, for any $i$ there is an isomorphism ([17], (149))

[TABLE]

Now the differential graded algebra $\mathrm{End}_{S[G]}(\mathbf{P},\mathbf{P})$ acts on $\mathrm{Hom}_{S[G]}(\mathbf{P},M_{\infty}^{\bullet})$. Passing to cohomology and applying Lemma 7.9, we get a graded action of the derived Hecke algebra for $(G,K)$ on $H^{*}(X_{K},\omega_{K})$. This action is specified by specifying, for each $x=Kg_{x}K\in K\backslash G/K$ as above, the corresponding action of $H^{*}(K_{x},S)$ on coherent cohomology. We can now restate Lemma A.10 of [17]:

Lemma 7.12**.**

The action of $h_{x}\in H^{*}(K_{x},S)$ on $\mathbb{H}^{*}(K,M_{\infty}^{\bullet})$ is given explicitly by the following composite:

[TABLE]

We obtain the derived Hecke operator $T_{q,z}$ described in §3, with the coefficient ring $S=\mathcal{O}/\mathfrak{p}^{n}$, by taking x=\left(\begin{array}[]{cc}q&0\\ 0&1\end{array}\right) and by taking the cohomology class $h_{x}\in H^{1}(K_{x},S)$ as the composite:

[TABLE]

where $z\in k\langle-1\rangle$ is regarded as a homomorphism $({\mathbb{Z}}/q)^{*}\rightarrow\mathcal{O}/\mathfrak{p}^{n}$. Indeed, to verify this, it only remains to show that the induced map

[TABLE]

given by cupping with the class $h_{x}$ is identified with

[TABLE]

that is to say the cup product with $z\mathfrak{S}$, i.e. the Shimura class multiplied by $z$, regarding as a class in the cohomology of $X(K_{x})$ with coefficients in $\mathcal{O}/\mathfrak{p}^{n}$. This follows easily from Lemma 7.10.

The following remark is due entirely to the first-named author (M.H); the second-named author disclaims both credit and responsibility for it.

Remark 7.1**.**

For the benefit of those millenials who believe the Godement resolution is one of the founding documents of the United Nations, here is a translation of the above construction into contemporary language. We thank Nick Rozenblyum for his patient guidance. We work in the DG category (or stable $\infty$-category) $C$ of complexes of quasicoherent sheaves on the scheme $X_{\infty}$, and consider the object $\omega_{\infty}$, all over $Spec(S)$. This object carries an action by $G=GL_{2}(\mathbf{Q}_{q})$. Therefore the object $R\Gamma(\omega_{\infty})$ in the DG category $Mod_{S}$ of complexes of $S$-modules carries an action of $G$. Everything up through Lemma 7.10 is automatic in this setting. The remaining observations are not strictly necessary to formulate the conjecture; however, they do provide the explicit computation of the derived Hecke operator, as in Lemma 7.12, needed in order to test the conjecture in specific applications.

8. Magma Code

What follows is a sample of Magma code which we used to compute the derived Hecke operator for the modular form of level $31$, with $q=139$ and $p=23$.

N := 31; Q := 139; L := 23; F := FiniteField(L); M := ModularForms(NQ); S := CuspidalSubspace(M); SQ := BaseExtend(S, RationalField()); SF := BaseExtend(S, F); V, h := VectorSpace(SF); time Tq := HeckeOperator(SF,N); time Wq := AtkinLehnerOperator(SF,N); Iq := IdentityMatrix(F, Dimension(S)); Qq := Iq +WqTq; /* Qq projects from level QN back down to level Q */ Pro := Dimension(S); Z<q> := PowerSeriesRing(IntegerRing()); QQ<q> := PowerSeriesRing(RationalField());

CUTOFF := Dimension(S)+3;

eps := KroneckerCharacter(-N); WeightOneSpace := ModularForms(eps, 1); etatemp := WeightOneSpace.2; etaprodA := qExpansion(etatemp, CUTOFF); etaprodB := Composition(etaprodA, q^Q+O(q^CUTOFF)); g := etaprodA * etaprodB + O(q^CUTOFF);

g0 := SF ! g; W := Vector(F, Inverse(h)(g0)); Wfin := W * Qq; /*denom := Denominator(Wfin); print(Factorization(denom)); */ M2 := ModularForms(Q); S2 := CuspidalSubspace(M2); S2Q := BaseExtend(S2, RationalField()); S2F := BaseExtend(S2, F); V2,h2 := VectorSpace(S2F); CUTOFF2 := Dimension(S2)+3; projform := S2F ! h(Wfin); projformcoeff := Vector(F, Inverse(h2)(projform)); normcoeffF := projformcoeff; randprime := 41; randT := HeckeOperator(S2F, randprime); charpoly := CharacteristicPolynomial(randT); P<u>,h3 := ChangeRing(PolynomialRing(IntegerRing()), F); Factorization(P! charpoly); unnormalizedredpoly := charpoly/(u-randprime-1); redpoly := unnormalizedredpoly/Evaluate(unnormalizedredpoly,randprime+1); print(Evaluate(redpoly, randprime+1)); projmatrix := Evaluate(P! redpoly, randT); finalanswerinbasis := normcoeffF * projmatrix; print(finalanswerinbasis);