*Symmetrizing the Computation of the Selmer Group*

Let be a field, , an elliptic curve with so that all -torsion of is rational. We have the exact sequence

Now let be a number field, the set of places of . We have the diagram

Over a local field, we have a pairing

**Fact. **[Tate]
is maximal isotropic for the above pairing.

If has good reduction at , then , which is , the maximal isotropic subgroup.

Suppose is a finite set of places, and suppose contains the primes above , and the primes of bad reduction. Then

**Fact. **
If
, then is injective.

This follows from class field theory. So we choose such that , and take . If , is an injection, and the image of is a maximal isotropic subgroup of , .

What we have achieved: the Selmer group is now a kernel of `a square matrix', since and have the same dimension over . Letting , we have

**Proposition. ***Assume
(containing primes above and those of bad reduction and such that
) is a finite set of places. Suppose that
is a maximal isotropic subgroup. Then there exist
for maximal isotropic such that
for
, and
*

This is purely a result in linear algebra.

Recall we have from and .

**Definition. **
We let
, where

for all ,

We let
**Proposition. ***The Selmer group is the kernel of . The map
is an isomorphism. For
, we define a map
*

**Example. **
For
,
,
(reduction is of type ). Then

*Algebraico-Geometric version of Selmer group*

Let be a field, , , so is defined over . To simplify, we assume that all are of the same even degree. We also assume that is separable, where , monic irreducible, so has reduction type .

Now assume is a totally imaginary number field. The Neron model has

Let consist of triples , squarefree, is a square which divides , and .

Put another way, we have

Corresponding to , we have the surface defined by the equations , . Then has a minimal model if and only if is locally isomorphic for the étale topology with .

We have a map

**Theorem. ***[Theorem A]
Suppose that
is the image of . (In particular, the generic rank is zero.) Assume Schinzel's hypothesis. Then there exist infinitely many
such that
.
*

**Theorem. ***[Theorem B]
Let
, and assume that
*

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