Poonen, Bjorn and Rains, Eric (2012) Random maximal isotropic subspaces and Selmer groups. Journal of the American Mathematical Society, 25 (1). pp. 245-269. ISSN 0894-0347 http://resolver.caltech.edu/CaltechAUTHORS:20111207-112922811
- Published Version
See Usage Policy.
Use this Persistent URL to link to this item: http://resolver.caltech.edu/CaltechAUTHORS:20111207-112922811
Under suitable hypotheses, we construct a probability measure on the set of closed maximal isotropic subspaces of a locally compact quadratic space over F_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_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.
|Additional Information:||© 2011 American Mathematical Society. The copyright for this article reverts to public domain after 28 years from publication. Received by editor(s): September 21, 2010; Received by editor(s) in revised form: April 20, 2011, and May 20, 2011; Posted: July 12, 2011. The first author was partially supported by NSF grant DMS-0841321. The authors thank Christophe Delaunay, Benedict Gross, Robert Guralnick, Karl Rubin, and the referee for comments.|
|Subject Keywords:||Selmer group; Shafarevich-Tate group; maximal isotropic; quadratic space; Weil pairing; theta characteristic|
|Other Numbering System:|
|Classification Code:||MSC (2010): Primary 11G10; Secondary 11G05, 11G30, 14G25, 14K15|
|Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Jason Perez|
|Deposited On:||08 Dec 2011 22:40|
|Last Modified:||26 Dec 2012 14:35|
Repository Staff Only: item control page