Two lemmas

In this section, we state two lemmas that will be used in the proofs of Theorems 3.5, 3.10, and 3.11. The following lemma is a standard fact; we state it as a lemma merely because it is used several times.

Lemma 4.1   Suppose $i: A\hookrightarrow B$ is an injective homomorphism of torsion-free abelian groups. If $p$ is a prime, then $B/i(A)$ has no nonzero $p$ -torsion if and only if the induced map $A\otimes {\bf {F}}_p \to B\otimes {\bf {F}}_p$ is injective.

Proof. Let $Q$ denote the quotient $B/i(A)$ . Tensor the exact sequence $0 \to A \to B \to Q \to 0$ with ${\bf {F}}_p$ . The associated long exact sequences reveal that $\ker(A\otimes {\bf {F}}_p \to B\otimes {\bf {F}}_p) \cong Q_{{\mathrm{tor}}}[p]$ . $\qedsymbol$

Suppose $\ell$ is a prime such that $\ell^2 \nmid N$ . In what follows, we will be stating some standard facts taken from [Maz78, §2(e)] (which in turn relies on [DR73]). Let  ${\mathcal{X}_{{{\bf {Z}}_{(\ell)}}}}$ be the minimal proper regular model for $X_0(N)$ over  ${{{\bf {Z}}_{(\ell)}}}$ , and let $\Omega_{\mathcal{X}/{{{\bf {Z}}_{(\ell)}}}}$ denote the relative dualizing sheaf of  ${\mathcal{X}_{{{\bf {Z}}_{(\ell)}}}}$ over  ${{\bf {Z}}_{(\ell)}}$ (it is the sheaf of regular differentials as in [MR91, §7]). The Tate curve over  ${{\bf {Z}}_{(\ell)}}[[q]]$ gives rise to a morphism from ${\rm Spec} {{\bf {Z}}_{(\ell)}}[[q]]$ to the smooth locus of ${\mathcal{X}_{{{\bf {Z}}_{(\ell)}}}} \rightarrow {\rm Spec} {{\bf {Z}}_{(\ell)}}$ . Since the module of completed Kahler differentials for  ${{\bf {Z}}_{(\ell)}}[[q]]$ over ${{\bf {Z}}_{(\ell)}}$ is free of rank $1$ on the basis $dq$ , we obtain a map $q$-exp $: H^0({\mathcal{X}_{{{\bf {Z}}_{(\ell)}}}},\Omega_{\mathcal{X}/{{{\bf {Z}}_{(\ell)}}}}) \rightarrow {{\bf {Z}}_{(\ell)}}[[q]]$ .

The natural morphism ${\rm Pic}^0_{\mathcal{X}/{{{\bf {Z}}_{(\ell)}}}} \rightarrow J_{{{\bf {Z}}_{(\ell)}}}$ identifies ${\rm Pic}^0_{\mathcal{X}/{{\bf {Z}}_{(\ell)}}}$ with the identity component of  $J_{{{\bf {Z}}_{(\ell)}}}$ (see, e.g., [BLR90, §9.4-9.5]). Passing to tangent spaces along the identity section over  ${{\bf {Z}}_{(\ell)}}$ , we obtain an isomorphism $H^1(\mathcal{X}_{{{\bf {Z}}_{(\ell)}}}, {\mathcal{O}}_{\mathcal{X}_{{{\bf {Z}}_{(\ell)}}}}) \cong {\rm Tan}(J_{{{\bf {Z}}_{(\ell)}}})$ . Using Grothendieck duality, one gets an isomorphism ${\rm Cot}(J_{{{\bf {Z}}_{(\ell)}}}) \stackrel{\co... ...mathcal{X}_{{{\bf {Z}}_{(\ell)}}}},\Omega_{\mathcal{X}/{{{\bf {Z}}_{(\ell)}}}})$ , where ${\rm Cot}(J_{{{\bf {Z}}_{(\ell)}}})$ is the cotangent space at the identity section. On the Néron model  $J_{{{\bf {Z}}_{(\ell)}}}$ , the group of global differentials is the same as the group of invariant differentials, which in turn is naturally isomorphic to  ${\rm Cot}(J_{{{\bf {Z}}_{(\ell)}}})$ . Thus we obtain an isomorphism $H^0({J_{{{\bf {Z}}_{(\ell)}}}},\Omega^1_{{J/{{{\b... ...mathcal{X}_{{{\bf {Z}}_{(\ell)}}}},\Omega_{\mathcal{X}/{{{\bf {Z}}_{(\ell)}}}})$ .

