2-Selmer Groups of Twists after Quadratic Extension
Ross Paterson (University of Glasgow)
Wednesday 18th November, 2020 16:00-17:00 online (zoom)
As E varies in a natural family of elliptic curves over the rational numbers, 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 of such curves 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 E varies over quadratic twists of a fixed curve, and give some interesting corollaries for the Mordell-Weil groups as a result.