Two lemmas

**Suppose
is a prime such that
.
In what follows, we will be stating some standard facts
taken from [Maz78, §2(e)] (which in turn relies
on [DR73]).
Let
be the
minimal proper regular model for
over
,
and let
denote the relative dualizing sheaf of
over
(it is the sheaf of regular differentials as in [MR91, §7]).
The Tate curve over
gives rise to a morphism from
to
the smooth locus of
.
Since the module of completed Kahler
differentials for
over
is free of rank
on the basis
,
we obtain a map
$q$-exp
.
**

**The natural morphism
identifies
with the identity
component of
(see, e.g., [BLR90, §9.4-9.5]).
Passing to tangent spaces along the identity section over
,
we obtain an isomorphism
.
Using Grothendieck duality, one gets
an isomorphism
, where
is the cotangent
space at the identity section. On the Néron model
,
the group of global differentials is the same as the group
of invariant differentials, which in turn is naturally
isomorphic to
. Thus we obtain
an isomorphism
.
**

**Let
be a
-module equipped with an injection
of
-modules such that
is annihilated by
.
If
, assume moreover that
is a
-module and that the inclusion in the previous
sentence is a homomorphism of
-modules.
As a typical example,
,
with the injection
.
Let
be the composition of the inclusions
**

**
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
of an abelian
group
is saturated (in
)
if the quotient
is torsion free.
**

obtained by tensoring (1) with is injective. Let denote the special fiber of and let denote the relative dualizing sheaf of over .

*First suppose that
does not divide
.*
Then
is smooth and proper over
.
Thus the formation of
is compatible with any base change on
(such as reduction
modulo
).
The injectivity of
now follows since by hypothesis
the induced map
is injective, and

is injective by the -expansion principle (which is easy in this setting, since is a smooth and geometrically connected curve).

*Next suppose that
divides
.*
First we verify that
is stable under
.
Suppose
. Choose
such that the
image of
in
is
, and let
.
Because
in
, there
exists
such that
.
Let
; then
is actually in
(since
).
Now
is annihilated by every element of
, hence
so is
; thus
.
By hypothesis,
.
Then

Reducing modulo , we get in . Thus , which proves that is stable under .

Since is an involution, and by hypothesis either is odd or is a scalar, the space breaks up into a direct sum of eigenspaces under with eigenvalues . It suffices to show that if is an element of either eigenspace, then . 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

In the following, we shall think of as a subgroup of , which we can do by the hypothesis that the induced map is injective and that

Suppose is in the eigenspace (we will treat the cases of and eigenspaces together). We will show that is trivial over , the base change of to an algebraic closure , which suffices for our purposes. Since , we have , and so the special fiber is as depicted on p. 177 of [Maz77]: it consists of the union of two copies of identified transversely at the supersingular points, and some copies of , each of which intersects exactly one of the two copies of and perhaps another , all of them transversally. All the singular points are ordinary double points, and the cusp lies on one of the two copies of .

In particular, is locally a complete intersection, hence Gorenstein, and so by [DR73, § I.2.2, p. 162], the sheaf is invertible. Since , the differential vanishes on the copy of containing the cusp by the -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 are swapped under the action of the Atkin-Lehner involution , and thus vanishes on the other copy that does not contain the cusp . Since , we see that is zero on both copies of . Also, by the description of the relative dualizing sheaf in [DR73, § I.2.3, p. 162], if is a normalization, then correponds to a meromorphic differential on which is regular outside the inverse images (under ) of the double points on and has at worst a simple pole at any point that lies over a double point on . Moreover, on the inverse image of any double point on , the residues of add to zero. For any of the 's, above a point of intersection of the with a copy of , the residue of on the inverse image of the copy of is zero (since is trivial on both copies of ), and thus the residue of on the inverse image of is zero. Thus restricted to the inverse image of is regular away from the inverse image of any point where the meets another (recall that there can be at most one such point). Hence the restriction of to the inverse image of the 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, restricted to the inverse image of the is regular, and therefore is zero. Thus is trivial on all the copies of as well. Hence , as was to be shown.