The Modularity of an Abelian Variety

Abstract

We introduce the concept of the modularity of an abelian variety defined over the rational number field extending the modularity of an elliptic curve. We discuss the modularity of an abelian variety over $\mathbb{Q}$. We conjecture that a simple abelian variety over $\mathbb{Q}$ is modular.

1. Introduction

An elliptic curve E over $\mathbb{Q}$ is said to be $\mathsf{modular}$ if the L-function $L(E,s)$ of E equals the L-function $L(f,s)$ for some eigenform f, equivalently if E has a finite covering by a modular curve of the form $X_0\left(N\right)$. At the Tokyo-Nikko conference held in 1955, Yutaka Taniyama (1927–1958) made a suggestion that every elliptic curve over $\mathbb{Q}$ is modular. At that time, his suggestion was not clear and hence was not accepted in the mathematics community. In the early 1960’s, Goro Shimura (1930–2019) refined Taniyama’s suggestion through private conversations with a number of mathematicians. In particular he discussed this subject with André Weil (1906–1998) seriously and intensively. Weil gave conceptual evidence for Taniyama’s suggestion in his famous paper [1] published in 1967. Through Weil’s paper, this suggestion was widely known as the so-called Shimura–Taniyama conjecture in the mathematics community.

The Shimura–Taniyama conjecture associates objects of representation theory to those of algebraic geometry. It states that the L-series of an elliptic curve over $\mathbb{Q}$, which measures the behavior of the curve mod p for all primes p, can be identified with an integral transform of the Fourier series defined from an eigenform.

In 1985, Frey [2] made the remarkable observation that the Shimura–Taniyama conjecture for a semi-stable elliptic curve over $\mathbb{Q}$ should imply Fermat’s Last Theorem. The precise mechanism relating the two was formulated by Serre [3] as the $ϵ$-conjecture and this conjecture was proved by Ribet [4] in the summer of 1986. Ribet’s result only requires proving the Shimura–Taniyama conjecture for a semi-stable elliptic curve over $\mathbb{Q}$ in order to reduce Fermat’s Last Theorem. As soon as Andrew Wiles learned Ribet’s result, he began to work the Shimura–Taniyama conjecture for a semi-stable elliptic curve over $\mathbb{Q}$ in the late summer of 1986. Here, the semi-stability of an elliptic curve E over $\mathbb{Q}$ is defined as follows:

Definition 1.

An elliptic curve E over $\mathbb{Q}$ is said to be $\mathsf{semi}$-$\mathsf{stable}$ at the prime q if it is isomorphic to an elliptic curve $\tilde{E}$ over $\mathbb{Q}$, where $\tilde{E}$ (mod q) is either nonsingular or has a node. An elliptic curve over $\mathbb{Q}$ is called $\mathsf{semi}$-$\mathsf{stable}$ if it is semi-stable at every prime.

On 21–23 June in 1993, Wiles had given a series of lectures under the title, “$\mathsf{Elliptic} \mathsf{curves}, \mathsf{modular} \mathsf{forms} \mathsf{and} \mathsf{Galois} \mathsf{representations}$” at the Issac Newton Institute for Mathematical Sciences in Cambridge, England. In this last lecture on 23 June, Wiles commented that he had proved a part of the Shimura–Taniyama conjecture to the effect that every semi-stable elliptic curve over $\mathbb{Q}$ is modular. With the aide of the works of Frey [2], Serre [3] and Ribet [4], his proof solved Fermat’s Last Theorem, which had been unsolved for more than 350 years. The news spread all over the world through well-known newspapers and magazines because Fermat’s Last Theorem is fascinating for amateurs and professionals alike. But in the fall that year, it turned out that Wiles’ proof was incomplete and flawed. Precisely, his construction of the Euler system used to extend Flach’s method was not complete. He was very concerned with filling the gaps in his flawed proof. At that time, he was completely isolated from the outside. In January 1994, he asked Dr. Richard Taylor, his former Ph.D. student in Cambridge University, UK, to join him in the attempt to repair the Euler system argument. Wiles was convinced that his method was correct. Dr. Taylor accepted his proposal and joined in that project. On 19 September 1994, Wiles was quite convinced that his method was correct and his gap could be filled up. After he invited Dr. Taylor to Princeton again, he completed his proof with the aid of Dr. Taylor in October 1994 and submitted his paper to Annals of Mathematics on 14 October 1994. Finally, his paper was accepted and was published in May 1995 (cf. [5,6]). We refer to [7] for more interesting stories.

The aim of this article is to introduce the concept of the modularity of an abelian variety defined over the rational number field extending the modularity of an elliptic curve and to discuss the modularity of an abelian variety over $\mathbb{Q}$. An abelian variety is defined as follows in [8] (p. 39):

Definition 2.

An abelian variety X is a complete (meaning, in particular, that it is irreducible) (see the footnote in [8] (p. 39)) algebraic variety over an algebraically closed field K with a group law $m_X:X\times X⟶X$ such that both $m_X$ and the inverse map are morphisms of varieties. An abelian variety over $\mathbb{Q}$ is a connected and complex projective manifold that is also a group variety generalizing elliptic curves over $\mathbb{Q}$.

