Correspondences in Arakelov geometry and applications to the case of Hecke operators on modular curves

TL;DR

This paper develops a framework for analyzing Hecke operators on modular curves within Arakelov geometry, introducing new invariants and proving self-adjointness properties using advanced intersection theory and L-series computations.

Contribution

It extends Arakelov geometry by defining Hecke actions on arithmetic Chow groups and introduces refined invariants linked to L-series, with applications to modular curves.

Findings

Hecke operators act self-adjointly on the arithmetic Chow groups of modular curves.

New numerical invariants are defined and computed using L-series and the Gross-Zagier formula.

Hecke correspondences on Jacobians are self-adjoint with respect to the Néron-Tate pairing.

Abstract

In the context of arithmetic surfaces, Bost defined a generalized Arithmetic Chow Group (ACG) using the Sobolev space L^2_1. We study the behavior of these groups under pull-back and push-forward and we prove a projection formula. We use these results to define an action of the Hecke operators on the ACG of modular curves and to show that they are self-adjoint with respect to the arithmetic intersection product. The decomposition of the ACG in eigencomponents which follows allows us to define new numerical invariants, which are refined versions of the self-intersection of the dualizing sheaf. Using the Gross-Zagier formula and a calculation due independently to Bost and Kuehn we compute these invariants in terms of special values of L series. On the other hand, we obtain a proof of the fact that Hecke correspondences acting on the Jacobian of the modular curves are self-adjoint with respect to the N\'eron-Tate height pairing.

Equations

$${\mu }_{Ar}:=\frac{i}{n}\sum _{j=1}^n{\omega }_j\wedge \overline{{\omega }_j}.$$

$$dd^{c}g+\delta_{D(\mathbb{C})}=\big{(}\deg D(\mathbb{C})\big{)}\mu_{Ar}.$$

$$g_1\ast g_2=g_1(d{d}^c{g}_2+{\delta }_{D_2(\mathbb{C})})+g_2{\delta }_{D_1(\mathbb{C})}.$$

$$g=-a_P\log |{\theta }_P(\cdot ){|}^2+b,$$

$${\phi }^{\ast }:\widehat{C{H}^1}(\mathcal{Y})⟶\widehat{C{H}^1}(\mathcal{X})$$

$${\phi }_{\ast }:\widehat{C{H}^1}(\mathcal{X})⟶\widehat{C{H}^1}(\mathcal{Y}).$$

$$<{\phi }^{\ast }y,x>=<y,{\phi }_{\ast }x>.$$

$$dd^{c}g+\delta_{D(\mathbb{C})}=\big{(}\deg D(\mathbb{C})\big{)}\mu$$

$${\phi }^{\ast }:\widehat{C{H}^1}(\mathcal{Y}{)}_{\nu }⟶\widehat{C{H}^1}(\mathcal{X}{)}_{\frac{{\phi }^{\ast }\nu }{\mathrm{deg}\phi }}$$

$${\phi }_{\ast }:\widehat{C{H}^1}(\mathcal{X}{)}_{\mu }⟶\widehat{C{H}^1}(\mathcal{Y}{)}_{{\phi }_{\ast }\mu }.$$

$${\widehat{CH}}_{\mathbb{R}}^{num}(N)=\text{Eis}\stackrel{\perp }{\oplus }J,$$

$$J\cong J_0(N)(\mathbb{Q})\otimes \mathbb{R}$$

$$J=\underset{f}{⨁}{J}_f,$$

$$\hat{\omega }={\hat{\omega }}_{\text{Eis}}+{\hat{\omega }}_J,{\hat{\omega }}_J=\sum _f{\hat{\omega }}_f.$$

$${\hat{\omega }}^2={\hat{\omega }}_{\text{Eis}}^2+{\hat{\omega }}_J^2,{\hat{\omega }}_J^2=\sum _f{\hat{\omega }}_f^2.$$

$$\displaystyle\frac{\hat{\omega}_{\emph{{Eis}}}^{2}}{(g_{N}-1)^{2}}=\frac{576}{[\Gamma_{0}(1):\Gamma_{0}(N)]}\Big{(}\frac{1}{2}\zeta(-1)+\zeta^{\prime}(-1)\Big{)}-\sum_{p|N}\frac{\log p}{g_{N}-2g_{\frac{N}{p}}+1},$$

$$\hat{\omega}_{f}^{2}=-\Big{(}h_{NT}\big{(}(H_{i})_{f}\big{)}^{1/2}+2h_{NT}\big{(}(H_{j})_{f}\big{)}^{1/2}\Big{)}^{2}.$$

$$h_{NT}(H_i{)}_f=\frac{1}{2{\pi }^2(f,f)}L(f,{\chi }_{-4},1)L^{\prime }(f,1),h_{NT}(H_j{)}_f=\frac{\sqrt{3}}{4{\pi }^2(f,f)}L(f,{\chi }_{-3},1)L^{\prime }(f,1),$$

$$(f,g{)}_{1,X}=i{\int }_X\partial f\wedge \overline{\partial g}.$$

$$\Vert f{\Vert }_{1,X}^2=i{\int }_X\partial f\wedge \overline{\partial f}\in {\mathbb{R}}_{\ge 0}.$$

$$(f,g)\mapsto {\int }_{X}f\bar{g}\nu +(f,g{)}_{1,X}.$$

$$(f,g{)}_{1,U}:=i{\int }_U\partial f\wedge \overline{\partial g}.$$

$$2\pi {\int }_{U}fd{d}^c\bar{g}=-(f,g{)}_{1,U}.$$

$${\phi }_{\ast }g(y)=\sum _{x\in {\phi }^{-1}(y)}{e}_{x}g(x),$$

$${\phi }_{\ast }g(z)=\sum _{{\rho }^n=1}g(\rho z^{1/n})\text{almost everywhere.}$$

$$\frac{\partial {\phi }_{\ast }g}{\partial z}(z)=\frac{z^{(1/n)-1}}{n}h(z^{1/n})\text{ almost eeverywhere, where }h(z):=\sum _{{\rho }^n=1}\rho \frac{\partial g}{\partial z}(\rho z).$$

$$\displaystyle i\int_{\mathbb{D}}\Big{|}\frac{\partial\varphi_{*}g}{\partial z}(z)\Big{|}^{2}dz\wedge d\bar{z}$$

$${\phi }^{\ast }:C^{\infty }(Y)\to C^{\infty }(X),{\phi }_{\ast }:C^{\infty }(X)\to L_1^2(Y)$$

$${\phi }^{\ast }:L_1^2(Y)\to L_1^2(X),{\phi }_{\ast }:L_1^2(X)\to L_1^2(Y).$$

$${\phi }^{\ast }:L_{-1}^2(Y)\to L_{-1}^2(X),{\phi }_{\ast }:L_{-1}^2(X)\to L_{-1}^2(Y),$$

$$({\phi }^{\ast }f,g{)}_{1,U}=(f,{\phi }_{\ast }g{)}_{1,\phi (U)}.$$

Full text

1. 1 Introduction

2. 2 Functoriality in $L^{2}_{1}$

3. 2.1 The space $L^{2}_{1}$ and related spaces

4. 2.2 Pull-back and push-forward in the space $L^{2}_{1}$

5. 2.3 Pull-back and push-forward of $L^{2}_{1}$-Green functions

6. 3 Pull-back and push-forward in Arakelov theory

7. 3.1 The $L^{2}_{1}$-arithmetic Chow group, the admissible arithmetic Chow group and intersection theory

