## Random maximal isotropic subspaces and Selmer groups

##### Author(s)

Poonen, Bjorn; Rains, Eric
DownloadPoonen_Random Maximal.pdf (482.6Kb)

OPEN_ACCESS_POLICY

# Open Access Policy

Creative Commons Attribution-Noncommercial-Share Alike

##### Terms of use

##### Metadata

Show full item record##### Abstract

Under suitable hypotheses, we construct a probability measure on the set of closed maximal isotropic subspaces of a locally compact quadratic space over F[subscript p]. A random subspace chosen with respect to this measure is discrete with probability 1, and the dimension of its intersection with a fixed compact open maximal isotropic subspace is a certain nonnegative-integer-valued random variable.
We then prove that the p-Selmer group of an elliptic curve is naturally the intersection of a discrete maximal isotropic subspace with a compact open maximal isotropic subspace in a locally compact quadratic space over F[subscript p]. By modeling the first subspace as being random, 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. Our model is compatible with Delaunay's heuristics for p-torsion in Shafarevich-Tate groups, and predicts that the average rank of elliptic curves over a fixed number field is at most 1/2. Many of our results generalize to abelian varieties over global fields.

##### Date issued

2012-07##### Department

Massachusetts Institute of Technology. Department of Mathematics##### Journal

Journal of the American Mathematical Society

##### Publisher

American Mathematical Society

##### Citation

Poonen, Bjorn, and Eric Rains. “Random Maximal Isotropic Subspaces and Selmer Groups.” Journal of the American Mathematical Society 25.1 (2012): 245–269. Web.

Version: Author's final manuscript

##### Other identifiers

MathSciNet review: 2833483

##### ISSN

1088-6834

0894-0347