The paper is organized as follows. In Section 2, we briefly outline the modularity of an elliptic curve over $\mathbb{Q}$. Wiles proved the modularity of a semi-stable elliptic curve over $\mathbb{Q}$ leading to solve Fermat’s Last Theorem (cf. [5,6]). In 2001, Breuil, Conrad, Diamond and Taylor proved that every elliptic curve over $\mathbb{Q}$ is modular (cf. [9]). In Section 3, we introduce the notion of the modularity of an abelian variety over $\mathbb{Q}$. We propose several conjectures and open problems. In Appendix A, we describe and survey the Hecke algebra for a symplectic group and p-Satake parameters for the readers.

Notations: 

$\mathbb{Q}, \mathbb{R}$ and $\mathbb{C}$ denote the field of rational numbers, the field of real numbers and the field of complex numbers, respectively. $\mathbb{Z}$ and ${\mathbb{Z}}^{+}$ denote the ring of integers and the set of all positive integers, respectively. For a prime p, ${\mathbb{Q}}_p$ denotes the field of p-adic numbers and ${\mathbb{Z}}_p$ denotes the ring of p-adic integers. $\overline{\mathbb{Q}}$ (resp. $\overline{{\mathbb{Q}}_p}$) denotes the algebraic closure of $\mathbb{Q}$ (resp. ${\mathbb{Q}}_p$). $G_{\mathbb{Q}}=\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ denotes the absolute Galois group of $\mathbb{Q}$. The symbol “:=” means that the expression on the right is the definition of that on the left. For two positive integers k and l, $F^{(k,l)}$ denotes the set of all $k\times l$ matrices with entries in a commutative ring F. For a square matrix $M\in F^{(k,k)}$ of degree k, $\mathrm{Tr}\left(M\right)$ denotes the trace of M. For any $M\in F^{(k,l)}, {}^{t}M$ denotes the transpose of a matrix M. $I_n$ denotes the identity matrix of degree n. For $A\in F^{(k,l)}$ and $B\in F^{(k,k)}$, we set $B\left[A\right]={}^t\mathit{ABA}$ (Siegel’s notation). For a number field F, ${\mathbb{A}}_F$ denotes the ring of adeles of F. If $F=\mathbb{Q}$, the subscript will be omitted.

$$Sp(2g,\mathbb{R})=\left\{M\in Sp(2g,\mathbb{R})|{}^{t}M{J}_{g}M=J_g\left[M\right]=J_g\right\}$$

denotes the symplectic group of degree g, where

$$J_g=\left(\begin{array}{cc}0& I_g\\ -I_g& 0\end{array}\right).$$

$${\Gamma }_g:=Sp(2g,\mathbb{Z})=\left\{\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)\in Sp(2g,\mathbb{R})|A, B, C, D \mathrm{integral}\right\}$$

denotes the Siegel modular group of degree g. For a prime ℓ and a positive integer n, ${\mathbb{F}}_{{\mathsf{\ell }}^n}$ denotes the field with ${\mathsf{\ell }}^n$ elements. The algebraic closure ${\overline{\mathbb{F}}}_{\mathsf{\ell }}$ of ${\mathbb{F}}_{\mathsf{\ell }}$ is given by

$${\overline{\mathbb{F}}}_{\mathsf{\ell }}=\bigcup _{n\ge 1}{\mathbb{F}}_{{\mathsf{\ell }}^n}.$$

If p is prime and $\lambda$ is a prime ideal dividing p in the ring of integers in $\overline{\mathbb{Q}}$, there exist a filtration

$$I_{\lambda }\subset D_{\lambda }\subset G_{\mathbb{Q}},$$

where the decomposition group $D_{\lambda }$ and the inertia group $I_{\lambda }$ are defined, respectively, by

$$D_{\lambda }:=\{\sigma \in G_{\mathbb{Q}}|\sigma \left(\lambda \right)=\lambda \}$$

and

$$I_{\lambda }:=\{\sigma \in D_{\lambda }|\sigma \left(x\right)\equiv x\left(\mathrm{mod}\lambda \right)\mathrm{for} \mathrm{all} \mathrm{algebraic} \mathrm{integers} x\}.$$

Then there are natural identifications:

$$D_{\lambda }\cong \mathrm{Gal}({\overline{\mathbb{Q}}}_p/{\mathbb{Q}}_p) \mathrm{and} D_{\lambda }/I_{\lambda }\cong \mathrm{Gal}({\overline{\mathbb{F}}}_p/{\mathbb{F}}_p).$$

