The Mordell-Weil Group of Curves of Genus 2

Abstract

In 1922 Mordell [2] proved Poirrcaré’s conjecture that the group of rational points on an abelian variety of dimension 1 (= elliptic curve with rational point) is finitely generated. His proof was somewhat indirect. In 1928 Weil [5] in his thesis generalized Mordell’s result to abelian varieties of any dimension and to any algebraic number field as ground field. At the same time, Weil [6] gave a very simple and elegant proof of Mordell’s original result. I observed some time ago that Weil’s simple proof admits a further simplification. He uses the explicit form of the duplication and addition theorems on the abelian variety, but these can be avoided by, roughly speaking, the observation that every element of a cubic field-extension k(0) of a field k can be put in the shape (a0 + b)/(c0 + d) (a, b, c, d ∈ k). This additional simplification has little interest in itself, but suggested that similar ideas might be usefully exploited in investigating the Mordell-Weil group of (the Jacobians of) curves of genus greater than 1. This is done here for curves of genus 2.