Proof of Theorem 3.10

Theorem 3.10 asserts that if $A=A_f$ is a quotient of $J=J_0(N)$ attached to a newform $f$ , and $\ell$ is a prime such that $\ell^2 \nmid N$ , then ${\mathrm{ord}}_{\ell}({c_{\scriptscriptstyle{A}}}) \leq \dim(A)$ . Our proof follows [AU96], except at the end we argue using lattice indices instead of multiples.

Let $B$ denote the kernel of the quotient map $J \rightarrow A$ . Consider the exact sequence $0\to B\to J\to A\to 0$ , and the corresponding complex $B_{{{\bf {Z}}_{(\ell)}}}\to J_{{{\bf {Z}}_{(\ell)}}} \to A_{J_{{{\bf {Z}}_{(\ell)}}}}$ of Néron models. Because $J_{{{\bf {Z}}_{(\ell)}}}$ has semiabelian reduction (since $\ell^2 \nmid N$ ), Theorem A.1 of the appendix of [AU96, pg. 279-280], due to Raynaud, implies that there is an integer $r$ and an exact sequence

$$0 \to {\mathrm{Tan}}(B_{{{\bf {Z}}_{(\ell)}}}) \to {\mathrm{Tan}}... ...thrm{Tan}}(A_{{{\bf {Z}}_{(\ell)}}}) \to ({\bf {Z}}/{\ell}{\bf {Z}})^r \to 0.$$

Here ${\mathrm{Tan}}$ is the tangent space at the 0 section; it is a finite free ${{\bf {Z}}_{(\ell)}}$ -module of rank equal to the dimension. In particular, we have $r\leq d=\dim(A)$ . Note that ${\mathrm{Tan}}$ is ${{\bf {Z}}_{(\ell)}}$ -dual to the cotangent space, and the cotangent space is isomorphic to the space of global differential $1$ -forms. The theorem of Raynaud mentioned above is the generalization to $e=\ell -1$ of [Maz78, Cor. 1.1], which we used above in the proof of Theorem 3.5.

Let $C$ be the cokernel of ${\mathrm{Tan}}(B_{{{\bf {Z}}_{(\ell)}}})\to {\mathrm{Tan}}(J_{{{\bf {Z}}_{(\ell)}}})$ . We have a diagram

$$\xymatrix @=0.15in{ 0 \ar[r] & {{\mathrm{Tan}}(B_{{{\bf {Z}}_{(\e... ...)}\ar[r]& {({\bf {Z}}/{\ell}{\bf {Z}})^r}\ar[r]& 0. & & & C\ar@{^(->}[ru] }$$ (3)

Since $C\subset {\mathrm{Tan}}(A_{{{\bf {Z}}_{(\ell)}}})$ , so $C$ is torsion free, we see that $C$ is a free ${{\bf {Z}}_{(\ell)}}$ -module of rank $d$ . Let $C^* = {\rm Hom}_{{{\bf {Z}}_{(\ell)}}}(C,{{\bf {Z}}_{(\ell)}})$ be the ${{\bf {Z}}_{(\ell)}}$ -linear dual of $C$ . Applying the ${\rm Hom}_{{{\bf {Z}}_{(\ell)}}}(-,{{\bf {Z}}_{(\ell)}})$ functor to the two short exact sequences in (3), we obtain exact sequences

$$0 \to C^* \to \H ^0(J_{{{\bf {Z}}_{(... ... \to \H ^0(B_{{{\bf {Z}}_{(\ell)}}},\Omega^1_{B/{{\bf {Z}}_{(\ell)}}}) \to 0,$$

and

$$0 \to \H ^0(A_{{{\bf {Z}}_{(\ell)}}}... ...ega^1_{A/{{\bf {Z}}_{(\ell)}}}) \to C^* \to ({\bf {Z}}/{\ell}{\bf {Z}})^r\to 0.$$ (4)

The $({\bf {Z}}/{\ell}{\bf {Z}})^r$ on the right in (4) occurs as ${\mathrm{Ext}}^1_{{{\bf {Z}}_{(\ell)}}}(({\bf {Z}}/{\ell}{\bf {Z}})^r,{{\bf {Z}}_{(\ell)}})$ .

Since $\H ^0(B_{{{\bf {Z}}_{(\ell)}}},\Omega^1_{B/{{\bf {Z}}_{(\ell)}}})$ is torsion free, by Lemma 4.1, the induced map

$$C^* \otimes {\bf {F}}_\ell \to \H ^0... ...\bf {Z}}_{(\ell)}}},\Omega^1_{J/{{\bf {Z}}_{(\ell)}}}) \otimes {\bf {F}}_\ell$$

is injective. Since $A$ is a newform quotient, if $\ell \mid N$ then $W_\ell$ acts as a scalar on $C^*$ and on  $S_2(\Gamma_0(N);{{{\bf {Z}}_{(\ell)}}})[I]$ . Using Lemma 4.2, with $G = C^*$ , we see that the image of $C^*$ in ${{\bf {Z}}_{(\ell)}}[[q]]$ under the composite of the maps in (1) is saturated. The Manin constant for $A$ at ${\ell}$ is the index of the image via $q$ -expansion of $\H ^0(A_{{{\bf {Z}}_{(\ell)}}},\Omega^1_{A/{{\bf {Z}}_{(\ell)}}})$ in ${{\bf {Z}}_{(\ell)}}[[q]]$ in its saturation. Since the image of $C^*$ in  ${{\bf {Z}}_{(\ell)}}[[q]]$ is saturated, the image of $C^*$ is the saturation of the image of  $\H ^0(A_{{{\bf {Z}}_{(\ell)}}},\Omega^1_{A/{{\bf {Z}}_{(\ell)}}})$ , so the Manin constant at ${\ell}$ is the index of  $\H ^0(A_{{{\bf {Z}}_{(\ell)}}},\Omega^1_{A/{{\bf {Z}}_{(\ell)}}})$ in $C^*$ , which is ${\ell}^r$ by (4), hence is at most ${\ell}^d$ .