Proof of Theorem 3.5

We continue to use the notation of Section 4.1.

First suppose that $\ell \mid N$ and $S_2({{\bf {Z}}_{(\ell)}})[I]$ is not stable under the action of $W_\ell$ . Relative differentials and Néron models are functorial, so $H^0(A_{{{\bf {Z}}_{(\ell)}}},\Omega^1_{A/{{{\bf {Z}}_{(\ell)}}}})$ is $W_{\ell}$ -stable. Thus the map $H^0(A_{{{\bf {Z}}_{(\ell)}}},\Omega^1_{A/{{{\bf {Z}}_{(\ell)}}}}) \to S_2({{\bf {Z}}_{(\ell)}})[I]$ is not surjective. But $c_A$ is the order of the cokernel, so $\ell \mid c_A$ .

Next we prove the other implication, namely that if $\ell \mid c_A$ , then $\ell \mid N$ and $S_2({{\bf {Z}}_{(\ell)}})[I]$ is not stable under $W_\ell$ . We will prove this by proving the contrapositive, i.e., that if either $\ell\nmid N$ or $S_2({{\bf {Z}}_{(\ell)}})[I]$ is stable under $W_\ell$ , then $\ell \nmid c_A$ .

We now follow the discussion preceding Lemma 4.2, taking $G = H^0(A_{{{\bf {Z}}_{(\ell)}}},\Omega^1_{A/{{{\bf {Z}}_{(\ell)}}}})$ . To show that $\ell \nmid {c_{\scriptscriptstyle{A}}}$ , we have to show that ${c_{\scriptscriptstyle{A}}}$ is a unit in  ${{{\bf {Z}}_{(\ell)}}}$ . For this, it suffices to check that in diagram (2), the image of $H^0(A_{{{\bf {Z}}_{(\ell)}}},\Omega^1_{A/{{{\bf {Z}}_{(\ell)}}}})$ in ${{{\bf {Z}}_{(\ell)}}}[[q]]$ under $\Phi$ is saturated, since the image of $S_2(\Gamma_0(N);{{{\bf {Z}}_{(\ell)}}})[I]$ under $F$ -exp is saturated in ${{{\bf {Z}}_{(\ell)}}}[[q]]$ . In view of Lemma 4.2, it suffices to show that the map

$$H^0({A_{{{\bf {Z}}_{(\ell)}}}},\Omeg... ...{Z}}_{(\ell)}}}},\Omega^1_{{J/{{{\bf {Z}}_{(\ell)}}}}})\otimes \mathbf{F}_\ell$$

is injective.

Since $A$ is an optimal quotient, $\ell\neq 2$ , and $J$ has good or semistable reduction at $\ell$ , [Maz78, Cor 1.1] yields an exact sequence

$$0 \rightarrow H^0(A_{{{\bf {Z}}_{(\e... ...H^0(B_{{{\bf {Z}}_{(\ell)}}},\Omega^1_{B/{{{\bf {Z}}_{(\ell)}}}}) \rightarrow 0$$

where $B = \ker(J\to A)$ . Since $H^0(B_{{{\bf {Z}}_{(\ell)}}},\Omega^1_{B/{{{\bf {Z}}_{(\ell)}}}})$ is torsion free, by Lemma 4.1 the map $H^0({A_{{{\bf {Z}}_{(\ell)}}}},\Omega^1_{{A/{{{\b... ...{Z}}_{(\ell)}}}},\Omega^1_{{J/{{{\bf {Z}}_{(\ell)}}}}})\otimes \mathbf{F}_\ell$ is injective, as was to be shown.