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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Random maximal isotropic subspaces and Selmer groups


Authors: Bjorn Poonen and Eric Rains
Journal: J. Amer. Math. Soc. 25 (2012), 245-269
MSC (2010): Primary 11G10; Secondary 11G05, 11G30, 14G25, 14K15
Published electronically: July 12, 2011
MathSciNet review: 2833483
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Abstract: Under suitable hypotheses, we construct a probability measure on the set of closed maximal isotropic subspaces of a locally compact quadratic space over $ \mathbb{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 $ \mathbb{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.


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Additional Information

Bjorn Poonen
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307
Email: poonen@math.mit.edu

Eric Rains
Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125
Email: rains@caltech.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-2011-00710-8
PII: S 0894-0347(2011)00710-8
Keywords: Selmer group, Shafarevich-Tate group, maximal isotropic, quadratic space, Weil pairing, theta characteristic
Received by editor(s): September 21, 2010
Received by editor(s) in revised form: April 20, 2011, and May 20, 2011
Published electronically: July 12, 2011
Additional Notes: The first author was partially supported by NSF grant DMS-0841321.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.