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Visible Elements of
In [Maz99], Mazur introduced the following definition.
Let
be an abelian variety over an arbitrary field
.
Definition 1.1
Let

be an embedding of

into an abelian variety

over

.
Then the
visible subgroup of
with respect
to the embedding 
is
The visible subgroup
depends on the choice of
embedding
, but we do not include
in the notation, as
it is usually clear from context.
The Galois cohomology group
has a geometric interpretation
as the group of classes of torsors
for
(see [LT58]).
To a cohomology class
, there is a corresponding
variety
over
and a map
that satisfies axioms
similar to those for a simply transitive group action. The set of
equivalence classes of such
forms a group, the Weil-Chatelet group
of
, which is canonically isomorphic to
.
There is a close relationship between visibility and the geometric
interpretation of Galois cohomology. Suppose
is an
embedding and
. We have an exact sequence of
abelian varieties
, where
. A piece of
the associated long exact sequence of Galois cohomology is
so there is an exact sequence
 |
(1.1) |
Thus there is a point
that maps to
. The fiber
over
is a subvariety of
, which, when equipped with its natural
action of
, lies in the class of torsors corresponding to
.
This is the origin of the terminology ``visible''. Also, we remark
that when
is a number field,
is finite
because it is torsion
and is the surjective image of the finitely generated group
.
Next: Visible Elements of
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William A Stein
2002-02-27