##

Generalization to quotients of arbitrary dimension

*
If
is a subring of
, let
denote the
-submodule of
of modular forms whose Fourier
expansions have all coefficients in
.
*

*Proof*.
If

, 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

stable.
For a general

, by [

DI95, (12.4.1)],
one just has to check in addition
that the diamond operators also leave

stable, which in turn
follows from [

DI95, Prop. 12.3.11].

**Lemma 3.2**
*We have
.*
*Proof*.
This is [

DI95, Thm. 12.3.2] when
our spaces

and

are replaced by their
algebraic analogues (see [

DI95, pg. 111]).
Our spaces and their algebraic analogues are
identified by the natural

-expansion maps
according to [

DI95, Thm. 12.3.7].

*If
is an abelian variety over
and
is a Dedekind domain
with field of fractions
, then
we denote by
the Néron model of
over
;
also, for ease of notation,
we will abbreviate
by
.
*

*The inclusion
that
sends the cusp
to 0
induces an isomorphism
*

*
Let
be the optimal
quotient map
. Then
induces an inclusion
,
and we have the following commutative diagram:
*

*
Theorem 3.4 below asserts that
, so we may also
consider the Manin module of
, which is the quotient
,
and the Manin ideal of
, which is the annihilator of
in
.
*
*If
is an elliptic curve, then
is the usual Manin constant.
The constant
as defined above was
also considered by Gross [Gro82, 2.5, p.222] and Lang [Lan91, III.5,
p.95]. The constant
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
-curves in [GL01].
*

William Stein
2006-06-25