${\mathrm{Frob}}_{\lambda }\in D_{\lambda }/I_{\lambda }$ denotes the inverse image of the canonical generator $x\mapsto x^p$ of $\mathrm{Gal}({\overline{\mathbb{F}}}_p/{\mathbb{F}}_p).$ If ${\lambda }^{\prime }$ is another prime lying above p, then ${\lambda }^{\prime }=\sigma \left(\lambda \right)$ for some $\sigma \in G_{\mathbb{Q}}$ and

$$D_{{\lambda }^{\prime }}=\sigma D_{\lambda }{\sigma }^{-1}, I_{{\lambda }^{\prime }}=\sigma I_{\lambda }{\sigma }^{-1} \mathrm{and} {\mathrm{Frob}}_{{\lambda }^{\prime }}=\sigma {\mathrm{Frob}}_{\lambda }{\sigma }^{-1}.$$

Since we care about these objects only up to conjugation, we will write $D_{\lambda }$ and $I_{\lambda }$. Now we will write ${\mathrm{Frob}}_p$ for any representative of a ${\mathrm{Frob}}_{\lambda }$. If $\rho$ is a representation of $G_{\mathbb{Q}}$, which is unramified at p, then $\mathrm{Tr}\left(\rho \left({\mathrm{Frob}}_p\right)\right)$ and $\det \left(\rho \left({\mathrm{Frob}}_p\right)\right)$ are well defined, that is, are independent of the choice of $\lambda$.

2. The Modularity of an Elliptic Curve

We set ${\Gamma }_1:=SL(2,\mathbb{Z})$. For a positive integer N, we let $\Gamma \left(N\right),{\Gamma }_1\left(N\right)$ and ${\Gamma }_0\left(N\right)$ be the congruence subgroups of ${\Gamma }_1$ such that $\Gamma \left(N\right)\subset {\Gamma }_1\left(N\right)\subset {\Gamma }_0\left(N\right)\subset {\Gamma }_1.$ We refer to [10] (pp. 13–14, 21) for the precise definitions and properties of $\Gamma \left(N\right),{\Gamma }_1\left(N\right)$ and ${\Gamma }_0\left(N\right)$. Let ${\mathbb{H}}_1$ be the Poincaré upper half plane. Let the quotient

$$Y_1\left(N\right):={\Gamma }_1\left(N\right)\setminus {\mathbb{H}}_1(\mathrm{resp}.Y_0\left(N\right):={\Gamma }_0\left(N\right)\setminus {\mathbb{H}}_1)$$

be the complex manifold, which has a natural model $Y_1\left(N\right)/\mathbb{Q}$ (resp. $Y_0\left(N\right)/\mathbb{Q}$). We let $X_1\left(N\right)$ (resp. $X_0\left(N\right)$) be the smooth projective curve, which contains $Y_1\left(N\right)$ (resp. $Y_0\left(N\right)$) as a dense Zariski open subset (cf. see [10] (pp. 45–60)).

Let $S_k\left(N\right)$ be the space of cusp forms of weight $k\ge 1$ and level $N\ge 1$. Here, let k and N be positive integers. We recall that if $f\in S_k\left(N\right)$, it satisfies the following properties:

(C1) $f\left((a\tau +b){(c\tau +d)}^{-1}\right)={(c\tau +d)}^{k}f\left(\tau \right)$ for all $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\Gamma }_1\left(N\right)$ and $\tau \in {\mathbb{H}}_1$;

(C2) ${\left|f\left(\tau \right)\right|}^2{\left(\mathrm{Im}\tau \right)}^k$ is bounded in ${\mathbb{H}}_1$;

(C3) The Fourier expansion of $f\left(\tau \right)$ is given by

$$f\left(\tau \right)=\sum _{n=1}^{\infty }{a}_n\left(f\right)q^n,\mathrm{where}q=e^{2\pi i\tau }.$$

We define the L-series of $f\in S_k\left(N\right)$ to be

$$L(f,s):=\sum _{n=1}^{\infty }{a}_n\left(f\right)n^{-s}.$$

For each prime $p\nmid N$, we recall that the Hecke operator $T_p:S_k\left(N\right)⟶S_k\left(N\right)$ is defined by

$$\left(T_{p}f\right)\left(\tau \right)=p^{-1}\sum _{i=0}^{p-1}f\left({\displaystyle \frac{\tau +i}{p}}\right)+p^{k-1}f\left({\displaystyle \frac{ap\tau +b}{cp\tau +d}}\right),f\in S_k\left(N\right)$$

for any $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\Gamma }_1$ with $c\equiv 0\left(\mathrm{mod}N\right)$ and $d\equiv p\left(\mathrm{mod}N\right)$. We refer to [9] (p. 844) or [10] (pp. 170–171) for more details. The Hecke operators $T_p(p\nmid N)$ can be simultaneously diagonalized on $S_k\left(N\right)$ and a simultaneous eigenvector is called a $\mathsf{Hecke} \mathsf{eigenform} \mathsf{or} \mathsf{simply} \mathsf{an} \mathsf{eigenform}$.

