Some auxiliary lemmas

We will use the following lemmas in the proof of Theorem 5.1.3. The notation is as in the previous section. In addition, if $f \in S_2(\Gamma_0(N))$ , we denote by $a_n(f)$ the $n$ -th Fourier coefficient of $f$ and by $K_f$ and $\mathcal{O}_f$ the Hecke eigenvalue field and its ring of integers, respectively.

Lemma 5.2.1   Suppose $A_f\subset J_0(N)$ and $A_g \subset J_0(pN)$ are attached to newforms $f$ and $g$ of level $N$ and $pN$ , respectively, with $p\nmid N$ . Suppose that there is a prime ideal $\lambda$ of residue characteristic $\ell\nmid 2pN$ in an integrally closed subring $\mathcal{O}$ of $\overline{\mathbb{Q}}$ that contains the ring of integers of the composite field $K = K_fK_g$ such that for $q\leq \nu(pN)$ ,

$$a_q(f) \equiv \begin{cases}a_q(g) \pmod{\lambda} & \text{i... ...$,}\\ (p+1)a_p(g)\pmod{\lambda} & \text{if $q=p$.} \end{cases}$$

Assume that $a_p(g) = -1$ . Let $\lambda_f = \mathcal{O}_f \cap \lambda$ and $\lambda_g = \mathcal{O}_g \cap \lambda$ and assume that $A_f[\lambda_f]$ is an irreducible $G_\mathbb{Q}$ -module. Then we have an equality

$$\varphi (A_f[\lambda_f]) = A_g[\lambda_g]$$

of subgroups of $J_0(pN)$ , where $\varphi$ is as in (3).

Proof. Our hypothesis that $a_p(f) \equiv -(p+1) \pmod{\lambda_f}$ implies, by the proofs in [Rib90b], that

$$\varphi (A_f[\lambda_f]) \subset \varphi (A_f) \cap J_0(pN)_{p\text{-new}},$$

where $J_0(pN)_{p\text{-new}}$ is the $p$-new abelian subvariety of $J_0(N)$ .

By [Rib90b, Lem. 1], the operator $U_p=T_p$ on $J_0(pN)$ acts as $-1$ on $\varphi (A_f[\lambda_f])$ . Consider the action of $U_p$ on the 2-dimensional vector space spanned by $\{f(q), f(q^p)\}$ . The matrix of $U_p$ with respect to this basis is

$$U_p = \left( \begin{matrix}a_p(f)&p\ -1&0 \end{matrix}\right).$$

In particular, neither of $f(q)$ and $f(q^p)$ is an eigenvector for $U_p$ . The characteristic polynomial of $U_p$ acting on the span of $f(q)$ and $f(q^p)$ is $x^2-a_p(f)x+p$ . Using our hypothesis on $a_p(f)$ again, we have

$$x^2 - a_p(f)x+p \equiv x^2 + (p+1) x + p \equiv (x + 1) (x + p)\pmod{\lambda}.$$

Thus we can choose an algebraic integer $\alpha$ such that

$$f_1(q) = f(q) + \alpha f(q^p)$$

is an eigenvector of $U_p$ with eigenvalue congruent to $-1$ modulo $\lambda$ . (It does not matter for our purposes whether $x^2+a_p(f)x+p$ has distinct roots; nonetheless, since $p\nmid N$ , [CV92, Thm. 2.1] implies that it does have distinct roots.) The cusp form $f_1$ has the same prime-indexed Fourier coefficients as $f$ at primes other than $p$ . Enlarge $\mathcal{O}$ if necessary so that $\alpha\in\mathcal{O}$ . The $p$ -th coefficient of $f_1$ is congruent modulo $\lambda$ to $-1$ and $f_1$ is an eigenvector for the full Hecke algebra. It follows from the recurrence relation for coefficients of the eigenforms that

$$a_n(g) \equiv a_n(f_1) \pmod{\lambda}$$

for all integers $n\leq \nu(pN)$ .

By [Stu87], we have $g\equiv f_1\pmod{\lambda}$ , so $a_q(g) \equiv a_q(f)\pmod{\lambda}$ for all primes $q\neq p$ . Thus by the Brauer-Nesbitt theorem [CR62], the 2-dimensional $G_\mathbb{Q}$ -representations $\varphi (A_f[\lambda_f])$ and $A_g[\lambda_g]$ are isomorphic.

