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The geometric distribution of Selmer groups of elliptic curves over function fields

Feng, Tony and Landesman, Aaron and Rains, Eric M. (2022) The geometric distribution of Selmer groups of elliptic curves over function fields. Mathematische Annalen . ISSN 0025-5831. doi:10.1007/s00208-022-02429-1. (In Press)

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Fix a positive integer n and a finite field F_q. We study the joint distribution of the rank rk (E), the n-Selmer group Sel_n (E), and the n-torsion in the Tate–Shafarevich group III(E)[n] as E varies over elliptic curves of fixed height d ≥ 2 over F_q (T). We compute this joint distribution in the large q limit. We also show that the "large q, then large height" limit of this distribution agrees with the one predicted by Bhargava–Kane–Lenstra–Poonen–Rains.

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Additional Information:It is our pleasure to thank Ravi Vakil for organizing the “What’s on My Mind” seminar, which led to the genesis of this paper. We thank Johan de Jong, Chao Li, Bjorn Poonen, Arul Shankar, Doug Ulmer, and Melanie Matchett Wood for helpful discussions. We thank Lisa Sauermann for help translating [24]. We also thank David Zureick-Brown and Jackson Morrow for help with writing and running MAGMA code. The first author was supported by a Stanford ARCS Fellowship and an NSF Postdoctoral Fellowship under Grant No. 1902927, and the second author was supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1656518. Open Access funding provided by the MIT Libraries.
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ID Code:117068
Deposited By: Melissa Ray
Deposited On:24 Sep 2022 01:32
Last Modified:24 Sep 2022 01:32

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