Proof of Theorem 5.1.3

Proof. [Proof of Theorem 5.1.3] By [BCDT01] $E$ is modular, so there is a rational newform $f \in S_2^{{\mathrm{new}}}(pN)$ which is an eigenform for the Hecke operators and an isogeny $E \to E_f$ defined over  $\mathbb{Q}$ , which by Hypothesis 4 can be chosen to have degree coprime to $\ell$ . Indeed, every cyclic rational isogeny is a composition of rational isogenies of prime degree, and $E$ admits no rational $\ell$ -isogeny since $\overline{\rho}_{E,\ell}$ is irreducible.

By Hypothesis 1 the Tamagawa numbers of $E$ are coprime to $\ell$ . Since $E$ and $E_f$ are related by an isogeny of degree coprime to $\ell$ , the Tamagawa numbers of $E_f$ are also not divisible by $\ell$ by Lemma 5.2.6. Moreover, note that

$$E(\mathbb{Q})\otimes \mathbb{F}_\ell \cong E_f(\mathbb{Q}) \otimes \mathbb{F}_\ell.$$

Let $\mathfrak{m}$ be the ideal of $\mathbb{T}(pN)$ generated by $\ell$ and $T_n-a_n(E)$ for all integers $n$ coprime to $p$ . Note that $\mathfrak{m}$ is maximal by Lemma 5.2.4.

Let $\varphi$ be as in (3), and let $A=\varphi (A_f)$ . Note that if $T_n\in \mathbb{T}(pN)$ then $T_n(E_f) \subset E_f$ since $E_f$ is attached to a newform, and if, moreover $p\nmid n$ , then $T_n(A) \subset A$ since the Hecke operators with index coprime to $p$ commute with the degeneracy maps. Lemma 5.2.1 implies that

$$E_f[\ell] = E_f[\mathfrak{m}] = \varphi (A_f[\lambda]) \subset A,$$

so $\Psi = E_f[\ell]$ is a subgroup of $A$ as a $G_{\mathbb{Q}}$ -module. Let

$$C= (A\times E_f)/\Psi,$$

where we embed $\Psi$ in $A\times E_f$ anti-diagonally, i.e., by the map $x\mapsto (x,-x)$ . The antidiagonal map $\Psi \to A\times E_f$ commutes with the Hecke operators $T_n$ for $p\nmid n$ , so $(A\times E_f)/\Psi$ is preserved by the $T_n$ with $p\nmid n$ . Let $R$ be the subring of ${\mathrm{End}}(C)$ generated by the action of all Hecke operators $T_n$ , with $p\nmid n$ . Also note that $T_p \in {\mathrm{End}}(J_0(pN))$ acts by Hypothesis 3 as $-1$ on $E_f$ , but $T_p$ need not preserve $A$ .

Suppose for the moment that we have verified that the hypothesis of Theorem 4.1.1 are satisfied with $A$ , $B = E_f$ , $C$ , $Q = C/B$ , $R$ as above and $K=\mathbb{Q}$ . Then we obtain an injective homomorphism

$$E(\mathbb{Q})/\ell E(\mathbb{Q}) \cong E_f(\mathbb{Q})/\ell E_f(\... ...yr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, C))[\mathfrak{m}].$$

We then apply Lemma 5.2.2 with $n=\ell$ , $A_f$ , $A$ , and $C$ , respectively, to see that

$$E_f(\mathbb{Q})/\ell E_f(\mathbb{Q}) \subset {\mathrm{Ker}}({\mbo... ...y{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, C))[\lambda].$$

That $E_f(\mathbb{Q})/\ell E_f(\mathbb{Q})$ lands in the $\lambda$ -torsion is because the subgroup of ${\mathrm{Vis}}_C({\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, E_f))$ that we constructed is $\mathfrak{m}$ -torsion.

Finally, consider $A\times E_f \to J_0(pN)$ given by $(x,y)\mapsto x+y$ . Note that $\Psi$ maps to 0 , since $(x,-x)\mapsto 0$ and the elements of $\Psi$ are of the form $(x,-x)$ . We have a (not-exact!) sequence of maps

$${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontsh... ...amily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, J_0(pN)),$$

hence inclusions

$$E_f(\mathbb{Q})/\ell E_f(\mathbb{Q}) \subseteq {\mathrm{Ker}}({\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fonts... ...}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, C))$$

$$\subseteq {\mathrm{Ker}}({\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fonts... ...amily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, J_0(pN))),$$

which gives the conclusion of the theorem.

