Random Maximal Isotropic Subspaces and Selmer Groups

JOINT IAS/PU NUMBER THEORY SEMINAR
Topic:Random Maximal Isotropic Subspaces and Selmer Groups
Speaker:Bjorn Poonen
Affiliation:Massachusetts Institute of Technology
Date:Thursday, March 24
Time/Room:4:30pm - 5:30pm/S-101

We show that the p-Selmer group of an elliptic curve is naturally the intersection of two maximal isotropic subspaces in an infinite-dimensional locally compact quadratic space over F_p. By modeling this intersection as the intersection of a random maximal isotropic subspace with a fixed compact open maximal isotropic subspace, we can explain the known phenomena regarding distribution of Selmer ranks, such as the theorems of Heath-Brown, Swinnerton-Dyer, and Kane for 2-Selmer groups in certain families of quadratic twists, and the average size of 2- and 3-Selmer groups as computed by Bhargava and Shankar. The random model is consistent with Delaunay's heuristics for Sha[p], and predicts that the average rank of elliptic curves is at most 1/2. This is joint work with Eric Rains.