Random maximal isotropic subspaces and Selmer groups
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...
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| Language: | en_US |
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American Mathematical Society
2012
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| Online Access: | http://hdl.handle.net/1721.1/71657 https://orcid.org/0000-0002-8593-2792 |
| _version_ | 1826191515482849280 |
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| author | Poonen, Bjorn Rains, Eric |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Poonen, Bjorn Rains, Eric |
| author_sort | Poonen, Bjorn |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T08:57:04Z |
| format | Article |
| id | mit-1721.1/71657 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T08:57:04Z |
| publishDate | 2012 |
| publisher | American Mathematical Society |
| record_format | dspace |
| spelling | mit-1721.1/716572022-09-26T09:25:09Z Random maximal isotropic subspaces and Selmer groups Poonen, Bjorn Rains, Eric Massachusetts Institute of Technology. Department of Mathematics Poonen, Bjorn Poonen, Bjorn Rains, Eric 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. National Science Foundation (U.S.) (DMS-0841321) 2012-07-17T18:25:23Z 2012-07-17T18:25:23Z 2012-07 2011-04 Article http://purl.org/eprint/type/JournalArticle 1088-6834 0894-0347 MathSciNet review: 2833483 http://hdl.handle.net/1721.1/71657 Poonen, Bjorn, and Eric Rains. “Random Maximal Isotropic Subspaces and Selmer Groups.” Journal of the American Mathematical Society 25.1 (2012): 245–269. Web. https://orcid.org/0000-0002-8593-2792 en_US http://dx.doi.org/10.1090/S0894-0347-2011-00710-8 Journal of the American Mathematical Society Creative Commons Attribution-Noncommercial-Share Alike 3.0 http://creativecommons.org/licenses/by-nc-sa/3.0/ application/pdf American Mathematical Society MIT web domain |
| spellingShingle | Poonen, Bjorn Rains, Eric Random maximal isotropic subspaces and Selmer groups |
| title | Random maximal isotropic subspaces and Selmer groups |
| title_full | Random maximal isotropic subspaces and Selmer groups |
| title_fullStr | Random maximal isotropic subspaces and Selmer groups |
| title_full_unstemmed | Random maximal isotropic subspaces and Selmer groups |
| title_short | Random maximal isotropic subspaces and Selmer groups |
| title_sort | random maximal isotropic subspaces and selmer groups |
| url | http://hdl.handle.net/1721.1/71657 https://orcid.org/0000-0002-8593-2792 |
| work_keys_str_mv | AT poonenbjorn randommaximalisotropicsubspacesandselmergroups AT rainseric randommaximalisotropicsubspacesandselmergroups |