8. 3.2 Functoriality with respect to pull-back and push-forward

9. 4 Hecke correspondences on modular curves

10. 4.1 Integral models of the modular curves $X_{0}(N)$ and Hecke correspondences

11. 4.2 Arithmetic Chow group of $\mathcal{X}_{0}(N)$

12. 4.3 Arithmetic Chow group of $\mathcal{X}_{0}(N)$ with real coefficients

13. 4.4 Self-intersection and refined self-intersection of the dualising sheaf

14. 4.4.1 Computation of $\hat{\omega}_{\textrm{Eis}}^{2}$

15. 5 Appendix: cohomology of $L^{2}_{1}$ spaces on compact Riemann surfaces

1. Introduction

Let $\mathcal{X}$ be an arithmetic surface, that is a projective, regular, 2-Krull dimensional scheme over Spec$(\mathbb{Z})$. Such an object arises naturally as a model over the integers of a smooth algebraic curve defined over a number field. Arakelov devised a formalism on $\mathcal{X}$ analogous to the classical intersection theory on an algebraic surface defined over a field (Ara (74)). The basic object of this formalism is a compactified divisor, that is a pair $(D,g)$, where $D$ is a Weil divisor on $\mathcal{X}$ and $g$ is a real-valued distribution on the compact Riemann surface $X=\mathcal{X}(\mathbb{C})$. Here, $g$ is a Green function for $D$, subject to the following condition. Suppose for simplicity that the genus of $X$ is positive, and take an orthonormal basis $\omega_{1},\ldots,\omega_{n}$ of the space of holomorphic 1-forms (where we take this space to be endowed with the inner product $(\alpha,\beta)\mapsto\frac{i}{2}\int_{X}\alpha\wedge\overline{\beta}$). The canonical Arakelov 2-form is defined by

[TABLE]

This is a $C^{\infty}$ volume form on $X$. It is independent of the choice of the orthonormal basis $\{\omega_{1},\ldots,\omega_{n}\}$. We impose that $g$ satisfies the equation of currents

[TABLE]

If this condition is satisfied, then $g$ is said to be admissible. We also impose the normalization $\int_{X}g\mu_{Ar}=0$.

The key analytic ingredient needed to define an intersection number from two compactified divisors $(D_{1},g_{1})$ and $(D_{2},g_{2})$ is the star product

[TABLE]

This is a well defined current if $D_{1}(\mathbb{C})$ and $D_{2}(\mathbb{C})$ do not have points in common. The integral $\int_{\mathcal{X}(\mathbb{C})}g_{1}*g_{2}$ is bilinear and symmetric in $(g_{1},g_{2})$.

In Gillet and Soulé’s subsequent treatment of Arakelov’s theory GS (90), the admissibility condition on $g$ is dropped. More precisely, $g$ is a real-valued $C^{\infty}$ function on the complement of the support of $D(\mathbb{C})=\sum_{P}a_{P}[P]$. It is required that for every point $P$ in this support, there is a local expansion of the form

[TABLE]

with $\theta_{P}$ a local chart vanishing at $P$ and $b$ a $C^{\infty}$ function. Such a distribution is called a Green current for $D$. Equation (1.1) implies the local expansion (1.3), making Gillet and Soulé’s theory a generalization of Arakelov’s theory. The star product (1.2) between normalized Green currents retains the symmetry and bilinearity properties.

In this paper we study the lack of functoriality problem in Arakelov theory. That is, let $\varphi:\mathcal{X}\rightarrow\mathcal{Y}$ be a finite morphism between arithmetic surfaces, and $(D,g)$ a compactified divisor on $\mathcal{X}$. If the induced holomorphic covering $\varphi_{\mathbb{C}}:\mathcal{X}(\mathbb{C})\rightarrow\mathcal{Y}(\mathbb{C})$ is ramified, then the pair $(\varphi_{*}D,\varphi_{*}g)$ induced from the the push-forward is not a compactified divisor on $\mathcal{Y}$, in either the Arakelov or the Gillet-Soulé sense. The reason is purely analytic in nature; $\varphi_{*}\mu_{Ar}$ has singularities at the branching points, hence it is not the Arakelov canonical form on $\mathcal{Y}(\mathbb{C})$. Furthermore, $\varphi_{*}g$ is not a Green current for $\varphi_{*}D$, because the push-forward operation does not preserve $C^{\infty}$ functions. In this sense, the theories of Arakelov and Gillet-Soulé are not functorial. In many situations, it is natural to consider correspondences on arithmetic surfaces. Unfortunately, the interaction with the intersection formalism is excluded by this phenomenon.

Bost brought to this circle of ideas the Sobolev space $L^{2}_{1}$, which consists of distributions in $L^{2}$ whose derivatives also belong to $L^{2}$. He proposed a generalization of Gillet-Soulé’s theory for arithmetic surfaces (Bos (99)). In this framework, $g$ is a real-valued element of $L^{2}_{1}$ away from the support of $D(\mathbb{C})$ and for every point $P$ in this support, there is a local expansion similar to (1.3) but where $b$ is in $L^{2}_{1}$. Such an objet is called a $L^{2}_{1}$-Green function. The integral of the star product (1.2) between $L^{2}_{1}$-Green functions is well defined. This theory is optimal in the sense that $L^{2}_{1}$ is the biggest space such that a useful (integral of a) star product can be defined. Bost’s theory also improves on the geometrical part, because it doest not require the scheme $\mathcal{X}$ to be regular but only integral and normal. We drop the regularity in the definition of arithmetic surface in what follows.

Let us write \big{(}\widehat{CH^{1}}(\mathcal{X}),\mathopen{<},\mathclose{>}\big{)} to denote the $L^{2}_{1}$ arithmetic Chow group defined in Bos (99). In this paper, we prove that these groups have a good functoriality. More precisely, we provide a proof of the following

Theorem 1.1**.**

Let $\varphi:\mathcal{X}\rightarrow\mathcal{Y}$ be a generically finite morphism between integral, normal, arithmetic surfaces.

1. (1)

The natural definitions of pull-back and push-forward induce homomorphisms

[TABLE] 2. (2)

Projection formula: for all $x\in\widehat{CH^{1}}(\mathcal{X})$ and $y\in\widehat{CH^{1}}(\mathcal{Y})$, we have that

[TABLE]

The main difficulty in proving this theorem is to control the $L^{2}$ norm of $\varphi^{*}f$ around the ramification points, where $f$ is $L^{2}_{1}$. This is achieved by establishing a Poincaré-like inequality (Lemma 2.10 in the main text), which we present as a variant of the Hardy inequality. On the other hand, the push-forward operation turns out to be easier to handle.

We point out that Theorem 1.1 was known to Bost before our investigation started (Bos (99), p. 245, Bos (98)), though he did not publish any details. The method described in the paragraph above, different from Bost’s approach, was found independently (cf. Remark 2.12).

We also consider a generalization of Arakelov’s condition (1.1). Let L^{2}_{-1}\big{(}\mathcal{X}(\mathbb{C})\big{)} be the space of 2-currents which are locally of the form $(f+\frac{\partial}{\partial z}h_{1}+\frac{\partial}{\partial\bar{z}}h_{2})dz\wedge d\bar{z}$, where $f,h_{1},h_{2}$ are $L^{2}$. Let $\mu$ be a real-valued element of L^{2}_{-1}\big{(}\mathcal{X}(\mathbb{C})\big{)} such that $\int_{\mathcal{X}(\mathbb{C})}\mu=1$. Given a Weil divisor $D$ on $\mathcal{X}$, we call a $\mu$-admissible Green function for $D$ an $L^{2}_{1}$-Green function $g$ for $D$ such that equation

[TABLE]

holds in the sense of currents. We denote by $\widehat{CH^{1}}(\mathcal{X})_{\mu}$ the corresponding $\mu$-admissible arithmetic Chow group. We show that pull-back and push-forward are well-behaved with respect to this notion, as well as study how the reference 2-current changes.