Let $\lambda$ be a place of the algebraic closure $\overline{\mathbb{Q}}$ of $\mathbb{Q}$ in $\mathbb{C}$ lying over a rational integer ℓ and ${\overline{\mathbb{Q}}}_{\lambda }$ denote the algebraic closure of ${\mathbb{Q}}_{\mathsf{\ell }}$ via $\lambda$. Let $G_{\mathbb{Q}}:=\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be the absolute Galois group of $\mathbb{Q}$. It is well-known that if $f\in S_k\left(N\right)$ is a normalized eigenform with $a_1\left(f\right)=1$, then there exists a unique continuous irreducible Galois representation

$${\rho }_{f,\lambda }:G_{\mathbb{Q}}⟶GL(2,{\overline{\mathbb{Q}}}_{\lambda })$$

such that ${\rho }_{f,\lambda }$ is unramified at p for all primes $p\nmid \mathsf{\ell }N$ and

$$\mathrm{Tr}\left({\rho }_{f,\lambda }\left({\mathrm{Frob}}_p\right)\right)=a_p\left(f\right)\mathrm{for}\mathrm{any}\mathrm{prime}p\nmid \mathsf{\ell }N.$$

The existence of ${\rho }_{f,\lambda }$ is due to Shimura if $k=2$ [11], due to Deligne if $k>2$ [12] and due to Deligne and Serre if $k=1$ [13]. We see that ${\rho }_{f,\lambda }$ is odd in the sense that det ${\rho }_{f,\lambda }$ of complex conjugation is −1. Moreover ${\rho }_{f,\lambda }$ is potentially semi-stable at ℓ in the sense of Fontaine [14].

We may choose a conjugate of ${\rho }_{f,\lambda }$, which is valued in $GL(2,{\mathcal{O}}_{{\overline{\mathbb{Q}}}_{\lambda }})$, and reducing modulo the maximal ideal and semi-simplyfing yields an irreducible continuous representation

$${\overline{\rho }}_{f,\lambda }:G_{\mathbb{Q}}⟶GL(2,{\overline{\mathbb{F}}}_{\mathsf{\ell }})$$

which, up to isomorphism, does not depend on the choice of conjugate of ${\rho }_{f,\lambda }$.

Definition 3.

Let

$$\rho :G_{\mathbb{Q}}⟶GL(2,{\overline{\mathbb{Q}}}_{\mathsf{\ell }})$$

be an irreducible continuous Galois representation, which is unramified outside finitely many primes, and for which the restriction of ρ to a decomposition group at ℓ is potentially semi-stable at ℓ in the sense of Fontaine. Then ρ is called $\mathsf{modular}$ if ρ is equivalent to ${\rho }_{f,\lambda }$ (denoted $\rho \sim {\rho }_{f,\lambda }$) for some normalized eigenform f and some place $\lambda |\mathsf{\ell }.$

Definition 4.

Let

$$\overline{\rho }:G_{\mathbb{Q}}⟶GL(2,{\overline{\mathbb{F}}}_{\mathsf{\ell }})$$

be a two-dimensional irreducible continuous representation of $G_{\mathbb{Q}}$. Then $\overline{\rho }$ is called $\mathsf{modular}$ if $\overline{\rho }\sim {\overline{\rho }}_{f,\lambda }$ for some normalized eigenform f and some place $\lambda |\mathsf{\ell }.$

Let E be an elliptic curve over $\mathbb{Q}$. We define

$$a_p\left(E\right):=p+1-\left|E\left({\mathbb{F}}_p\right)\right|\mathrm{for}\mathrm{a}\mathrm{prime}p.$$

The L-function $L(E,s)$ of E is defined by the product of the local L-factors

$$L(E,s):=\prod _{p|D}\left({\displaystyle \frac{1}{1-a_p\left(E\right)p^{-s}}}\right)\prod _{p\nmid D}\left({\displaystyle \frac{1}{1-a_p\left(E\right)p^{-s}+p^{1-2s}}}\right).$$

Then $L(E,s)$ converges absolutely for Re $s>{\displaystyle \frac{3}{2}}$ and extends to an entire function by [9].

Definition 5.

An elliptic curve E over $\mathbb{Q}$ is called $\mathsf{modular}$ if there exists a Hecke eigenform $f\in S_2\left(N\right)$ such that

$$L(E,s)=L(f,s).$$

Let E be an elliptic curve over $\mathbb{Q}$ with its conductor $N\left(E\right)$. Let

$${\rho }_{E,\mathsf{\ell }}:G_{\mathbb{Q}}⟶GL(2,{\overline{\mathbb{Q}}}_{\mathsf{\ell }})$$

be the ℓ-adic representation of $G_{\mathbb{Q}}$ with the Tate module $T_{\mathsf{\ell }}\left(E\right)$ as its representation space. Let

$$J_1\left(N\right):={\Omega }^1{\left(X_1\left(N\right)\right)}^{\vee }/H_1(X_1\left(N\right),\mathbb{Z})\cong S_2{\left({\Gamma }_1\left(N\right)\right)}^{\vee }/H_1(X_1\left(N\right),\mathbb{Z})$$

be the Jacobian variety of the modular curve $X_1\left(N\right)$. Here ${\Omega }^1\left(X_1\left(N\right)\right)$ denotes the complex vector space of holomorphic 1-forms on $X_1\left(N\right)$ and $W^{\vee }$ denotes the dual space of a complex vector space W. It is known that the following statements are equivalent:

(a)

E is modular.

