Random maximal isotropic subspaces and Selmer groups
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Poonen_Random Maximal.pdf
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482.69 KB
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Author(s)
Poonen, Bjorn
Rains, Eric
Date Issued
July 2012
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
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
https://doi.org/10.1090/S0894-0347-2011-00710-8
