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
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
Back to the main index for Rational and integral points on higher dimensional varieties.