Let $G$ be a ${\bf {T}}$ -module equipped with an injection $G \hookrightarrow H^0({J_{{{\bf {Z}}_{(\ell)}}}},\Omega^1_{{J/{{{\bf {Z}}_{(\ell)}}}}})$ of ${\bf {T}}$ -modules such that $G$ is annihilated by $I$ . If $\ell \mid N$ , assume moreover that $G$ is a ${\bf {T}}[W_\ell]$ -module and that the inclusion in the previous sentence is a homomorphism of ${\bf {T}}[W_\ell]$ -modules. As a typical example, $G = H^0(A_{{{\bf {Z}}_{(\ell)}}},\Omega^1_{A/{{{\bf {Z}}_{(\ell)}}}})$ , with the injection $\pi^*: H^0(A_{{{\bf {Z}}_{(\ell)}}},\Omega^1_{A/{... ...ightarrow H^0({J_{{{\bf {Z}}_{(\ell)}}}},\Omega^1_{{J/{{{\bf {Z}}_{(\ell)}}}}})$ . Let $\Phi$ be the composition of the inclusions

$$G \hookrightarrow H^0({J_{{{\bf {Z}}... ...f {Z}}_{(\ell)}}}}) \xrightarrow{\text{ $q$-exp }} {{{\bf {Z}}_{(\ell)}}}[[q]],$$ (1)

and let $\psi'$ be the composition of

$$G \hookrightarrow H^0({J_{{{\bf {Z}}... ...{{J/{{{\bf {Z}}_{(\ell)}}}}})[I] \hookrightarrow S_2({{\bf {Z}}_{(\ell)}})[I],$$

where the last inclusion follows from a ``local'' version of Theorem 3.4. The maps $\Phi$ and $\psi'$ are related by the commutative diagram

$$\xymatrix{ & {S_2({{{\bf{Z}}_{(\ell)}}})[I]}\ar[dr]^{\text{$F$-exp}} G \ar[ur]^{\psi'} \ar[rr]^{\Phi} & & {{{{\bf{Z}}_{(\ell)}}}[[q]]}, }$$ (2)

where $F$ -exp is the Fourier expansion map (at infinity), as before.

where the map tex2html_wrap_inline$q$-exp is the tex2html_wrap_inline$q$-expansion map on differentials as in [Maz78, §2(e)] (actually, Mazur works over tex2html_wrap_inline$Z$; our map is obtained by tensoring with tex2html_wrap_inline$Z_(&ell#ell;)$).

We say that a subgroup $B$ of an abelian group $C$ is saturated (in $C$ ) if the quotient $C/B$ is torsion free.

Lemma 4.2   Recall that $\ell$ is a prime such that $\ell^2 \nmid N$ . If $\ell$ divides $N$ , suppose that $S_2({{\bf {Z}}_{(\ell)}})[I]$ is stable under the action of $W_\ell$ ; if $\ell = 2$ assume moreover that $W_{\ell}$ acts as a scalar on $A$ . Consider the map

$$G \otimes \mathbf{F}_\ell \rightarro... ...}_{(\ell)}}}},\Omega^1_{{J/{{{\bf {Z}}_{(\ell)}}}}}) \otimes \mathbf{F}_\ell ,$$

which is obtained by tensoring the inclusion $G \hookrightarrow H^0({J_{{{\bf {Z}}_{(\ell)}}}},\Omega^1_{{J/{{{\bf {Z}}_{(\ell)}}}}})$ with $\mathbf{F}_\ell$ . If this map is injective, then the image of $G$ under the map $\Phi$ of (2) is saturated in  ${{{\bf {Z}}_{(\ell)}}}[[q]]$ .

Proof. By Lemma 4.1, it suffices to prove that the map

$$\Phi_\ell: G \otimes \mathbf{F}_\ell \rightarrow {{{\bf {Z}}_{(\ell)}}}[[q]] \otimes \mathbf{F}_\ell = \mathbf{F}_\ell [[q]]$$

obtained by tensoring (1) with $\mathbf{F}_\ell$ is injective. Let $\mathcal{X}_{{\bf {F}}_\ell}$ denote the special fiber of  $\mathcal{X}_{{{\bf {Z}}_{(\ell)}}}$ and let $\Omega_{\mathcal{X}/{{\bf {F}}_\ell}}$ denote the relative dualizing sheaf of  $\mathcal{X}_{\mathbf{F}_\ell }$ over  $\mathbf{F}_\ell$ .