Let $\mathfrak{m}$ be a maximal ideal of the Hecke algebra $\mathbb{T}(pN)$ that annihilates the module $A_g[\lambda_g]$ . Note that $A_g[\mathfrak{m}]=A_g[\lambda_g]$ since $A_g[\mathfrak{m}] \subset A_g[\lambda_g]$ and $A_g[\lambda_g]\cong \varphi (A_f[\lambda_f])$ is irreducible as a $G_\mathbb{Q}$ -module. The maximal ideal $\mathfrak{m}$ gives rise to a Galois representation $\overline{\rho}_\mathfrak{m}: G_\mathbb{Q}\rightarrow {\mathrm{GL}}_2(\mathbb{T}(pN)/\mathfrak{m})$ isomorphic to $A_g[\lambda_g]$ , which is irreducible since the Galois module $A_f[\lambda_f]$ is irreducible. Finally, we apply [Wil95, Thm. 2.1(i)] for $H = (\mathbb{Z}/ N\mathbb{Z})^\times$ (i.e., $J_H = J_0(N)$ ) to conclude that $J_0(N)(\overline{\mathbb{Q}})[\mathfrak{m}] \cong (\mathbb{T}(pN)/\mathfrak{m})^2$ , i.e., the representation $\overline{\rho}_\mathfrak{m}$ occurs with multiplicity one in $J_0(pN)$ . Thus

$$A_g[\lambda_g] = \varphi (A_f[\lambda_f]).$$

$\qedsymbol$

Lemma 5.2.2   Suppose $\varphi : A\to B$ and $\psi:B \to C$ are homomorphisms of abelian varieties over a number field $K$ , with $\varphi$ an isogeny and $\psi$ injective. Suppose $n$ is an integer that is relatively prime to the degree of $\varphi$ . If $G={\mathrm{Vis}}_C({\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, B))[n^{\infty}]$ , then there is some injective homomorphism

$$f: G \hookrightarrow {\mathrm{Ker}}\left\{(\psi\circ\varphi )_*:{... ...ily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, C)\right\},$$

such that $\varphi _*(f(G)) = G$ .

Proof. Let $m$ be the degree of the isogeny $\varphi : A\to B$ . Consider the complementary isogeny $\varphi ':B\to A$ , which satisfies $\varphi \circ \varphi ' = \varphi ' \circ \varphi = [m]$ . By hypothesis $m$ is coprime to $n$ , so $\gcd(m,\char93 G)=\gcd(m,n^{\infty})=1$ , hence

$$\varphi _* (\varphi '_*(G)) = [m]G = G.$$

Thus $\varphi '_*(G)$ maps, via $\varphi _*$ , to $G\subset {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, B)$ , which in turn maps to 0 in ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, C)$ . $\qedsymbol$

Lemma 5.2.3   Let $M$ be an odd integer coprime to $N$ and let $R$ be the subring of $\mathbb{T}(N)$ generated by all Hecke operators $T_n$ with $\gcd(n,M) = 1$ . Then $R = \mathbb{T}(N)$ .

Proof. See the lemma on page 491 of [Wil95]. (The condition that $M$ is odd is necessary, as there is a counterexample when $N=23$ and $M=2$ .) $\qedsymbol$

proof Let tex2html_wrap_inline$J = J_0(N)$ and tex2html_wrap_inline$T=T(N)$. Suppose tex2html_wrap_inline$p&mid#mid;M$ is a prime. We first show that there is a prime tex2html_wrap_inline$q&ne#neq;p$ such that tex2html_wrap_inline$T_q$ and tex2html_wrap_inline$T_p$ act in the same way on tex2html_wrap_inline$J[&ell#ell;]$. Consider the Galois representation displaymath &rho#rho;_&ell#ell;: G_Q&rarr#to;(Tate_&ell#ell;(J)). View tex2html_wrap_inline$Tate_&ell#ell;(J)$ as a tex2html_wrap_inline$2$-dimensional tex2html_wrap_inline$T&otimes#otimes;Q_&ell#ell;$-module, and the elements of tex2html_wrap_inline$(Tate_&ell#ell;(J))$ as tex2html_wrap_inline$2×2$-matrices with entries in tex2html_wrap_inline$T&otimes#otimes;Q_&ell#ell;$. For each prime tex2html_wrap_inline$q&nmid#nmid;N$, we have displaymath Tr(&rho#rho;_&ell#ell;(Frob_q)) = T_q &isin#in;T&otimes#otimes;Z_&ell#ell;. Since tex2html_wrap_inline$J[&ell#ell;]$ is finite, the representation displaymath &rho#rho;_&ell#ell;: G_Q&rarr#to;(J[&ell#ell;]) factors through the Galois group of a finite extension of tex2html_wrap_inline$Q$. The Chebotarev density theorem implies that there exists a prime tex2html_wrap_inline$q&ne#neq;p$ such that displaymath Tr(&rho#rho;_&ell#ell;(Frob_p)) = Tr(&rho#rho;_&ell#ell;(Frob_q)). Thus displaymath T_p &equiv#equiv;T_q &ell#ell;, where congruence modulo tex2html_wrap_inline$&ell#ell;$ means their difference is in tex2html_wrap_inline$&ell#ell;T$. Hence tex2html_wrap_inline$T_q$ acts on tex2html_wrap_inline$J[&ell#ell;]$ in the same way as tex2html_wrap_inline$T_p$.

