Random maximal isotropic subspaces and Selmer groups
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- by Bjorn Poonen and Eric Rains
- J. Amer. Math. Soc. 25 (2012), 245-269
- DOI: https://doi.org/10.1090/S0894-0347-2011-00710-8
- Published electronically: July 12, 2011
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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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Bibliographic 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