Theorem 1.2**.**

Let \mu\in L^{2}_{-1}\big{(}\mathcal{X}(\mathbb{C})\big{)} and \nu\in L^{2}_{-1}\big{(}\mathcal{Y}(\mathbb{C})\big{)} be real-valued 2-currents such that $\int_{\mathcal{X}(\mathbb{C})}\mu=\int_{\mathcal{Y}(\mathbb{C})}\nu=1$. The natural definitions of pull-back and push-forward induce homomorphisms

[TABLE]

Here, the main issue is the question of the existence of such $\mu$-admissible Green functions (cf. Proposition 3.2). Such a distibution always exists locally, a result coming from the theory of elliptic PDEs. This reduces our problem to patching together local solutions to obtain a global one, which in turn is an exercise in $L^{2}_{1}$-cohomology of Riemann surfaces. This exercise is probably well known to the experts, though for lack of suitable reference in the literature, we collect in the appendix the necessary facts and deductions to complete it.

We mention that Burgos, Kramer and Kühn BGKK (07) have developed an arithmetic intersection theory which, in the case of arithmetic surfaces considered here, lies strictly in between Gillet-Soulé’s and Bost’s. They show that their theory is functorial. This theory has the advantage of being able to handle higher dimensional arithmetic varieties, where no optimal theory analogous to Bost’s is known (though there are partial results in this direction due to A. Moriwaki Mor (98)).

As an application, which is also our main motivation to study functoriality in this context, we consider the case of the modular curves $X_{0}(N)=\Gamma_{0}(N)\backslash\mathbb{H}^{*}$ and their Hecke correspondences. We use the integral model $\mathcal{X}_{0}(N)$ obtained by the modular interpretation in terms of $\Gamma_{0}(N)$-structures as in KM (85). On the other hand, the hyperbolic measure on $\mathbb{H}$ descends on $X_{0}(N)$ to a finite measure. We denote by $\mu_{N}$ the normalized 2-current on $X_{0}(N)$. This is an element of L^{2}_{-1}\big{(}X_{0}(N)\big{)}. We define $\widehat{CH}(N):=\widehat{CH^{1}}(\mathcal{X}_{0}(N))_{\mu_{N}}$.

Theorem 1.3**.**

The Hecke correspondences $T_{l}$ with $l\nmid N$ and the involutions $w_{d}$ with $d|N$ induce homomorphisms $\hat{T}_{l}$ and $\hat{w}_{d}$ on $\widehat{CH^{1}}(\mathcal{X}_{0}(N))$. These homomorphism form a commutative algebra, and the induced operator $\hat{T}_{l}$ is self-adjoint. Moreover, this algebra preserves $\widehat{CH}(N)$.

We remark that, as the singularities of an admissible Green function with respect to $\mu_{N}$ are of $\log\log$ type, the part of this statement concerning functoriality can also be obtained via the Burgos-Kramer-Kühn theory.

From now on, we suppose that $N$ is squarefree. We denote by $\widehat{CH}_{\mathbb{R}}^{num}(N)$ the variant of the arithmetic Chow group $\widehat{CH}(N)$ obtained by taking divisors (compactified, $\mu_{N}$-admissible) with real coefficients, and then the quotient by the numerical equivalence relation (cf. Section 4.3 for a precise definition). This is a real vector space of finite dimension, and the Hecke operators $\hat{T}_{l}$ and $\hat{w}_{d}$ act on it. The Hodge index theorem in this context allows us to decompose $\widehat{CH}_{\mathbb{R}}^{num}(N)$ as a sum of two Hecke-invariant subspaces:

[TABLE]

where Eis is the space spanned by compactified irreducible components of fibers, and

[TABLE]

as real vector spaces. It is possible to choose the isomorphism in (1.5) such that the Hecke actions on both sides are compatible (Corollary 4.11). Hence we may further decompose

[TABLE]

where $f$ runs through a basis of S_{2}\big{(}\Gamma_{0}(N)\big{)} consisting of eigenforms.

Let $\omega$ be the dualising sheaf on $\mathcal{X}_{0}(N)$. By taking the divisor induced by a section of this sheaf with an appropriate choice of $L^{2}_{1}$-Green function (cf. Section 4.4), we obtain a class $\hat{\omega}\in\widehat{CH}(N)$. Following the decompositions (1.4) and (1.6) we can then write

[TABLE]

We consider the self-intersection of each component as a refined invariant:

[TABLE]

Using a calculation due to U. Kühn (Küh (01)), we then obtain

Theorem 1.4**.**

Suppose that $N$ is squarefree. Let $g_{M}:=\textrm{genus of }X_{0}(M)$. We have that

[TABLE]

where $\zeta(\cdot)$ is the Riemann zeta function.

One of the divisors in the class $\omega_{J}$ is an explicit linear combination of Heegner points of discriminants -3 and -4. Let $H_{i}$ (resp. $H_{j}$) be the sum of all divisors on $X_{0}(N)$ of the form \frac{1}{2}\big{(}[P]-[\infty]\big{)} (resp. \frac{1}{3}\big{(}[P]-[\infty]\big{)}), with $P$ a Heegener point of discriminant -4 (resp. -3). Using the Gross-Zagier formula, we obtain that

Theorem 1.5**.**

Suppose that $f$ is a normalized newform, and that $N$ is a squarefree integer such that $(6,N)=1$. Then,

[TABLE]

where $h_{NT}$ is the Néron-Tate height on $J_{0}(N)(\mathbb{Q})$. Furthermore, if $N$ is prime, then

[TABLE]

where $\chi_{-4}$ (resp. $\chi_{-3}$) is the quadratic Dirichlet character associated to $\mathbb{Q}(i)/\mathbb{Q}$ (resp. $\mathbb{Q}(\sqrt{-3})/\mathbb{Q}$) and $(f,f)$ is the Petersson norm of $f$.

Though we have treated in detail only the case of $\Gamma_{0}(N)$, an analogous analysis can be made for Hecke operators on other congruence groups, as well as on Shimura curves.

Acknowledgements: This work is based in part on the author’s Ph.D. thesis at Université d’Orsay, written under the supervision of E. Ullmo. It was A. Chambert-Loir who suggested viewing the results of the thesis as a functoriality problem on an appropiate function space. We have benefited from useful concrete suggestions by J.-B. Bost, U. Kühn and A. Thuillier.

2. Functoriality in $L^{2}_{1}$

2.1. The space $L^{2}_{1}$ and related spaces

We recall the basic definitions from Bos (99). Let $X$ be a compact Riemann surface. The space $L^{2}_{1}(X)$ is the space of distributions $f\in L^{2}(X)$ such that $\frac{\partial}{\partial z}f$ and $\frac{\partial}{\partial\bar{z}}f$ are also in $L^{2}(X)$. We have a hermitian inner product $(\cdot,\cdot)_{1,X}$ on $L^{2}_{1}(X)$ given by

[TABLE]

Its associated seminorm is given by

[TABLE]

Let $\nu$ be a 2-form on $X$. We consider formally the pairing

[TABLE]

If $\nu$ is positive and continuous, then the pairing (2.1) is a non degenerate, hermitian inner product on $L^{2}_{1}(X)$, endowing it with a Hilbert space structure. We denote by $\|\cdot\|_{\nu}$ the associated norm. As $X$ is compact, the topology on $L^{2}_{1}(X)$ induced by this norm does not depend on the choice of $\nu$. The subspace $C^{\infty}(X)$ is dense in $L^{2}_{1}(X)$ under this topology.

