# Colliot-Thelene 1: Rational points on surfaces with a pencil of curves of genus one

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

where . The long exact sequence in Galois cohomology gives

where classifies -coverings: that is, given , we have the -cover defined by the equations: , , . The group classifies principal homogeneous spaces.

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

so gives -covers with points everywhere locally, and measures the difference.

Over a local field, we have a pairing

induced by the Weil pairing, which is nondegenerate and alternating, so that .

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

Then is the right kernel of .

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

where

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

We then have a pairing

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

via , the new pairing is symmetric.

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

Here , .

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 be .

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

where . We see that .

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

Assume Schinzel's hypothesis. Assume . Then there exist infinitely many such that is of rank one and .

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