Strongly visible subgroups

Let $A_{/\mathbb{Q}}$ be an abelian subvariety of $J_0(N)_{/\mathbb{Q}}$ and let $p\nmid N$ be a prime. Let

$$\varphi = \delta_1^* + \delta_p^* : J_0(N) \to J_0(pN),$$ (3)

where $\delta_1^*$ and $\delta_p^*$ are the pullback maps on equivalence classes of degree-zero divisors of the degeneracy maps $\delta_1, \delta_p : X_0(pN) \rightarrow X_0(N)$ . Let $\H ^1(\mathbb{Q},A)^{{\mathrm{odd}}}$ be the prime-to-2-part of the group $\H ^1(\mathbb{Q},A)$ .

Definition 5.1.1 (Strongly Visibility)   The strongly visible subgroup of $\H ^1(\mathbb{Q},A)$ for $J_0(pN)$ is

$${\mathrm{Vis}}_{pN}\H ^1(\mathbb{Q},A) = {\mathrm{Ker}}\left(\H ^... ...rrow{\varphi _*} \H ^1(\mathbb{Q},J_0(pN))\right) \subset \H ^1(\mathbb{Q},A).$$

Also,

$${\mathrm{Vis}}_{pN}{\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\f... ...}\selectfont Sh}}}(\mathbb{Q}, A) \cap {\mathrm{Vis}}_{pN}\H ^1(\mathbb{Q},A).$$

The reason we replace $\H ^1(\mathbb{Q},A)$ by $\H ^1(\mathbb{Q},A)^{{\mathrm{odd}}}$ is that the kernel of $\varphi$ is a $2$ -group (see [Rib90b]).

Remark 5.1.2   We could obtain more visible subgroups by considering the map $\delta_1^* - \delta_p^*$ in Definition 5.1.1. However, the methods of this paper do not apply to this map.

For a positive integer $N$ , let

$$\nu(N) = \frac{1}{6} \cdot \prod_{q^r\Vert N} (q^r+q^{r-1}) = \frac{1}{6} \cdot [{\mathrm{SL}}_2(\mathbb{Z}) : \Gamma_0(N)].$$

We call the number $\nu(N)$ the Sturm bound (see [Stu87]).

Theorem 5.1.3   Let $A_{/\mathbb{Q}}=A_f$ be a newform abelian subvariety of $J_0(N)$ for which $L(A_{/\mathbb{Q}},1)\neq 0$ and let $p\nmid N$ be a prime. Suppose that there is a maximal ideal $\lambda \subset \mathbb{T}(N)$ and an elliptic curve  $E_{/\mathbb{Q}}$ of conductor $pN$ such that:

1. [Nondivisibility]The residue characteristic $\ell$ of $\lambda$ satisfies
   $$\ell\nmid 2 \cdot N \cdot p \cdot \prod_{q\mid N} c_{E,q}.$$

2. [Component Groups] For each prime $q\mid N$ ,
   $$\Phi_{A,q}(\mathbb{F}_q)[\lambda] = 0.$$

3. [Fourier Coefficients] Let $a_n(E)$ be the $n$ -th Fourier coefficient of the modular form attached to $E$ , and $a_n(f)$ the $n$ -th Fourier coefficient of $f$ . Assume that $a_p(E) = -1$ ,
   $$a_p(f) \equiv -(p+1)\pmod{\lambda}$$    and $$\quad a_q(f) \equiv a_q(E) \pmod{\lambda},$$
   for all primes $q\neq p$ with $q\leq \nu(pN)$ .

4. [Irreducibility] The mod $\ell$ representation $\overline{\rho}_{E,\ell}$ is irreducible.

Then there is an injective homomorphism

$$E(\mathbb{Q})/\ell E(\mathbb{Q}) \hookrightarrow {\mathrm{Vis}}_{... ...wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, A_f))[\lambda].$$

Remark 5.1.4   In fact, we have

$$E(\mathbb{Q})/\ell E(\mathbb{Q}) \hookrightarrow {\mathrm{Ker}}({... ...wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, A_f))[\lambda],$$

where $C\subset J_0(pN)$ is isogenous to $A_f\times E$ .