We can also make $L^{2}_{1}$ into a sheaf. For an open set $U\subseteq X$, we denote by $\nu_{U}$ the restriction to $U$ of the given continuous volume form. We define $L^{2}_{1}(U)$ as the space of distributions on $U$ which are $L^{2}$ with respect to $\nu_{U}$ and such that $\partial f,\bar{\partial}f$ are $L^{2}$. Again, $L^{2}_{1}(U)$ is independent of the choice of $\nu$, as well as the choice of ambient Riemann surface, cf Bos (99), p.255.

We define $L^{2}_{-1}(X)$ to be the space of $2$-currents $\mu$ that can be locally written on open sets $U$ as $\mu=dd^{c}f$ with $f\in L^{2}_{1}(U)$.

Remark 2.1**.**

This definition of $L^{2}_{-1}$ is equivalent to the one given in the introduction (cf. Theorem 5.1 in the appendix).

Let $\mathcal{D}(U)$ denote the space of $C^{\infty}$, complex valued functions on $X$ whose support is a compact set contained in $U$. We define $L^{2}_{1}(U)_{0}$ to be the closure of the image of the natural map $\mathcal{D}(U)\rightarrow L^{2}_{1}(X)$. As $X$ is compact, we have that $L^{2}_{1}(X)_{0}=L^{2}_{1}(X)$.

Similarly, we define $L^{1}(U)$ as the space of distributions on $U$ which are $L^{1}$ with respect to $\nu_{U}$, with $L^{1}(U)_{0}$ the closure of the image of the natural map $\mathcal{D}(U)\rightarrow L^{1}(X)$.

We define formally

[TABLE]

Let us record for latter use the following result.

Lemma 2.2**.**

Assume either of the following hypotheses:

• •

$f\in L^{2}_{1}(U)_{0},g\in L^{2}_{1}(U)$**

• •

$f\in L^{1}(U)_{0},\frac{\partial}{\partial z}f\in L^{1}(U),g\in C^{\infty}(U)$.

Then, we have an equality of absolutely convergent integrals:

[TABLE]

Proof: It suffices to consider $f\in\mathcal{D}(U)$. Then, the identity $d(f\overline{\partial g})=\partial f\wedge\overline{\partial g}-2\pi ifdd^{c}\bar{g}$ and the vanishing of $f$ on $\partial U$ prove (2.2). Since either of the hypotheses ensure that $(f,g)_{1,U}$ is absolutely convergent, this finishes the proof. $\blacksquare$

2.2. Pull-back and push-forward in the space $L^{2}_{1}$

Let $\varphi:X\rightarrow Y$ be a non-constant holomorphic map between compact Riemann surfaces. For a function $g\in C^{\infty}(X)$ (resp. $f\in C^{\infty}(Y)$), the functions $\varphi^{*}f$ and $\varphi_{*}g$ are given by $\varphi^{*}f=f\circ\varphi$ and

[TABLE]

where $e_{x}$ is the ramification index of $\varphi$ at $x$. We have that $\varphi^{*}f\in C^{\infty}(X)$. Although $\varphi_{*}g$ is continuous, it is not $C^{\infty}$ at branched points. However, we have the following result for the latter function.

Lemma 2.3**.**

Let $U\subset X$ be an open set. For any $g\in C^{\infty}(U)$ the function $\varphi_{*}g$ given by (2.3) belongs to L^{2}_{1}\big{(}\varphi(U)\big{)}.

Proof: As $\varphi_{*}g$ is continuous, it suffices to check that the derivative of $\varphi_{*}g$ is $L^{2}$ around the finite set of branched points contained in $\varphi(U)$. Using local coordinates, we reduce the problem further to the following situation: $g$ is a $C^{\infty}$ function on $\mathbb{D}=\{z\in\mathbb{C},|z|<1\}$, and $\varphi:\mathbb{D}\rightarrow\mathbb{D}$ is the map $\varphi(z)=z^{n}$. Fixing a branch of the logarithm on $\mathbb{D}$, we then have that

[TABLE]

Then,

[TABLE]

Hence,

[TABLE]

as can be seen by an elementary change of variables. The last integral converges because $h$ is smooth. $\blacksquare$

Proposition 2.4**.**

The linear maps

[TABLE]

are continuous in the $L^{2}_{1}$ topology.

We postpone the proof of this statement to the end of this Section.

As the spaces $C^{\infty}(X),C^{\infty}(Y)$ are dense in the respective $L^{2}_{1}$ topologies, Proposition 2.4 allows us to extend the pull-back and push-forward to the respective $L^{2}_{1}$ spaces,

[TABLE]

We also have well-defined maps

[TABLE]

given locally by $\varphi^{*}dd^{c}f:=dd^{c}\varphi^{*}f$ and $\varphi_{*}dd^{c}g:=dd^{c}\varphi_{*}g$.

Lemma 2.5**.**

Fix an open set $U\subset X$. Assume either of the following hypotheses:

• •

f\in L^{2}_{1}\big{(}\varphi(U)\big{)},g\in L^{2}_{1}(U)**

• •

\frac{\partial}{\partial z}\varphi^{*}f\in L^{1}(U),\frac{\partial}{\partial z}f\in L^{1}\big{(}\varphi(U)\big{)},g\in C^{\infty}(U).**

Then, we have that

[TABLE]

Proof: Either of the hypotheses above allows us to reduce to the case where $f$ and $g$ are $C^{\infty}$ (by a well known density argument). Then, using a partition of unity, the problem is further reduced to the case where $\varphi$ is the covering $z\mapsto z^{n}$ of the complex unit disc. In this situation, the asserted equality boils down to an elementary change of variables. $\blacksquare$

For a 2-current $\mu$ and a distribution $f$, we define formally the symbol

[TABLE]

Lemma 2.6**.**

The integral in (2.4) is absolutely convergent under either of the following hypotheses:

• (i)

$f\in L^{2}_{1}(X),\mu\in L^{2}_{-1}(X)$**

• (ii)

$f\in L^{1}(X),\frac{\partial}{\partial z}f\in L^{1}(X),\mu$* a $C^{\infty}$ 2-form.*

Moreover, we have that

1. (1)

$\varphi^{*}s\cdot\mu=s\cdot\varphi_{*}\mu$, for all $s\in L^{2}_{1}(Y)$ and $\mu\in L^{2}_{-1}(X)$ 2. (2)

$\varphi_{*}t\cdot\nu=t\cdot\varphi^{*}\nu$, for all $t\in L^{2}_{1}(X)$ and $\nu\in L^{2}_{-1}(Y)$.

Proof: Suppose hypothesis (i) holds. Using a partition of unity and the compactness of $X$, the integral in (2.4) splits as a finite sum of integrals over open sets $U$ such that $\mu=dd^{c}h$, with $h\in L^{2}_{1}(U)$ and $f\in L^{2}_{1}(U)_{0}$. Then, every integral on the sum converges by Lemma 2.2, which shows that (2.4) is absolutely convergent. The same argument works under hypothesis (ii), taking $h\in C^{\infty}(U)$ and $f\in L^{1}(U)_{0}$.

