Title: Quadratic Twists as Random Variables
Speaker: Ross Paterson (University of Bristol)
Abstract: Let E/Q be an elliptic curve and K be a quadratic field of discriminant d. It is well known that the rank of E(Q) and the rank of the quadratic twist E_d(Q) sum to the rank of E(K) — but do these groups generate E(K)? The answer to this question is sometimes yes and sometimes no, which leads to the natural question: how common is each answer? We will provide partial answers, in particular showing that one answer is a lot more likely than the other!
In studying this question we are drawn to consider the behaviour of the 2-Selmer groups of E and E_d. There is a heuristic model of Poonen-Rains for the behaviour of these 2-Selmer groups individually, as E varies, but how independent are they? We'll describe a heuristic in this direction, and some results in support of it.
URL associated with Seminar: https://research.math.osu.edu/numbertheory/
