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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Random maximal isotropic subspaces and Selmer groups
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by Bjorn Poonen and Eric Rains PDF
J. Amer. Math. Soc. 25 (2012), 245-269 Request permission

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.

References
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Additional Information
  • Bjorn Poonen
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307
  • MR Author ID: 250625
  • ORCID: 0000-0002-8593-2792
  • Email: poonen@math.mit.edu
  • Eric Rains
  • Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125
  • MR Author ID: 311477
  • Email: rains@caltech.edu
  • 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.
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 25 (2012), 245-269
  • MSC (2010): Primary 11G10; Secondary 11G05, 11G30, 14G25, 14K15
  • DOI: https://doi.org/10.1090/S0894-0347-2011-00710-8
  • MathSciNet review: 2833483