Holly Green

Parity of ranks of elliptic curves

In the 1920s, Mordell and Weil asserted that the rational points on an elliptic curve form a finitely generated abelian group. The structure of this group is determined by the rank and torsion subgroup; we will discuss the first (and more mysterious) of these pieces of data. We will focus on determining whether the rank is odd or even, beginning with an overview of known results. We will then look at practical methods to compute this parity, which only involve the arithmetic of curves over local fields (the reals and p-adics). As an application, we will show that the famous Birch and Swinnerton—Dyer conjecture correctly predicts the parity of the rank in a new case. This is all joint work with Céline Maistret.