The Average Size of 2-Selmer Groups of Elliptic Curves in Characteristic 2

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.”

Name thesis:degree_name

Doctoral

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Mathematics

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2025

Author dc:creator

• Achenjang, Niven

Advisor dc:contributor.advisor

• Poonen, Bjorn

Statement dc:rights

• In Copyright - Educational Use Permitted
• Copyright retained by author(s)

Licence dc:rights.uri

• https://rightsstatements.org/page/InC-EDU/1.0/

Handle dc:identifier.uri

https://hdl.handle.net/1721.1/159884

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/159884