To prove (1), we choose a finite collection of open sets $U$ as in the paragraph above. As $\varphi$ is a proper map, we may suppose that $\varphi^{*}s\in L^{2}_{1}(U)_{0}$. Using Lemma 2.2 and Lemma 2.5, we obtain that

[TABLE]

The identity (2) is proved similarly. $\blacksquare$

Now, we establish three lemmas leading to the proof of Proposition 2.4. The first one will be used to control the $L^{2}$ norm of $\varphi^{*}$ (resp. $\varphi_{*}$) away from the ramification (resp. branched) points.

Lemma 2.7**.**

Let $\mu_{X}$ (resp. $\mu_{Y}$) be a nonnegative, continuous 2-form on $X$ (resp. on $Y$). Let $\varphi:X\rightarrow Y$ be a non constant holomorphic map between compact Riemann surfaces, and let $f\in C^{\infty}(Y),g\in C^{\infty}(X)$. Then, the following relations hold:

[TABLE]

Remark 2.8**.**

We stress that, as $\varphi_{*}\mu_{X}$ is not continuous if $\varphi$ is ramified, this lemma does not immediately imply Proposition 2.4 (cf. Remark 2.12).

Proof: The space of $C^{\infty}$ functions is dense in $L^{2}_{1}$. In particular, we can approximate in the $L^{2}_{1}$ sense continuous functions by $C^{\infty}$ functions. Hence, we may suppose without loss of generality that $\mu_{X}\in L^{2}_{-1}(X)$ and $\mu_{Y}\in L^{2}_{-1}(Y)$. We use Lemma 2.6 and Lemma 2.5 to find that

[TABLE]

This proves the first assertion. We will now deduce the second from the first:

[TABLE]

Applying the Cauchy-Schwarz inequality to the inner product

[TABLE]

we obtain that

[TABLE]

Lemma 2.3 ensures that $\|\varphi_{*}g\|_{\mu_{Y}}$ is a real number. Hence, by the above inequality, we conclude that $\|\varphi_{*}g\|_{\mu_{Y}}\leq\deg\varphi\|g\|_{\varphi^{*}\mu_{Y}}.\blacksquare$

Lemma 2.9**.**

Let $\mathbb{D}:=\{|z|<1\}\subset\mathbb{C}$ and let $S=\{f\in C^{\infty}(\overline{\mathbb{D}}):f=0$ on $\partial\mathbb{D}$}. Then, for all $f\in S$, we have that

[TABLE]

Proof: Using Lemma 2.2, we have that

[TABLE]

Similarly,

[TABLE]

The identity $d(f\overline{\partial f}+\bar{f}\partial f)=2\pi i(\bar{f}dd^{c}f-fdd^{c}\bar{f})$, along with the vanishing on the boundary allow us to deduce the result. $\blacksquare$

The following lemma can be seen as a weighted Poincaré inequality. It will be used in the proof of Proposition 2.4 to control the $L^{2}$ norm of $\varphi^{*}$ around the ramification points.

Lemma 2.10**.**

Let $S$ be the set defined in Lemma 2.9. For every $\delta>0$ there exists $C_{\delta}>0$ such that for all $f\in S$,

[TABLE]

Proof: Observe that the integral on the left hand side is convergent. Indeed,

[TABLE]

Let $\mathbb{D}_{\varepsilon}=\{\varepsilon<|z|<1\}$. We have that

[TABLE]

We remark that

[TABLE]

Then, using (2.8),

[TABLE]

But

[TABLE]

where we have used the relation $\frac{\partial}{\partial z}|f|^{2}=\bar{f}\frac{\partial}{\partial z}f+f\overline{\frac{\partial}{\partial\bar{z}}f}$. We bound the first integral using the Cauchy-Schwarz inequality. That is,

[TABLE]

Using Lemma 2.9, the same bound applies to the second integral in (2.9). Putting everything together, we obtain that

[TABLE]

This implies (2.7) with $C_{\delta}=(4/\delta)^{2}.\blacksquare$

Remark 2.11**.**

Lemma 2.10 can be viewed as a variant of the Hardy inequality (HLP (88), Theorem 327). In both cases, the main idea is to express the weighted $L^{2}$ norm of $f$ as an integral involving the derivative of $f$. In our case, this is done through the equality (2.8).

Proof of Proposition 2.4 : the endomorphism of the unit disk $\mathbb{D}$ given by $z\mapsto z^{n}$ pulls back $dz\wedge d\bar{z}$ to $n^{2}|z|^{2(n-1)}dz\wedge d\bar{z}$. Hence, the form $\varphi^{*}\mu_{Y}$ on Lemma 2.7 is continuous, non negative, with zeroes at the ramification points. This proves the assertion about the push-forward in Proposition 2.4.

Concerning the pull-back, we must prove that we can choose a positive, continuous 2-form $\mu=\mu(\varphi,\mu_{X})$ on $Y$ as well as a constant $C=C(\varphi,\mu_{X})>0$, such that

[TABLE]

holds for all $f\in C^{\infty}(Y)$.

Lemma 2.5 implies the equality $(\varphi^{*}f,\varphi^{*}f)_{1,X}=\deg\varphi(f,f)_{1,Y}$, so it suffices to bound the $L^{2}$ norm of $\varphi^{*}f$.

Let $E\subset Y$ be the set of branched points of $\varphi$. We select a convenient neighborhood $U$ of $E$ as follows: for each ramification point $P\in X$ of $\varphi$ with ramification index $n_{P}$, there exists a neighborhood $V_{P}\subset X$ (resp. $U_{P}\subset Y$) of $P$ (resp. of the branched point $\varphi(P)$) and local charts $\theta_{P}:\mathbb{D}\rightarrow V_{P}$, $\eta_{P}:\mathbb{D}\rightarrow U_{P}$ such that the following diagram commutes:

[TABLE]

We may assume without loss of generality (shrinking the neighborhoods if required) that $\overline{V}_{P}\cap\overline{V}_{Q}=\emptyset$ if $P\neq Q$. Let us put $U:=\cup_{P\textrm{ ramified}}U_{P}$.

Let $U_{E}$ be another neighborhood of $E$ such that $\overline{U}_{E}\subset U$. Let $\psi_{1},\psi_{2}$ be a partition of unity subordinate to the covering $\{Y\backslash\overline{U}_{E},U\}$. Let

[TABLE]

We split $f\in C^{\infty}(Y)$ as $f=f_{1}+f_{2}$, where $f_{1}:=f\psi_{1}$ and $f_{2}:=f\psi_{2}$. As

[TABLE]

it suffices to prove (2.10) for $f_{1}$ and $f_{2}$.