(b)

There is a non-constant holomorphic mapping $X_1\left(N\right)⟶E\left(\mathbb{C}\right)$ for some positive integer N.

(c)

There is a non-constant holomorphic mapping $J_1\left(N\right)⟶E\left(\mathbb{C}\right)$ for some positive integer N.

(d)

${\rho }_{E,\mathsf{\ell }}$ is modular for a prime ℓ.

The above statements have been called the Shimura–Taniyama conjecture. We refer to [15] for the historical story of this conjecture. The implication (a) ⟹ (b) follows from a construction of Shimura [11] and a theorem of Faltings [16]. The implication (b) ⟹ (d) is due to Mazur [17]. The implication (d) ⟹ (a) follows from a theorem of Carayol [18]. The implication (c) ⟹ (b) is obvious. Wiles [5,6] proved that a semi-stable elliptic curve over $\mathbb{Q}$ is modular by proving the statement (d). Thereafter Breuil, Conrad, Diamond and Taylor [9] proved that every elliptic curve over $\mathbb{Q}$ is modular.

Serre [3] conjectured the following:

• Serre’s Modularity Conjecture: Let $\overline{\rho }:G_{\mathbb{Q}}⟶GL(2,\mathbb{F})$ be a two-dimensional absolutely irreducible, continuous, odd representation of $G_{\mathbb{Q}}$. Here $\mathbb{F}$ is a finite field of characteristic p. Then $\overline{\rho }$ is modular, i.e., arises from (with respect to some fixed embedding $\mathit{ı}:\overline{\mathbb{Q}}\hookrightarrow \overline{{\mathbb{Q}}_p})$ a newform f of some weight $k\ge 2$ and level N prime to p.

In 2009, Khare and Wintenberger [19,20] proved that Serre’s Modularity Conjecture is true.

3. The Modularity of an Abelian Variety

Let $G:=Sp(2g,\mathbb{R})$ and $K=U\left(g\right).$ Let

$${\mathbb{H}}_g:=\{\Omega \in {\mathbb{C}}^{(g,g)}|\Omega ={}^t\Omega ,\mathrm{Im}\Omega >0\}$$

be the Siegel upper half-plane of degree g. Then G acts on ${\mathbb{H}}_g$ transitively by

$$\alpha ·\Omega =(A\Omega +B){(C\Omega +D)}^{-1},$$

(1)

where $\alpha =\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)\in G$ and $\Omega \in {\mathbb{H}}_g.$ The stabilizer of the action (1) at $i{I}_g$ is

$$\left\{\left(\begin{array}{cc}A& B\\ -B& A\end{array}\right)|A+iB\in U\left(g\right)\right\}\cong U\left(g\right).$$

Thus we get the biholomorphic map

$$G/K⟶{\mathbb{H}}_g,\alpha K⟼\alpha ·i{I}_g,\alpha \in G.$$

It is known that ${\mathbb{H}}_g$ is an Einstein–Kähler Hermitian symmetric space.

Let ${\Gamma }_g:=Sp(2g,\mathbb{Z})$ be the Siegel modular group of degree g. For a positive integer N, we let

$${\Gamma }_g\left(N\right):=\left\{\gamma \in {\Gamma }_g|\gamma \equiv I_{2g}\left(\mathrm{mod}N\right)\right\}$$

be the the principal congruence subgroup of ${\Gamma }_g$ of level N. Let

$${\Gamma }_{g,0}\left(N\right):=\left\{\gamma \in {\Gamma }_g|\gamma =\left(\begin{array}{cc}A& B\\ C& D\end{array}\right),C\equiv 0\left(\mathrm{mod}N\right)\right\}$$

and

$${\Gamma }_{g,1}\left(N\right):=\left\{\gamma \in {\Gamma }_g|\gamma =\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)\equiv \left(\begin{array}{cc}{I}_g& \ast \\ 0& I_g\end{array}\right)\left(\mathrm{mod}N\right)\right\}$$

be the congruence subgroups of Level N. Then we have the relation

$${\Gamma }_g\left(N\right)\subset {\Gamma }_{g,1}\left(N\right)\subset {\Gamma }_{g,0}\left(N\right)\subset {\Gamma }_g.$$

Definition 6.

Let Γ be a congruence subgroup of ${\Gamma }_g$ and let k be a non-negative integer k. A function $F:{\mathbb{H}}_g⟶\mathbb{C}$ is called a $\mathsf{Siegel} \mathsf{modular} \mathsf{form}$ of degree g and weight k with respect to Γ if it satisfies the following conditions:

($\mathsf{S1}$) $F(\Omega )$ is holomorphic on ${\mathbb{H}}_g$;

($\mathsf{S2}$) $F(\gamma ·\Omega )={(C\Omega +D)}^{k}F(\Omega )$ for all $\gamma =\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)\in \Gamma$ and $\Omega \in {\mathbb{H}}_g$;

($\mathsf{S3}$) $F(\Omega )$ is bounded in any domain $Y\ge Y_0>0$ in the case $g=1$.

