Generalization to quotients of arbitrary dimension

If $R$ is a subring of ${\bf {C}}$ , let $S_2(R)=S_2(\Gamma;R)$ denote the ${\bf {T}}$ -submodule of  $S_2(\Gamma;{\bf {C}})$ of modular forms whose Fourier expansions have all coefficients in $R$ .

Lemma 3.1   The Hecke operators leave $S_2(R)$ stable.

Proof. If $\Gamma = \Gamma_0(N)$ , then by the explicit description of the Hecke operators on Fourier expansions (e.g., see [DI95, Prop. 3.4.3]), it is clear that the Hecke operators leave $S_2(R)$ stable. For a general $\Gamma$ , by [DI95, (12.4.1)], one just has to check in addition that the diamond operators also leave $S_2(R)$ stable, which in turn follows from [DI95, Prop. 12.3.11]. $\qedsymbol$

Lemma 3.2   We have $S_2(R)\cong S_2({\bf {Z}})\otimes R$ .

Proof. This is [DI95, Thm. 12.3.2] when our spaces $S_2(R)$ and $S_2({\bf {Z}})$ are replaced by their algebraic analogues (see [DI95, pg. 111]). Our spaces and their algebraic analogues are identified by the natural $q$ -expansion maps according to [DI95, Thm. 12.3.7]. $\qedsymbol$

If $B$ is an abelian variety over ${\bf {Q}}$ and $S$ is a Dedekind domain with field of fractions ${\bf {Q}}$ , then we denote by $B_S$ the Néron model of $B$ over $S$ ; also, for ease of notation, we will abbreviate $H^0(B_S, \Omega^1_{B_S/S})$ by  $H^0(B_S, \Omega^1_{B/S})$ .

The inclusion $X \hookrightarrow J$ that sends the cusp $\infty$ to 0 induces an isomorphism

$$H^0(X,\Omega^1_{X/{{\bf {Q}}}}) \cong H^0(J,\Omega^1_{J/{{\bf {Q}}}}).$$

Let $\phi_2$ be the optimal quotient map $J \rightarrow A$ . Then $\phi_2^*$ induces an inclusion $\psi: H^0(A_{{\bf {Z}}},\Omega^1_{A/{{\bf {Z}}}}) \hookrightarrow H^0(J,\Omega^1_{J/{{\bf {Q}}}})[I] \cong S_2({\bf {Q}})[I]$ , and we have the following commutative diagram:

$$\xymatrix{ {H^0(A,\Omega^1_{A/{{\bf{... ...}}})}\ar@{^(->}[u]\ar@{^(->}[urr]_{\psi} & & {S_2({\bf{Z}})[I]}\ar@{^(->}[u] }$$

Definition 3.3   The Manin constant of $A$ is the (lattice) index

$${c_{\scriptscriptstyle{A}}}= [S_2({\bf {Z}})[I]: \psi(H^0(A_{{\bf {Z}}},\Omega^1_{A/{\bf {Z}}}))].$$

Theorem 3.4 below asserts that ${c_{\scriptscriptstyle{A}}}\in{\bf {Z}}$ , so we may also consider the Manin module of $A$ , which is the quotient $M = S_2({\bf {Z}})[I] / \psi(H^0(A_{{\bf {Z}}},\Omega^1_{A/{\bf {Z}}}))$ , and the Manin ideal of $A$ , which is the annihilator of $M$ in ${\bf {T}}$ .

If $A$ is an elliptic curve, then ${c_{\scriptscriptstyle{A}}}$ is the usual Manin constant. The constant $c$ as defined above was also considered by Gross [Gro82, 2.5, p.222] and Lang [Lan91, III.5, p.95]. The constant  ${c_{\scriptscriptstyle{A}}}$ was defined for the winding quotient in [Aga99], where it was called the generalized Manin constant. A Manin constant is defined in the context of ${\bf {Q}}$ -curves in [GL01].