First suppose that $\ell$ does not divide $N$ . Then ${\mathcal{X}_{{{\bf {Z}}_{(\ell)}}}}$ is smooth and proper over ${{\bf {Z}}_{(\ell)}}$ . Thus the formation of $\H ^0({\mathcal{X}_{{{\bf {Z}}_{(\ell)}}}}, \Omega_{{\mathcal{X}_{{{\bf {Z}}_{(\ell)}}}}})$ is compatible with any base change on  ${{\bf {Z}}_{(\ell)}}$ (such as reduction modulo $\ell$ ). The injectivity of $\Phi_\ell$ now follows since by hypothesis the induced map $G \otimes \mathbf{F}_\ell \rightarrow H^0({J_{{{\... ...}}_{(\ell)}}}},\Omega^1_{{J/{{{\bf {Z}}_{(\ell)}}}}}) \otimes \mathbf{F}_\ell$ is injective, and

$$H^0({J_{{{\bf {Z}}_{(\ell)}}}},\Omeg... ...ell },\Omega_{\mathcal{X}/{\mathbf{F}_\ell }}) \rightarrow {\bf {F}}_\ell[[q]]$$

is injective by the $q$ -expansion principle (which is easy in this setting, since $\mathcal{X}_{\mathbf{F}_\ell }$ is a smooth and geometrically connected curve).

Next suppose that $\ell$ divides $N$ . First we verify that $\ker(\Phi_\ell)$ is stable under $W_\ell$ . Suppose $\omega \in \ker(\Phi_\ell)$ . Choose $\omega' \in G$ such that the image of $\omega'$ in $G \otimes {\bf {F}}_\ell$ is $\omega$ , and let $f = \psi'(\omega')$ . Because $\Phi_\ell(\omega) = 0$ in  ${\bf {F}}_\ell[[q]]$ , there exists $h \in {{\bf {Z}}_{(\ell)}}[[q]]$ such that $\ell h = {\text{$F$-exp}}(f)$ . Let $f' = f/\ell \in S_2({\bf {Q}})$ ; then $f'$ is actually in  $S_2({{\bf {Z}}_{(\ell)}})$ (since ${\text{$F$-exp}}(f/\ell) = h \in {{\bf {Z}}_{(\ell)}}[[q]]$ ). Now $\ell f' = f$ is annihilated by every element of $I$ , hence so is $f'$ ; thus $f' \in S_2({{\bf {Z}}_{(\ell)}})[I]$ . By hypothesis, $W_{\ell}(f') \in S_2({{\bf {Z}}_{(\ell)}})[I]$ . Then

$$\Phi(W_\ell \omega') = {\text{$F$-ex... ... f) = \ell \cdot {\text{$F$-exp}}(W_\ell f') \in \ell{{\bf {Z}}_{(\ell)}}[[q]].$$

Reducing modulo $\ell$ , we get $\Phi_\ell (W_\ell \omega) = 0$ in  ${\bf {F}}_\ell[[q]]$ . Thus $W_\ell \omega \in \ker(\Phi_\ell)$ , which proves that $\ker(\Phi_\ell)$ is stable under $W_{\ell}$ .

Since $W_\ell$ is an involution, and by hypothesis either $\ell$ is odd or $W_\ell$ is a scalar, the space $\ker(\Phi_\ell)$ breaks up into a direct sum of eigenspaces under $W_\ell$ with eigenvalues $\pm 1$ . It suffices to show that if $\omega \in \ker(\Phi_\ell)$ is an element of either eigenspace, then $\omega = 0$ . For this, we use a standard argument that goes back to Mazur (see, e.g., the proof of Prop. 22 in [MR91]); we give some details to clarify the argument in our situation.

Following the proof of Prop. 3.3 on p. 68 of [Maz77], we have

$$H^0(\mathcal{X}_{{{\bf {Z}}_{(\ell)}... ...ng H^0(\mathcal{X}_{\mathbf{F}_\ell },\Omega_{\mathcal{X}/{\mathbf{F}_\ell }}).$$

In the following, we shall think of $G \otimes \mathbf{F}_\ell$ as a subgroup of  $H^0(\mathcal{X}_{\mathbf{F}_\ell },\Omega_{\mathcal{X}/{\mathbf{F}_\ell }})$ , which we can do by the hypothesis that the induced map $G \otimes \mathbf{F}_\ell \rightarrow H^0({J_{{{... ...Z}}_{(\ell)}}}},\Omega^1_{{J/{{{\bf {Z}}_{(\ell)}}}}}) \otimes \mathbf{F}_\ell$ is injective and that

$$H^0({J_{{{\bf {Z}}_{(\ell)}}}},\Omeg... ...ng H^0(\mathcal{X}_{\mathbf{F}_\ell },\Omega_{\mathcal{X}/{\mathbf{F}_\ell }}).$$