Let $\mu$ be a positive continuous volume form such that $\mu=\varphi_{*}\mu_{X}$ on $Y\backslash\overline{U}_{E}$. Note that such a form exists because $\varphi_{*}\mu_{X}$ is continuous on $Y\backslash E$. We have that $\|f_{1}\|_{\varphi_{*}\mu_{X}}=\|f_{1}\|_{\mu}$. Then, since supp$f_{1}\subset Y\backslash\overline{U}_{E}$, it follows from Lemma 2.7 that

[TABLE]

We may suppose that the support of $f_{2}$ is contained in some $U_{P}$. We then have that

[TABLE]

The last inequality, valid for some $\lambda>0$ depending only on the choice of charts, follows from the continuity of $\theta_{Q}^{*}\mu_{X}$ on $\overline{\mathbb{D}}$. Putting $n:=n_{P}$ we then find that

[TABLE]

where the last follows by application of Lemma 2.10 to $f_{2}\circ\eta_{Q}\in\mathcal{D}(\mathbb{D})$. Finally,

[TABLE]

This achieves the proof of (2.10) for $f_{2}$. $\blacksquare$

Remark 2.12**.**

J.-B. Bost explained to us another way to handle the problem of the ramification points. Let $C_{\varphi}^{\infty}(Y)$ be the subspace of $C^{\infty}(Y)$ consisting of functions whose support does not contain the set of branched points of $\varphi$. A result by Deny and Lions (DL (54), Théorème II.2.2), using Cartan’s theory of balayage, ensures that $C_{\varphi}^{\infty}(Y)$ is dense in $L^{2}_{1}(Y)$. Then to obtain a well defined pull-back map $\varphi^{*}:L^{2}_{1}(Y)\rightarrow L^{2}_{1}(X)$, it suffices to show the continuity of $\varphi^{*}$ on $C_{\varphi}^{\infty}(Y)$, which can be easily done (e.g. using Lemma 2.7 and the classical Poincaré inequality). This approach avoids the use of Lemma 2.10, though is less elementary.

2.3. Pull-back and push-forward of $L^{2}_{1}$-Green functions

Let $X$ be a compact Riemann surface, and let $D=\sum_{P}a_{P}[P]$ be a divisor with real coefficients on $X$. By definition, a Green function with $C^{\infty}$ regularity for $D$ is a real-valued distribution $h$ on $X$ such that the current $dd^{c}h+\delta_{D}$ is a $C^{\infty}$ 2-form111This is what Gillet and Soulé call a Green current. Following Bost, we use a different terminology better suited to the discussion of regularity issues. .

Remark 2.13**.**

The above definition is equivalent to the following two properties:

• •

$h$ is $C^{\infty}$ outside of the finite set $|D|:=\{P$ such that $a_{P}\neq 0\}$

• •

for every $P\in|D|$, there exists a local chart $(U,\theta_{P})$ centered at $P$ such that

[TABLE]

on $U$, where $b\in C^{\infty}(U)$.

The first property is a consequence of the ellipticity of the $dd^{c}$ operator. The second one comes from the fact that $\log|\cdot|^{2}$ is a fundamental solution of $dd^{c}f=\delta_{0}$ on the complex unit disc.

Definition 2.14**.**

An $L^{2}_{1}$-Green function for $D$ is a real-valued distribution $g$ on $X$ such that there exists a Green function with $C^{\infty}$ regularity $h$ for $D$, along with an element $\psi\in L^{2}_{1}(X)$ such that

[TABLE]

We call this equality a regular splitting of $g$.

Remark 2.15**.**

A distribution $g$ satisfies the above definition if and only if $dd^{c}g+\delta_{D}$ is a 2-current in $L^{2}_{-1}(X)$ (cf. Proposition 3.2 below). In particular, the 2-current $dd^{c}g$ is a Radon measure, that is, it extends to a continuous functional on the space $C^{0}(X)$ of continuos functions (and not merely on $C^{\infty}(X)$).

Let $\varphi:X\rightarrow Y$ be a non-constant holomorphic map between compact Riemann surfaces. Let $D$ be a divisor on $X$ and let $h$ be a Green function with $C^{\infty}$ regularity for $D$. We define the distribution $\varphi_{*}h$ on $Y$ by

[TABLE]

where $e_{x}$ is the ramification index of $\varphi$ at $x$.

The definition above has to be understood in the following sense. Using Remark 2.13, we see that the right side of (2.12) defines a function on the complement of the finite set

[TABLE]

It can be proven that the singularities of $\varphi_{*}h$ on this set are logarithmic (cf. (2.15) below for a precise verification), thus defining a distribution on $X$. Finally, we define

[TABLE]

Similarly, let $E\in$Div$(Y)$ and let $k$ be a Green function with $C^{\infty}$ regularity for $E$. We define the distribution $\varphi^{*}k$ on $X$ by

[TABLE]

This definition is made rigorous in the same way as we did it with $\varphi_{*}h$ above.

Let $f$ be a $L^{2}_{1}$-Green function for $E$, with a regular splitting

[TABLE]

where $\xi\in L^{2}_{1}(Y)$. We define

[TABLE]

Theorem 2.16**.**

The distribution $\varphi_{*}g$ (resp. $\varphi^{*}f$) is a $L^{2}_{1}$-Green function for $\varphi_{*}D$ (resp. for $\varphi^{*}E$).

Proof: We first prove the assertion concerning the push-forward. It suffices to consider the case $D=[P]$. Let $n$ be the ramification index of $\varphi$ at $P$, and let $Q:=\varphi(P)$. We choose local charts $(U,\theta_{P})$ centered at $P$ and $(V,\theta_{Q})$ centered at $Q$ such that the following diagram commutes:

[TABLE]

Here, $\alpha(z)=z^{n}$. We may assume that the ramification index is 1 for every point in $U$ different from $P$, and that we have

[TABLE]

on $U$, with $b\in C^{\infty}(U)$. We may also assume that supp $h\subset U$222If this is not the case, we take a $C^{\infty}$ function $s$ with support contained in $U$ and such that $s\equiv 1$ in a neighborhood of $P$. Then, g=sh+\big{(}\psi+(1-s)h\big{)} is also a regular splitting as in (2.11)..

As $\varphi_{*}\psi\in L^{2}_{1}(Y)$, it suffices to prove that $\varphi_{*}h$ can be expanded around $Q$ as

[TABLE]

where \tilde{b}\in L^{2}_{1}\big{(}\varphi(U)\big{)}. Using our assumptions on the support of $h$, we have that for all $y\in V\backslash\{Q\}$,

[TABLE]

Using Lemma 2.3, we have \tilde{b}:=\varphi_{*}b\in L^{2}_{1}\big{(}\varphi(U)\big{)}.

Now, we prove the assertion concerning the pull-back. As before, we may assume that $E=[Q]$. We have that

[TABLE]

We fix $P\in|\varphi^{*}[Q]|$ and choose local charts $(U,\theta_{P})$ centered at $P$ and $(V,\theta_{Q})$ centered at $Q$ such that the diagram (2.14) commutes, with $\alpha(z)=z^{e_{P}}$. Let us write $f=k+\xi$ as in (2.13). This reduces the problem to a study of the expansion of $\varphi^{*}k$ around $P$. We have that

[TABLE]

As $\varphi^{*}b$ is $C^{\infty}$, this concludes the proof. $\blacksquare$

The following two Lemmas will be used in the proof of Theorem 3.5.

We adopt the following notation. For a 2-current $T$ and a test function $\phi\in C^{\infty}$, we denote by $T[\phi]$ the value of $T$ at $\phi$.

Lemma 2.17**.**

Using the notations of Theorem 2.16, we have that

[TABLE]

Remark 2.18**.**

As $\varphi_{*}\phi$ is continuous, the right hand side of (2.16) is well defined (cf. Remark 2.15).

Proof: we write $f=k+\xi$. Using Lemma 2.6, we have that

[TABLE]

We put $w:=dd^{c}k+\delta_{E}$, which is a $C^{\infty}$ 2-form. Using Lemma 2.6, we have that

[TABLE]

thus proving (2.16). Equation $\eqref{laotrafuntorialidad}$ is proved similarly. $\blacksquare$

The following lemma is a variant of Lemma 2.6 for Green functions.