It remains to verify the hypotheses of Theorem 4.1.1. That $C=A+B$ is clear from the definition of $C$ . Also, $A\cap E_f = E_f[\ell]$ , which is finite. We explained above when defining $R$ that each of $A$ and $E_f$ is preserved by $R$ . Since $K=\mathbb{Q}$ and $\ell$ is odd the condition $1=e<\ell-1$ is satisfied. That $A(\mathbb{Q})$ is finite follows from our hypothesis that $L(A_f,1)\neq 0$ (by [KL89]).

It remains is to verify that the groups

$$Q(\mathbb{Q})[\mathfrak{m}] ,\quad E_f(\mathbb{Q})[\mathfrak{m}] ,\quad \Phi_{A,q}(\mathbb{F}_q)[\mathfrak{m}] ,$$    and $$\Phi_{E_f,q}(\mathbb{F}_q)[\ell] ,$$

are 0 for all primes $q\mid pN$ . Since $\ell\in \mathfrak{m}$ , we have by Hypothesis 4 that

$$E_f(\mathbb{Q})[\mathfrak{m}] = E_f(\mathbb{Q})[\ell] = 0.$$

We will now verify that $Q(\mathbb{Q})[\mathfrak{m}]=0$ . From the definition of $C$ and $\Psi$ we have $Q \cong A/\Psi.$ Let $\lambda_p$ be as in Lemma 5.2.4 with $a_n = a_n(E)$ . The map $\varphi$ induces an isogeny of $2$ -power degree

$$A_f/(A_f[\lambda]) \to A/\Psi.$$

Thus there is $\lambda_p$ -torsion in $(A_f/(A_f[\lambda]))(\mathbb{Q})$ if and only if there is $\mathfrak{m}$ -torsion in $(A/\Psi)(\mathbb{Q})$ . Thus it suffices to prove that $(A_f/A_f[\lambda])(\mathbb{Q})[\lambda_p] = 0$ .

By Lemma 5.2.4, we have $\lambda_p = \lambda$ , and by Lemma 5.2.5,

$$(A_f/A_f[\lambda])[\lambda] \cong A_f[\lambda^2]/A_f[\lambda].$$

By [Maz77, § II.14], the quotient $A_f[\lambda^2]/A_f[\lambda]$ injects into a direct sum of copies of $A_f[\lambda]$ as Galois modules. But $A_f[\lambda] \cong E[\ell]$ is irreducible, so $(A_f[\lambda^2]/A_f[\lambda])(\mathbb{Q}) = 0$ , as required.

By Hypothesis 2, we have $\Phi_{A_f,q}(\mathbb{F}_q)[\lambda] = 0$ for each prime divisor $q\mid N$ .Since $A$ is $2$ -power isogenous to $A_f$ and $\ell$ is odd, this verifies the Tamagawa number hypothesis for $A$ . Our hypothesis that $a_p(E) = -1$ implies that ${\mathrm{Frob}}_p$ acts on $\Phi_{E_f,p}(\overline{\mathbb{F}}_p)$ as $-1$ . Thus $\Phi_{E_f,p}(\mathbb{F}_p)[\ell] = 0$ since $\ell$ is odd. This completes the proof. $\qedsymbol$

Remark 5.3.1   An essential ingrediant in the proof of the above theorem is the multiplicity one result used in the paper of Wiles (see [Wil95, Thm. 2.1.]). Since this result holds for Jacobians $J_H$ of the curves $X_H(N)$ that are intermediate covers for the covering $X_1(N) \rightarrow X_0(N)$ corresponding to subgroups $H \subseteq (\mathbb{Z}/N\mathbb{Z})^\times$ (i.e., the Galois group of $X_1(N) \rightarrow X_H$ is $H$ ), one should be able to give a generalization of Theorem 5.1.3 which holds for newform subvarieties of $J_H$ . This requires generalizing some results from [Rib90b] to arbitrary $H$ .