Ross Paterson

Statistics for Selmer Groups as Galois Modules

If we let E vary in a natural family of elliptic curves over the rational numbers, then the average size of the 2-Selmer group of E has been well studied, e.g. in the work of Heath-Brown, Swinnerton-Dyer, Kane, Poonen--Rains, Bhargava--Shankar and many others. If we fix a Galois number field K, and look instead at the 2-Selmer group over K, then the size is no longer the only interesting structure at hand; in fact the 2-Selmer group over K has a natural action of the Galois group of K. It is then natural to ask the more refined question: what are the statistical properties of this Galois module?

I will report on joint work with Adam Morgan, in which we consider this question in the case that K is a quadratic field and give some interesting corollaries for the Mordell-Weil groups as a result.