Lemma 2.19**.**

Let $h$ be a Green function with $C^{\infty}$ regularity for a divisor $D$ on $Y$ such that the support of $\varphi^{*}h$ is contained in a neighborhood $U$ of $|\varphi^{*}D|$. Let $s\in C^{\infty}(U)$. Then

[TABLE]

Proof: Theorem 2.16 ensures that $\varphi^{*}h$ and $\frac{\partial}{\partial z}\varphi^{*}$ are $L^{1}$ on $U$ (i.e., apply the local expansion in Remark 2.13 to $\varphi^{*}h$). Then, by Lemma 2.2 and Lemma 2.5, we have that

[TABLE]

3. Pull-back and push-forward in Arakelov theory

3.1. The $L^{2}_{1}$-arithmetic Chow group, the admissible arithmetic Chow group and intersection theory

Let $\mathcal{X}$ be an arithmetic surface, that is, a projective, integral, normal scheme over Spec$(\mathbb{Z})$ of Krull dimension 2. The ring $O_{\mathcal{X}}(\mathcal{X})$ is the ring of integers of a number field $K$. We denote by Div$(\mathcal{X})$ the set of Weil divisors on $\mathcal{X}$.

A compactified divisor (in the sense of Bost) is a pair $(D,g)$, where $D\in$Div$(\mathcal{X})$ and $g$ is a $L^{2}_{1}$-Green function for the divisor $D(\mathbb{C})$ on the compact Riemann surface $\mathcal{X}(\mathbb{C})$. In addition, $g$ must be invariant under complex conjugation on $\mathcal{X}(\mathbb{C})$. A principal divisor is a compactified divisor of the form $(\textrm{div }f,-\log|f_{\mathbb{C}}|^{2})$, where $f$ is a rational function on $\mathcal{X}$. This is a well defined notion, as can be seen from the Poincaré-Lelong equation (GH (78), p. 388)

[TABLE]

Let $\widehat{\textrm{Div}}(\mathcal{X})$ be the set of compactified divisors. It is an abelian group under the operation $(D,g)+(D^{\prime},g^{\prime})=(D+D^{\prime},g+g^{\prime})$. From remark 2.13 it is easily verified that this operation is well defined. The neutral element is $(0,0)$, that is, the empty divisor together with the zero function. The set of principal divisors is a subgroup of the group of compactified divisors.

The $L^{2}_{1}$-arithmetic Chow group $\widehat{CH^{1}}(\mathcal{X})$ is defined as the quotient of the group of compactified divisors by the subgroup of principal divisors.

Let $X=\mathcal{X}(\mathbb{C})$. Let $\mu$ be a real-valued 2-current in $L^{2}_{-1}(X)$ such that

[TABLE]

A compactified divisor $(D,g)$ is said to be admissible with respect to $\mu$ (or $\mu$-admissible) if the following equality of currents holds:

[TABLE]

In this situation we will also say that $g$ is a $\mu$-admissible $L^{2}_{1}$-Green function for $D$.

The $L^{2}_{1}$-Green function $g$ is determined by $D$ up to a function which is constant on every connected component of $X$. Indeed, if $(D,h)$ is $\mu$-admissible, then $g-h$ is an harmonic function on a compact Riemann surface. Hence, it must be constant on each connected component of $X$ by the maximum modulus principle.

We remark that (3.1) is a necessary condition for the existence of admissible compactified divisors $(D,g)$ with $\deg D(\mathbb{C})\neq 0$. To give a sufficient condition we use an existence result coming from the theory of elliptic operators:

Lemma 3.1**.**

Let $\mu\in L^{2}_{-1}(X)$. The equation

[TABLE]

has a solution $u\in L^{2}_{1}(X)$ if and only if $\int_{X}\mu=0$. If $\mu$ is real-valued then $u$ can be taken to be real-valued.

If we take $\mu$ to be a $C^{\infty}$ form (in which case $u\in C^{\infty}(X)$), this is a well known consequence of the Hodge theorem in differential geometry (cf. War (71), 6.8).

Equation (3.2) can be solved locally via standard techniques from the theory of elliptic PDE’s, namely the Lax-Milgram theorem and the Fredholm alternative. To prove that it is possible to patch together these local solutions to get a global solution, we shall use the cohomology machinery. We explain the precise steps giving a proof of Lemma 3.1 in the Appendix.

Proposition 3.2**.**

Let $\mu$ be a real-valued 2-current in $L^{2}_{-1}(X)$ such that $\int_{X}\mu=1$. Then for every Weil divisor $D$ on $\mathcal{X}$, there exists a $L^{2}_{1}$-Green function $g$ such that $(D,g)$ is $\mu$-admissible.

Proof: let $\mu_{0}$ be the canonical Arakelov 2-form if the genus of $X$ is positive and the Fubini-Study form if it is zero. Let $\tilde{g}$ be a real-valued solution to the equation of currents (Lan (88), Theorem 1.4)

[TABLE]

Let \tilde{\mu}=\big{(}\deg D(\mathbb{C})\big{)}(\mu_{0}-\mu). Then $\tilde{\mu}$ belongs to $L^{2}_{-1}(X)$ and $\int_{X}\tilde{\mu}=0$. Using Lemma 3.1, we have a real-valued $u\in L^{2}_{1}(X)$ such that $dd^{c}u=\tilde{\mu}$. Then $g:=\tilde{g}-u$ is a $\mu$-admissible, $L^{2}_{1}$-Green function for $D(\mathbb{C}).\blacksquare$

Principal divisors are admissible with respect to any 2-current. The set of $\mu$-admissible compactified divisors forms a group. We denote by $\widehat{CH^{1}}(\mathcal{X})_{\mu}$ the quotient of this group by the subgroup of principal divisors. The natural map $\widehat{CH^{1}}(\mathcal{X})_{\mu}\rightarrow\widehat{CH^{1}}(\mathcal{X})$ is an embedding of abelian groups.

Let $x_{1},x_{2}\in\widehat{CH^{1}}(\mathcal{X})$ be represented by the compactified divisors $(D_{1},g_{1})$ and $(D_{2},g_{2})$. We suppose that the representatives have been chosen without common irreducible components. The intersection pairing is defined as

[TABLE]

where the first pairing is the finite intersection pairing as defined in Bos (99), Section 5.3. We recall from loc. cit. the definition of the second pairing: we choose regular splittings

[TABLE]

as in (2.11), and we define $w_{i}=dd^{c}h_{i}+\delta_{D_{i}(\mathbb{C})}$. Then,

[TABLE]

where $h_{1}*h_{2}=h_{1}w_{2}+h_{2}\delta_{D_{1}(\mathbb{C})}$. This number does not depend of the choice of $h_{i},\psi_{i}$.

Example 3.3**.**

For every compactified divisor $\hat{D}=(D,g)$ and $c\in\mathbb{R}$, we have that

[TABLE]

Let $\varphi:\mathcal{X}\rightarrow\mathcal{Y}$ be a generically finite morphism between arithmetic surfaces. If we have a regular splitting $g=h+\psi$, then by definition $\varphi_{*}g=\varphi_{*}h+\varphi_{*}\psi$. As $\varphi_{*}h$ is not a Green function with $C^{\infty}$ regularity when $\varphi$ is ramified, this is not a regular splitting of $\varphi_{*}g$. However, we show in the following lemma that this decomposition can still be used to compute intersections.