We denote the space of all Siegel modular forms of degree g and weight k with respect to $\Gamma$ by $[\Gamma ,k]$.

We define the so-called $\mathsf{Siegel}\mathsf{operator}$

$${\Phi }_g:[{\Gamma }_g,k]⟶[{\Gamma }_{g-1},k]$$

by

$$\left({\Phi }_g\left(F\right)\right)\left({\Omega }_1\right):=\underset{t⟶\infty }{\lim }F\left(\begin{array}{cc}{\Omega }_1& 0\\ 0& it\end{array}\right),{\Omega }_1\in {\mathbb{H}}_{g-1}.$$

Then ${\Phi }_g$ is a well-defined linear mapping (cf. [21] (pp. 187–189)). A Siegel modular form $F\in [{\Gamma }_g,k]$ is called a $\mathsf{Siegel}\mathsf{cusp}\mathsf{form}$ if ${\Phi }_g\left(F\right)=0$ (cf. [21] (p. 198)). ${[{\Gamma }_g,k]}_0$ denotes the space of all Siegel cusp forms in $[{\Gamma }_g,k]$.

Let $\Gamma$ be a congruence subgroup of ${\Gamma }_g$. If $F\in [\Gamma ,k]$, then F has a Fourier expansion

$$F(\Omega )=\sum _{T}a(T;F)e^{2\pi i\mathrm{Tr}(T\Omega )},$$

where T runs through all $g\times g$ half-integral semi-positive symmetric matrices. Here $\mathrm{Tr}\left(M\right)$ denotes the trace of a square matrix M. Following Hecke’s method, Maass [21] (pp. 202–210) associated with $F(\Omega )$ the Dirichlet series

$$D(F,s):=\sum _{\left\{T\right\}}{\displaystyle \frac{a(T;F)}{\epsilon \left(T\right)}}{\left(\det T\right)}^{-s},$$

where the summation indicates that T runs through a complete set of representatives of the sets

$$\left\{T\left[U\right]|U\mathrm{unimodular}\right\},T>0$$

and $\epsilon \left(T\right)$ denotes the number of unimodular matrices U which satisfy the equation $T\left[U\right]=T.$ We note that the numbers $\epsilon \left(T\right)$ are finite.

Definition 7.

Let F be a nonzero Siegel–Hecke eigenform in ${[{\Gamma }_g,k]}_0$. Let ${\alpha }_{p,0},{\alpha }_{p,1}, \cdots ,$${\alpha }_{p,g}$ be the p-Satake parameters of F at a prime p (cf. see Appendix A). We define the $\mathsf{local}\mathsf{spinor}\mathsf{zeta}\mathsf{function}$ $Z_{F,p}\left(t\right)$ of F at p by

$$Z_{F,p}\left(t\right):=(1-{\alpha }_{p,0}t)\sum _{r=1}^g\sum _{1\le i_1<\cdots <i_r\le g}(1-{\alpha }_{p,0}{\alpha }_{p,i_1}\cdots {\alpha }_{p,i_r}t).$$

The $\mathsf{spinor}\mathsf{zeta}\mathsf{function}$ $Z_F\left(s\right)$ of F is defined to be the following function

$$Z_F\left(s\right):=\sum _{p:\mathrm{prime}}{Z}_{F,p}{\left(p^{-s}\right)}^{-1},\mathrm{Re}s\gg 0.$$

Secondly one has the so-called $\mathsf{standard}\mathsf{zeta}\mathsf{function}$ $D_F\left(s\right)$ of a Siegel–Hecke eigenform F in ${[{\Gamma }_g,k]}_0$ defined by

$$D_F\left(s\right):=\sum _{p:\mathrm{prime}}{D}_{F,p}{\left(p^{-s}\right)}^{-1},\mathrm{Re}s\gg 0,$$

where

$$D_{F,p}\left(t\right)=(1-t)\sum _{i=1}^g(1-{\alpha }_{p,i}t)(1-{\alpha }_{p,i}^{-1}t).$$

We refer to [22] (p. 249).

Remark 1.

(1) If $g=1$, the spinor zeta function $Z_f\left(s\right)$ of a Hecke eigenform f in $S_k\left({\Gamma }_1\right)$ is nothing but the Hecke L-function $L(f,s)$ of f.

• (2) If $g=1$, the standard zeta function $D_f\left(s\right)$ of a Hecke eigenform $f\left(\tau \right)={\sum }_{n=1}^{\infty }a\left(n\right)e^{2\pi in\tau }$ in $S_k\left({\Gamma }_1\right)$ has the following equation

  $$D_f(s-k+1)=\sum _{p:\mathrm{prime}}{(1+p^{k-s-1})}^{-1}·\sum _{n=1}^{\infty }a\left(n^2\right)n^{-s}.$$

Let A be a g-dimensional simple abelian variety defined over $\mathbb{Q}$. For a prime ℓ, we set

$$A\left[{\mathsf{\ell }}^n\right]:=\{x\in A\left(\overline{\mathbb{Q}}\right)|{\mathsf{\ell }}^n·x=0\}.$$

