Notation

1. Abelian varieties. For a number field $K$ , $A_{/K}$ denotes an abelian variety over $K$ . We denote the dual of $A$ by $A^\vee_{/K}$ . If $\varphi : A\to B$ is an isogeny of degree $n$ , we denote the complementary isogeny by $\varphi '$ ; this is the isogeny $\phi' : B \rightarrow A$ , such that $\varphi \circ \varphi ' = \varphi ' \circ \varphi = [n]$ , the multiplication-by-$n$ map on $A$ . Unless otherwise specified, Néron models of abelian varieties will be denoted by the corresponding caligraphic letters, e.g., $\mathcal{A}$ denotes the Néron model of $A$ .

2. Galois cohomology. For a fixed algebraic closure $\overline{K}$ of $K$ , $G_K$ will be the Galois group ${\mathrm{Gal}}(\overline{K}/K)$ . If $v$ is any non-archimedean place of $K$ , $K_v$ and $k_v$ will always mean the completion and the residue field of $K$ at $v$ , respectively. By $K_v^{{\mathrm{ur}}}$ we always mean the maximal unramified extension of the completion $K_v$ . Given a $G_K$ -module $M$ , we let $\H ^1(K, M)$ denote the Galois cohomology group $\H ^1(G_K, M)$ .

3. Component groups. The component group of $A$ at $v$ is the finite group $\Phi_{A,v} = \mathcal{A}_{k_v} / \mathcal{A}_{k_v}^0$ which also has a structure of a finite group scheme over $k_v$ . The Tamagawa number of $A$ at $v$ is $c_{A,v} = \char93 \Phi_{A,v}(k_v)$ , and the component group order of $A$ at $v$ is $\overline{c}_{A,v} = \char93 \Phi_{A,v}(\overline{k}_v)$ .

4. Modular abelian varieties. Let $h = 0$ or $1$ . A $J_h$ -modular abelian variety is an abelian variety $A_{/K}$ which is a quotient of $J_h(N)$ for some $N$ , i.e., there exists a surjective morphism $J_h(N) \twoheadrightarrow A$ defined over $K$ . We define the level of a modular abelian variety $A$ to be the minimal $N$ , such that $A$ is a quotient of $J_h(N)$ . The modularity theorem of Wiles et al. (see [BCDT01]) implies that all elliptic curves over  $\mathbb{Q}$ are modular. Serre's modularity conjecture implies that the modular abelian varieties over $\mathbb{Q}$ are precisely the abelian varieties over  $\mathbb{Q}$ of ${\mathrm{GL}}_2$ -type (see [Rib92, §4]).

5. Shimura construction. Let $$f=\sum_{n=1}^\infty a_n q^n\in S_2(\Gamma_0(N))$$ be a newform of level $N$ and weight 2 for $\Gamma_0(N)$ which is an eigenform for all Hecke operators in the Hecke algebra $\mathbb{T}(N)$ . Shimura (see [Shi94, Thm. 7.14]) associated to $f$ an abelian subvariety ${A_f}_{/\mathbb{Q}}$ of $J_0(N)$ , simple over $\mathbb{Q}$ , of dimension $d = [K : \mathbb{Q}]$ , where $K = \mathbb{Q}(\dots, a_n, \dots)$ is the Hecke eigenvalue field. More precisely, if $I_f = {\mathrm{Ann}}_{\mathbb{T}(N)}(f)$ then $A_f$ is the connected component containing the identity of the $I_f$ -torsion subgroup of $J_0(N)$ , i.e., $A_f = J_0(N)[I_f]^0 \subset J_0(N)$ . The quotient $\mathbb{T}(N)/I_f$ of the Hecke algebra $\mathbb{T}(N)$ is a subalgebra of the endomorphism ring ${\mathrm{End}}_\mathbb{Q}(A_{/\mathbb{Q}})$ . Also $$L(A_f,s) = \prod_{i = 1}^d L(f_i,s)$$ , where the $f_i$ are the $G_\mathbb{Q}$ -conjugates of $f$ . We also consider the dual abelian variety $A_f^\vee$ which is a quotient variety of $J_0(N)$ .

theorem_type[remark][theorem][][remark][][] In this paper tex2html_wrap_inline$A_f$ always denotes an abelian subvariety of tex2html_wrap_inline$J_0(N)$. By abuse of notation, it is also common to denote by tex2html_wrap_inline$A_f$ the dual of the subvariety tex2html_wrap_inline$A_f$, which is a quotient of tex2html_wrap_inline$J_0(N)$ (see e.g. []).

6. $I$ -torsion submodules. If $M$ is a module over a commutative ring $R$ and $I$ is an ideal of $R$ , let

$$M[I] = \{ x \in M : mx = 0$$    all $$m \in I\}$$

be the $I$ -torsion submodule of $M$ .

7. Hecke algebras. Let $S_2(\Gamma)$ denote the space of cusp forms of weight $2$ for any congruence subgroup $\Gamma$ of ${\mathrm{SL}}_2(\mathbb{Z})$ . Let

$$\mathbb{T}(N) = \mathbb{Z}[\ldots, T_n, \ldots]\subseteq {\mathrm{End}}_{\mathbb{Q}}(J_0(N))$$

be the Hecke algebra, where $T_n$ is the $n$ th Hecke operator. $\mathbb{T}(N)$ also acts on $S_2(\Gamma_0(N))$ and the integral homology $H_1(X_0(N),\mathbb{Z})$ .

8. Modular degree. If $A$ is an abelian subvariety of $J_0(N)$ , let

$$\theta:A\to J_0(N) \cong J_0(N)^{\vee} \to A^{\vee}$$

be the induced polarization. The modular degree of $A$ is

$$m_A = \sqrt{\char93 {\mathrm{Ker}}(A\xrightarrow{\theta} A^{\vee})}.$$

See [AS02] for why $m_A$ is an integer and for an algorithm to compute it.