##

Proof of Theorem 3.5

**
We continue to use the notation of Section 4.1.
**
**First suppose that
and
is not stable
under the action of
. Relative differentials and
Néron models are functorial, so
is
-stable.
Thus the map
is not surjective. But
is the order of the cokernel,
so
.
**

**Next we prove the other implication, namely that if
, then
and
is not stable under
. We will prove
this by proving the contrapositive, i.e., that
if either
or
is stable under
, then
.
**

**We now follow the discussion preceding Lemma 4.2,
taking
.
To show that
, we have to show that
is a unit in
. For this, it
suffices to check that in diagram (2),
the image of
in
under
is saturated,
since the image of
under
-exp
is saturated in
.
In view of Lemma 4.2,
it suffices to show that the map
**

**
is injective.
**
**Since
is an optimal quotient,
, and
has good or semistable reduction at
,
[Maz78, Cor 1.1] yields an exact sequence
**

**
where
. Since
is torsion free, by Lemma 4.1 the map
is injective, as was to be shown.
**

William Stein
2006-06-25