Then $A\left[{\mathsf{\ell }}^n\right]\cong {(\mathbb{Z}/{\mathsf{\ell }}^n\mathbb{Z})}^g\times {(\mathbb{Z}/{\mathsf{\ell }}^n\mathbb{Z})}^g$ (cf. [8]). Then the Tate module of A is given by

$$T_{\mathsf{\ell }}\left(A\right):=\underset{⟵}{\lim }A\left[{\mathsf{\ell }}^n\right]\cong {\mathbb{Z}}_{\mathsf{\ell }}^g\times {\mathbb{Z}}_{\mathsf{\ell }}^g\cong {\mathbb{Z}}_{\mathsf{\ell }}^{2g}.$$

Therefore we have the $2g$-dimensional ℓ-adic Galois representation of $G_{\mathbb{Q}}$

$${\rho }_{A,\mathsf{\ell }}:G_{\mathbb{Q}}⟶GL(2g,{\mathbb{Z}}_{\mathsf{\ell }})\subset GL(2g,{\mathbb{Q}}_{\mathsf{\ell }}).$$

Definition 8.

A $2g$-dimensional ℓ-adic Galois representation ρ of $G_{\mathbb{Q}}$ given by

$$\rho :G_{\mathbb{Q}}⟶GL(2g,{\mathbb{Z}}_{\mathsf{\ell }})\subset GL(2g,{\mathbb{Q}}_{\mathsf{\ell }})$$

is called $\mathsf{modular}$ if there is a Siegel–Hecke eigenform $F(\Omega )\in {[{\Gamma }_{g,0}\left(N\right),g+1]}_0$ of weight $g+1$ with respect to ${\Gamma }_{g,0}\left(N\right)$ such that

$$\mathrm{Tr}\left(\rho \left({\mathrm{Frob}}_p\right)\right)=a(p{I}_g;F)\mathrm{and}\det \left(\rho \left({\mathrm{Frob}}_p\right)\right)=p^g\mathrm{for}\mathrm{any}\mathrm{prime}p\nmid \mathsf{\ell }N,$$

where

$$F(\Omega )=\sum _{T}a(T;F)e^{2\pi i\mathrm{Tr}(T\Omega )}$$

is a Fourier expansion of $F(\Omega )$.

Definition 9.

Let A be a g-dimensional simple abelian variety defined over $\mathbb{Q}$ and let ℓ be a prime. For a prime p, we let

$$L_p(A,s):={\left\{\det \left(I_{2g}-p^{-s}·{\rho }_{A,\mathsf{\ell }}\left({\mathrm{Frob}}_p\right){|}_{T_{\mathsf{\ell }}\left(A\right)}\right)\right\}}^{-1}$$

be the local L-function of A at p. We define the L-function $L(A,s)$ of A by

$$L(A,s)=\prod _{p:\mathrm{prime}}{L}_p(A,s).$$

Definition 10.

Let A be a g-dimensional simple abelian variety defined over $\mathbb{Q}$. A is called $\mathsf{modular}$ if there exists a Siegel–Hecke eigenform $F(\Omega )\in {[{\Gamma }_{g,0}\left(N\right),g+1]}_0$ of weight $g+1$ with respect to ${\Gamma }_{g,0}\left(N\right)$ such that

$$L(A,s)=D(F,s),Z_F\left(s\right)\mathrm{or}{D}_F\left(s\right).$$

For two positive integers g and N, we let

$${\mathcal{A}}_{g,0}\left(N\right):={\Gamma }_{g,0}\left(N\right)\setminus {\mathbb{H}}_g$$

be the Siegel modular variety of level structure N and let ${\mathcal{A}}_{g,0}^{\mathrm{tor}}\left(N\right)$ be a smooth toroidal compactification of ${\mathcal{A}}_{g,0}\left(N\right)$ (cf. [23,24]).

$${\Omega }^i\left({\mathcal{A}}_{g,0}^{\mathrm{tor}}\left(N\right)\right),0\le i\le {\displaystyle \frac{g(g+1)}{2}}$$

denotes the complex vector space of holomorphic i-forms on ${\mathcal{A}}_{g,0}^{\mathrm{tor}}\left(N\right)$. The Jacobian variety $\mathrm{Jac}({\mathcal{A}}_{g,0}^{\mathrm{tor}}\left(N\right))$ of ${\mathcal{A}}_{g,0}^{\mathrm{tor}}\left(N\right)$ is defined to be the abelian variety

$$\mathrm{Jac}\left({\mathcal{A}}_{g,0}^{\mathrm{tor}}\left(N\right)\right):={\Omega }^{\nu }{\left({\mathcal{A}}_{g,0}^{\mathrm{tor}}\left(N\right)\right)}^{\vee }/H_{\nu }\left({\mathcal{A}}_{g,0}^{\mathrm{tor}}\left(N\right),\mathbb{Z}\right),\nu ={\displaystyle \frac{g(g+1)}{2}}.$$

(2)

