2026-09-20T00:19:17Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/1598842025-07-08T03:03:55Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Poonen, Bjorn
Achenjang, Niven
Massachusetts Institute of Technology. Department of Mathematics
2025-07-07T17:36:54Z
2025-07-07T17:36:54Z
2025-05
2025-05-13T13:31:10.577Z
https://hdl.handle.net/1721.1/159884
https://orcid.org/0000-0001-9551-5821
Let K be the function field of a smooth curve B over a finite field k of arbitrary characteristic. We prove that the average size of the 2-Selmer groups of elliptic curves E/K is at most 1 + 2ζʙ(2)ζʙ(10), where ζʙ is the zeta function of B. In particular, in the limit as q = #k ! ∞ (with the genus g(B) fixed), we see that the average size of 2-Selmer is bounded above by 3, even in “bad” characteristics. This completes the proof that the average rank of elliptic curves, over any fixed global field, is finite. Handling the case of characteristic 2 requires us to develop a new theory of integral models of 2-Selmer elements, dubbed “hyper-Weierstrass curves.”
Ph.D.
Massachusetts Institute of Technology
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The Average Size of 2-Selmer Groups of Elliptic Curves in Characteristic 2
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<Abstract>Let K be the function field of a smooth curve B over a finite field k of arbitrary characteristic. We prove that the average size of the 2-Selmer groups of elliptic curves E/K is at most 1 + 2ζʙ(2)ζʙ(10), where ζʙ is the zeta function of B. In particular, in the limit as q = #k ! ∞ (with the genus g(B) fixed), we see that the average size of 2-Selmer is bounded above by 3, even in “bad” characteristics. This completes the proof that the average rank of elliptic curves, over any fixed global field, is finite. Handling the case of characteristic 2 requires us to develop a new theory of integral models of 2-Selmer elements, dubbed “hyper-Weierstrass curves.”</Abstract>
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