Chapter 6 - ℭ/2ℭ

Summary

Introduction. The Mordell- Weil group ℭ of an abelian variety A defined over a field k is the group of points of A defined over k. We shall be concerned with the case when A is the jacobian of a curve C defined over k, and then an equivalent definition of ℭ is as the group of divisor classes of degree 0 on C defined over k. If necessary, the ground field under consideration will be denoted by a subscript, e.g. ℭ_k.

A decisive tool in the investigation of the Mordell-Weil group ℭ of a curve of genus 1 is a homomorphism with kernel 2ℭ into an easily treated group. It generalizes to abelian varieties of higher dimension, but existing versions are not well adapted to the explicit treatment of special cases. In this chapter, we give a version for the jacobian of a curve of genus 2 which can be so used. It is noteworthy that it brings in not just the Kummer but its dual.

We first set up and treat the homomorphism by a simple-minded generalization of an elementary version of the genus 1 homomorphism. Next, we see that it has a natural interpretation in terms of the Kummer, and that it essentially does not distinguish between the different curves C_d: dY² = F(X), where d ∈ k*. Further, essentially only six of the tropes are used. The need to separate the C_d leads to the jacobian as a cover of the Kummer, which we started to look at in Chapter 3, Section 8. This brings in the remaining ten tropes in a minor way.