The geometric genus of ${\mathcal{A}}_{g,0}^{\mathrm{tor}}\left(N\right)$ is the dimension of the Jacobian variety $\mathrm{Jac}({\mathcal{A}}_{g,0}^{\mathrm{tor}}\left(N\right))$. It is known that the following two vector spaces are isomorphic:

$${[{\Gamma }_{g,0}\left(N\right),g+1]}_0\cong {\Omega }^{\nu }\left({\mathcal{A}}_{g,0}^{\mathrm{tor}}\left(N\right)\right),\nu ={\displaystyle \frac{g(g+1)}{2}}.$$

(3)

More precisely, for a coordinate $\Omega =\left({\omega }_{ij}\right)\in {\mathbb{H}}_g$, we let

$${\omega }_0:=d{\omega }_{11}\wedge d{\omega }_{12}\wedge d{\omega }_{13}\wedge \cdots \wedge d{\omega }_{gg}$$

be a holomorphic $\nu$-form on ${\mathbb{H}}_g$. If $\omega =F(\Omega ){\omega }_0$ is a ${\Gamma }_{g,0}\left(N\right)$-invariant holomorphic form on ${\mathbb{H}}_g$, then

$$F(\gamma ·\Omega )=\det {(C\Omega +D)}^{g+1}F(\Omega )$$

for all $\gamma =\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)\in {\Gamma }_{g,0}\left(N\right)$ and $\Omega \in {\mathbb{H}}_g.$ Thus $F\in [{\Gamma }_{g,0}\left(N\right),g+1].$ It was shown by Freitag [25] that $\omega$ can be extended to a holomorphic $\nu$-form on ${\mathcal{A}}_{g,0}^{\mathrm{tor}}\left(N\right)$ if and only if F is a cusp form in ${[{\Gamma }_{g,0}\left(N\right),g+1]}_0.$ Indeed, the mapping

$${[{\Gamma }_{g,0}\left(N\right),g+1]}_0⟶{\Omega }^{\nu }\left({\mathcal{A}}_{g,0}^{\mathrm{tor}}\left(N\right)\right),F⟼F{\omega }_0$$

is an isomorphism as complex vector spaces. We observe that if ${\omega }_k:=G(\Omega ){\omega }_0^{\otimes k}$ is a ${\Gamma }_{g,0}\left(N\right)$-invariant holomorphic form on ${\mathbb{H}}_g$ of degree $k\nu$, then $G(\Omega )\in {[{\Gamma }_{g,0}\left(N\right),k(g+1)]}_0$ is a cusp form of weight $k(g+1).$

Therefore according to (2) and (3), we have

$$\mathrm{Jac}\left({\mathcal{A}}_{g,0}^{\mathrm{tor}}\left(N\right)\right)\cong {[{\Gamma }_{g,0}\left(N\right),g+1]}_0^{\vee }/H_{\nu }\left({\mathcal{A}}_{g,0}^{\mathrm{tor}}\left(N\right),\mathbb{Z}\right),\nu ={\displaystyle \frac{g(g+1)}{2}}.$$

(4)

If there is no confusion, we simply set

$$J_{g,0}\left(N\right):=\mathrm{Jac}\left({\mathcal{A}}_{g,0}^{\mathrm{tor}}\left(N\right)\right).$$

We propose the following conjectures.

Conjecture 1.

A simple abelian variety of dimesion g defined over $\mathbb{Q}$ is modular.

Conjecture 2.

Let A be a simple abelian variety of dimension g defined over $\mathbb{Q}$. The following statements are equivalent:

• ($\mathsf{MAV1}$)

  A is modular.

  ($\mathsf{MAV2}$)

  There exists a non-constant holomorphic mapping ${\mathcal{A}}_{g,0}^{\mathrm{tor}}\left(N\right)⟶A$ for some positive integer N.

  ($\mathsf{MAV3}$)

  There exists a non-constant holomorphic mapping $J_{g,0}\left(N\right)⟶A$ for some positive integer N.

  ($\mathsf{MAV4}$)

  ${\rho }_{A,\mathsf{\ell }}$ is modular for any prime ℓ.

We propose the following problems.

Problem 1.

Let $F\in {[{\Gamma }_{g,0}\left(N\right),g+1]}_0$ be a Siegel–Hecke eigenform of weight $g+1$. Associate to F a $2g$-dimensional continuous irreducible Galois representation of $G_{\mathbb{Q}}$.

Problem 2.

Let k be a positive integer. Let $F\in {[{\Gamma }_{g,0}\left(N\right),k]}_0$ be a Siegel–Hecke eigenform of weight k. Associate to F a $2g$-dimensional continuous irreducible Galois representation of $G_{\mathbb{Q}}$.

Remark 2.

As mentioned in Section 2, in the case $g=1$, to a Hecke eigenform of weight $k\ge 1$, Shimura, Deligne and Serre [11,12,13] associated a two-dimensional continuous irreducible Galois representation. For the case $g=2$, Taylor [26,27] tried to associate the four dimensional continuous Galois representation of $G_{\mathbb{Q}}$ to a Siegel–Hecke eigenform of small weight. But he did not specify the precise weight.