The Manin Constant

When trying to compute the conjectural order of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A)$, we try to compute the quotient $L(A,1)/\Omega _A$, but find that it is easier to compute $c_A\cdot L(A,1)/\Omega_A$ where $c_A$ is the Manin constant of $A$.

Definition 4.1 (Manin constant)   The Manin constant of $A$ is

$$c_A = \char93 \left(\frac{S_2(\Gamma_0(N),\mathbf{Z})[I_f]} {H^0(\mathcal{A},\Omega_{\mathcal{A}/\mathbf{Z}})}\right)\in\mathbf{Z},$$

where we consider $H^0(\mathcal{A},\Omega_{\mathcal{A}/\mathbf{Z}})$ as a submodule of $S_2(\Gamma_0(N),\mathbf{Q})$ using

$$H^0(\mathcal{A},\Omega_{\mathcal{A}/\mathbf{Z}}) \rightarrow H^0... ...0(J,\Omega_{J/\mathbf{Q}})[I_f] \rightarrow S_2(\Gamma_0(N),\mathbf{Q})[I_f],$$

where $\mathcal{J}$ is the Néron model of $J_0(N)$. (See [AS02a] for a discussion of why the image of $H^0(\mathcal{A},\Omega_{\mathcal{A}/\mathbf{Z}})$ is contained in $S_2(\Gamma_0(N),\mathbf{Z})$.)

Theorem 4.2   If $\ell\mid c_A$ is a prime then $\ell^2 \mid 4N$.

Proof. Mazur proved this when $\dim A=1$ in [Maz78, §4], and we generalized his proof in [AS02a]. $\qedsymbol$

When $\dim A=1$, Edixhoven [Edi91] obtained strong results towards the folklore conjecture that $c_A=1$, and when $A$ has arbitrary dimension the authors have made the following conjecture (see [AS02a] for evidence):

Conjecture 4.3   $c_A=1$.