##

Proof of Theorem 3.11

**
We continue to use the notation and hypotheses of Section 4.1
(so
)
and assume in addition that
is a newform quotient, and
that
. We have to show that then
.
Just as in the previous proof,
it suffices to check that the image of
in
is saturated.
Since
is a newform quotient, if
, then
acts as a scalar on
and on
.
So again, using Lemma 4.2,
it suffices to show that the map
is injective.
**
**The composition of pullback and pushforward in the following diagram
is multiplication by the modular exponent of
:
**

**
Since
, the map
is a
section to the map
up to a unit and hence its reduction modulo
is
injective, which is what was left to be shown.
Let tex2html_wrap_inline$&pi#pi;_*$ and tex2html_wrap_inline$&pi#pi;^*$denote the maps obtained by tensoring the diagram above with
tex2html_wrap_inline$****F_&ell#ell;$. Then tex2html_wrap_inline$&pi#pi;_*&cir#circ;&pi#pi;^*$ is
multiplication by an integer coprime to tex2html_wrap_inline$&ell#ell;$ from the finite dimension
tex2html_wrap_inline$****F_&ell#ell;$-vector space tex2html_wrap_inline$H^0(A_****Z_(&ell#ell;), &Omega#Omega;^1_A/****Z_(&ell#ell;))&otimes#otimes;****F_&ell#ell;$ to
itself, hence an isomorphism. In particular, tex2html_wrap_inline$&pi#pi;^*$ is
injective, which is what was left to show.
**
**
theorem_type[rmk][lem][][definition][][]
Adam Joyce observed that one can also obtain injectivity
of tex2html_wrap_inline$&pi#pi;^*$ as a
consequence of Prop. 7.5.3(a) of [BLR90]. **

**
**

**
William Stein
2006-06-25
**