Suppose $\omega \in \ker(\Phi_\ell)$ is in the $\pm 1$ eigenspace (we will treat the cases of $+1$ and $-1$ eigenspaces together). We will show that $\omega$ is trivial over $\mathcal{X}_{\overline{\mathbf{F}}_\ell}$ , the base change of  ${\mathcal{X}}_{\mathbf{F}_\ell }$ to an algebraic closure  ${\overline{\mathbf{F}}_\ell}$ , which suffices for our purposes. Since $\ell^2 \nmid N$ , we have $\ell \mid\mid N$ , and so the special fiber $\mathcal{X}_{{\overline{\mathbf{F}}_\ell}}$ is as depicted on p. 177 of [Maz77]: it consists of the union of two copies of  $X_0(N/\ell)_{{\overline{\mathbf{F}}_\ell}}$ identified transversely at the supersingular points, and some copies of  ${\bf {P}}^1$ , each of which intersects exactly one of the two copies of  $X_0(N/\ell)_{{\overline{\mathbf{F}}_\ell}}$ and perhaps another  ${\bf {P}}^1$ , all of them transversally. All the singular points are ordinary double points, and the cusp $\infty$ lies on one of the two copies of  $X_0(N/\ell)_{{\overline{\mathbf{F}}_\ell}}$ .

In particular, $\mathcal{X}_{{\overline{\mathbf{F}}_\ell}} \rightarrow {\mathrm{Spec}}{{\overline{\mathbf{F}}_\ell}}$ is locally a complete intersection, hence Gorenstein, and so by [DR73, § I.2.2, p. 162], the sheaf  $\Omega_{\mathcal{X}/{{\overline{\mathbf{F}}_\ell} }} = \Omega_{\mathcal{X}/{\mathbf{F}_\ell }} \otimes {\overline{\mathbf{F}}_\ell}$ is invertible. Since $\omega\in\ker(\Phi_\ell)$ , the differential $\omega$ vanishes on the copy of  $X_0(N/\ell)_{{\overline{\mathbf{F}}_\ell}}$ containing the cusp $\infty$ by the $q$ -expansion principle (which is easy in this case, since all that is being invoked here is that on an integral curve, the natural map from the group of global sections of an invertible sheaf into the completion of the sheaf's stalk at a point is injective). The two copies of  $X_0(N/\ell)_{{\overline{\mathbf{F}}_\ell}}$ are swapped under the action of the Atkin-Lehner involution $W_\ell$ , and thus $W_\ell(\omega)$ vanishes on the other copy that does not contain the cusp $\infty$ . Since $W_\ell(\omega) = \pm \omega$ , we see that $\omega$ is zero on both copies of  $X_0(N/\ell)_{{\overline{\mathbf{F}}_\ell}}$ . Also, by the description of the relative dualizing sheaf in [DR73, § I.2.3, p. 162], if $\pi: \widetilde{\mathcal{X}}_{{\overline{\mathbf{F}}_\ell}} \rightarrow \mathcal{X}_{{\overline{\mathbf{F}}_\ell}}$ is a normalization, then $\omega$ correponds to a meromorphic differential  $\widetilde{\omega}$ on  $\widetilde{\mathcal{X}}_{{\overline{\mathbf{F}}_\ell}}$ which is regular outside the inverse images (under $\pi$ ) of the double points on  $\mathcal{X}_{{\overline{\mathbf{F}}_\ell}}$ and has at worst a simple pole at any point that lies over a double point on  $\mathcal{X}_{{\overline{\mathbf{F}}_\ell}}$ . Moreover, on the inverse image of any double point on  $\mathcal{X}_{{\overline{\mathbf{F}}_\ell}}$ , the residues of  $\widetilde{\omega}$ add to zero. For any of the  ${{\bf {P}}}^1$ 's, above a point of intersection of the  ${\bf {P}}^1$ with a copy of  $X_0(N/\ell)_{{\overline{\mathbf{F}}_\ell}}$ , the residue of  $\widetilde{\omega}$ on the inverse image of the copy of  $X_0(N/\ell)_{{\overline{\mathbf{F}}_\ell}}$ is zero (since $\omega$ is trivial on both copies of  $X_0(N/\ell)_{{\overline{\mathbf{F}}_\ell}}$ ), and thus the residue of  $\widetilde{\omega}$ on the inverse image of  ${\bf {P}}^1$ is zero. Thus $\widetilde{\omega}$ restricted to the inverse image of  ${\bf {P}}^1$ is regular away from the inverse image of any point where the  ${\bf {P}}^1$ meets another  ${\bf {P}}^1$ (recall that there can be at most one such point). Hence the restriction of $\widetilde{\omega}$ to the inverse image of the  ${\bf {P}}^1$ is either regular everywhere or is regular away from one point where it has at most a simple pole; in the latter case, the residue is zero by the residue theorem. Thus in either case, $\widetilde{\omega}$ restricted to the inverse image of the  ${\bf {P}}^1$ is regular, and therefore is zero. Thus $\omega$ is trivial on all the copies of  ${\bf {P}}^1$ as well. Hence $\omega=0$ , as was to be shown. $\qedsymbol$