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

Creators: Feng, Tony; Landesman, Aaron; Rains, Eric M.

Abstract

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.

Additional Information

This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 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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