Random maximal isotropic subspaces and Selmer groups
Wednesday, January 5, 2011 - 10:30am - 11:30am
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 only distribution on Mordell-Weil ranks compatible with both our random model and Delaunay's heuristics for Sha[p] is the distribution in which 50% of elliptic curves have rank 0, and 50% have rank 1. We generalize many of our results to abelian varieties over global fields. This is joint work with Eric Rains.