Lemma 3.4**.**

Let $(D_{x},g_{x})$ (resp. $(D_{y},g_{y})$) be a compactified divisor on $\mathcal{X}$ (resp.$\mathcal{Y}$). We decompose $g_{x}=h_{x}+\psi_{x}$ (resp. $g_{y}=h_{y}+\psi_{y}$) as in (2.11) and we put $w(\varphi_{*}h_{x}):=dd^{c}\varphi_{*}h_{x}+\delta_{\varphi_{*}D_{x}(\mathbb{C})}$ (resp. $w(h_{y}):=dd^{c}h_{y}+\delta_{D_{y}(\mathbb{C})}$). we have that

[TABLE]

Moreover, all of the integrals above are absolutely convergent.

Proof: Let $Y=\mathcal{Y}(\mathbb{C})$. Using Theorem 2.16, we obtain an auxiliary regular splitting

[TABLE]

where $h$ is a Green function for $\varphi_{*}D_{x}$ with $C^{\infty}$ regularity, and $\psi\in L^{2}_{1}(Y)$. Hence, we have that

[TABLE]

We deduce from (3.4) that

[TABLE]

As $w(h_{y})$ is $C^{\infty}$ and $\psi-\varphi_{*}\psi_{x}$ is $L^{2}$, this equality shows the absolute convergence of the integral on the left. Thus, we find that

[TABLE]

The last integral is absolutely convergent because

[TABLE]

and the functions on the right are $L^{2}_{1}$. This shows that the left hand side of (3.1) is absolutely convergent. From this equality we deduce that

[TABLE]

Together with (3.5), this proves both the absolute convergence of the first integral on the left and the desired formula. $\blacksquare$

3.2. Functoriality with respect to pull-back and push-forward

Let $\varphi:\mathcal{X}\rightarrow\mathcal{Y}$ be a generically finite morphism between arithmetic surfaces. Let $(D,g_{D})$ (resp. $(E,g_{E})$) be a compactified divisor on $\mathcal{X}$ (resp. on $\mathcal{Y}$). We define

[TABLE]

Theorem 3.5**.**

1. (1)

The formulas above induce homomorphisms

[TABLE] 2. (2)

Let \mu\in L^{2}_{-1}\big{(}\mathcal{X}(\mathbb{C})\big{)} and \nu\in L^{2}_{-1}\big{(}\mathcal{Y}(\mathbb{C})\big{)} be real-valued and such that $\int_{\mathcal{X}(\mathbb{C})}\mu=\int_{\mathcal{Y}(\mathbb{C})}\nu=1$. The formulas above induce homomorphisms

[TABLE] 3. (3)

Projection formula: for $x\in\widehat{CH^{1}}(\mathcal{X})$ and $y\in\widehat{CH^{1}}(\mathcal{Y})$, we have that

[TABLE]

Remark 3.6**.**

The functorial properties of the pairing in (2.4) ensure that

[TABLE]

Hence, the arithmetic Chow groups appearing to the right in Theorem 3.5 (2) contain compactified divisors of nonzero generic degree (cf. Proposition 3.2).

A correspondence $T$ on $\mathcal{X}$ is, by definition, an arithmetic surface $\mathcal{Y}$ together with two ordered finite morphisms $p,q:\mathcal{Y}\rightarrow\mathcal{X}$. The correspondence $T$ is said to be symmetric if $p_{*}q^{*}=q_{*}p^{*}$ on $Div(\mathcal{X})$.

For a compactified divisor $(D,g)$ on $\mathcal{X}$ and \mu\in L^{2}_{-1}\big{(}\mathcal{X}(\mathbb{C})\big{)}, we define

[TABLE]

The following statement is a straightforward consequence of Theorem 3.5 and the definitions above.

Corollary 3.7**.**

1. Let $T$ be a correspondence on the arithmetic surface $\mathcal{X}$.
2. (1)

The formula (3.7) induces a homomorphism

[TABLE] 3. (2)

If $T$ is symmetric, then the morphism $\hat{T}$ is self-adjoint, i.e. for all $x,y\in\widehat{CH^{1}}(\mathcal{X})$,

[TABLE] 4. (3)

Formula (3.7) induces a homomorphism

[TABLE]

Corollary 3.8**.**

Let $S$ be another correspondence on $\mathcal{X}$. If $S$ and $T$ commute as endomorphisms of Div$(\mathcal{X})$, then $\hat{S}$ and $\hat{T}$ commute as endomorphisms of $\widehat{\textrm{Div}}(\mathcal{X})$.

Proof: let $(D,g)$ be a compactified divisor. Let $g=h+\psi$ be a decomposition as in (2.11). Let $\psi_{n}$ be a sequence of $C^{\infty}$ functions converging to $\psi$ in L^{2}_{1}\big{(}\mathcal{X}(\mathbb{C})\big{)}. The hypothesis implies that $S$ and $T$ commute as correspondences on $\mathcal{X}(\mathbb{C})$, so $ST\psi_{n}-TS\psi_{n}\equiv 0$. By Proposition 2.4, we have that $ST\psi=TS\psi$.

By Theorem 2.16, we deduce that

[TABLE]

The function $h$ is $C^{\infty}$ outside of the finite set $F:=\{x\textrm{ such that }|ST[x]|\cap|D(\mathbb{C})|\neq\emptyset\}$. For any point $x\in\mathcal{X}(\mathbb{C})-F$, we have $STh(x)=TSh(x)$ by hypothesis, so $STh=TSh$ in L^{2}_{1}\big{(}\mathcal{X}(\mathbb{C})\big{)}. This shows that $ST(D,g)=TS(D,g)$, as required. $\blacksquare$

Proof of Theorem 3.5: To prove (1), it suffices by Theorem 2.16 to show that principal divisors are sent into principal divisors. Let $f$ be a rational function on $\mathcal{X}$. Let $N_{\varphi}:K(\mathcal{X})\rightarrow K(\mathcal{Y})$ be the norm map between function fields induced by $\varphi$. We have that

[TABLE]

On the other hand, if $f$ is a rational function on $\mathcal{Y}$, we have that

[TABLE]

This finishes the proof of (1).

Now we prove the first statement of (2). Let $(E,g)$ be a compactified divisor on $\mathcal{Y}$ which is admissible with respect to $\nu$. Theorem 2.16 shows that $\varphi^{*}(E,g)$ is a compactified divisor on $\mathcal{X}$. We need to check the equality of currents

[TABLE]

Let $\phi\in C^{\infty}(X)$. Using Lemma 2.17, we have that

[TABLE]

Using Lemma 2.6 and the fact that $\deg\varphi^{*}E(\mathbb{C})=\deg\varphi\deg E(\mathbb{C})$, this finishes the proof of the statement concerning $\varphi^{*}$ in (2).

To prove the part concerning $\varphi_{*}$ in (2), let $(D,g)$ be a compactified divisor on $\mathcal{X}$ which is admissible with respect to $\mu$. Theorem 2.16 shows that $\varphi_{*}(D,g)$ is a compactified divisor on $\mathcal{Y}$. As \deg\big{(}\varphi_{*}D(\mathbb{C})\big{)}=\deg\big{(}D(\mathbb{C})\big{)}, we need to check the equality of currents

[TABLE]

Let $\phi$ be a $C^{\infty}$ function on $Y$. Using Lemma 2.17, we have that

[TABLE]

As before, we conclude by Lemma 2.6. This finishes the proof of (2).

Now, we prove (3). Let $(D_{x},g_{x})$ (resp. $(D_{y},g_{y})$) be a member in the class $x$ (resp. $y$), chosen such that $|D_{x}(\mathbb{C})|\cap|\varphi^{*}D_{y}(\mathbb{C})|=\emptyset$