Define tex2html_wrap_inline$S$ by the exact sequence displaymath 0 &rarr#to;R &rarr#to;T(N) &rarr#to;S &rarr#to;0. Let tex2html_wrap_inline$&ell#ell;$ be any prime. Then we have an exact sequence displaymath R&otimes#otimes;F_&ell#ell;&rarr#to;T(N)&otimes#otimes;F_&ell#ell;&rarr#to;S&otimes#otimes;F_&ell#ell;&rarr#to;0. Using what we did above,for each prime tex2html_wrap_inline$p&mid#mid;M$ we find a prime tex2html_wrap_inline$q&nmid#nmid;M$ such that tex2html_wrap_inline$T_q &equiv#equiv;T_p$ on tex2html_wrap_inline$J[&ell#ell;]$. Thus tex2html_wrap_inline$R&otimes#otimes;F_&ell#ell;&rarr#to;T&otimes#otimes;F_&ell#ell;$is surjective, hence tex2html_wrap_inline$S &otimes#otimes;F_&ell#ell;=0$. Since tex2html_wrap_inline$T$ is a finitely generated abelian group, so is tex2html_wrap_inline$S$, so we must have tex2html_wrap_inline$S=0$.

Lemma 5.2.4   Suppose $\lambda$ is a maximal ideal of $\mathbb{T}(N)$ with generators a prime $\ell$ and $T_n - a_n$ (for all $n\in\mathbb{Z}$ ), with $a_n \in \mathbb{Z}$ . For each integer $p\nmid N$ , let $\lambda_p$ be the ideal in $\mathbb{T}(N)$ generated by $\ell$ and all $T_n - a_n$ , where $n$ varies over integers coprime to $p$ . Then $\lambda = \lambda_p$ .

Proof. Since $\lambda_p\subset\lambda$ and $\lambda$ is maximal, it suffices to prove that $\lambda_p$ is maximal. Let $R$ be the subring of $\mathbb{T}(N)$ generated by Hecke operators $T_n$ with $p\nmid n$ . The quotient $R/\lambda_p$ is a quotient of $\mathbb{Z}$ since each generator $T_n$ is equivalent to an integer. Also, $\ell \in \lambda_p$ , so $R/\lambda_p = \mathbb{F}_\ell$ . But by Lemma 5.2.3, $R = \mathbb{T}(N)$ , so $\mathbb{T}(N)/\lambda_p = \mathbb{F}_\ell$ , hence $\lambda_p$ is a maximal ideal. $\qedsymbol$

Lemma 5.2.5   Suppose that $A$ is an abelian variety over a field $K$ . Let $R$ be a commutative subring of ${\mathrm{End}}(A)$ and $I$ an ideal of $R$ . Then

$$(A/A[I])[I] \cong A[I^2]/A[I],$$

where the isomorphism is an isomorphism of $R[G_K]$ -modules.

Proof. Let $a + A[I]$ , for some $a \in A$ , be an $I$ -torsion element of $A/A[I]$ . Then by definition, $xa \in A[I]$ for each $x \in I$ . Therefore, $a \in A[I^2]$ . Thus $a + A[I] \mapsto a + A[I]$ determines a well-defined homomorphism of $R[G_K]$ -modules

$$\varphi : (A/A[I])[I] \rightarrow A[I^2]/A[I].$$

Clearly this homomorphism is injective. It is also surjective as every element $a + A[I] \in A[I^2]/A[I]$ is $I$ -torsion as an element of $A/A[I]$ , as $I{}a \in A[I]$ . Therefore, $\varphi$ is an isomorphism of $R[G_K]$ -modules. $\qedsymbol$

Lemma 5.2.6   Let $\ell$ be a prime and let $\phi : E \rightarrow E'$ be an isogeny of elliptic curves of degree coprime to $\ell$ defined over a number field $K$ . If $v$ is any place of $K$ then $\ell \mid c_{E, v}$ if and only if $\ell \mid c_{E', v}$ .

Proof. Consider the complementary isogeny $\phi' : E' \rightarrow E$ . Both $\phi$ and $\phi'$ induce homomorphisms $\phi : \Phi_{E, v}(k_v) \rightarrow \Phi_{E', v}(k_v)$ and $\phi' : \Phi_{E',v}(k_v) \rightarrow \Phi_{E, v}(k_v)$ and $\phi \circ \phi'$ and $\phi' \circ \phi$ are multiplication-by-$n$ maps. Since $(n, \ell) = 1$ then $\char93 \ker \phi$ and $\char93 \ker \phi'$ must be coprime to $\ell$ which implies the statement. $\qedsymbol$