Sun Woo Park : On the prime Selmer ranks of cyclic prime twist families of elliptic curves over global function fields

Posted by Zhehao Li , part of the Graduate Number Theory Seminar.

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April 12, 2023, 4 p.m.

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Abstract

Fix a prime number p. Let $\mathbb{F}_q$ be a finite field of characteristic coprime to 2, 3, and p, which also contains the primitive $p$-th root of unity. Based on the works by Swinnerton-Dyer and Klagsbrun, Mazur, and Rubin, we prove that the probability distribution fo the sizes of prime Selmer groups over a family of cyclic prime twists of non-isotrivial elliptic curves over $\mathbb{F}_q(t)$ satisfying a number of mild constraints conforms to the distribution conjectured by Bhargava, Kane, Lenstra, Poonen, and Rains with explicit error bounds. The key tools used in proving these results are the Riemann hypothesis over global function fields, the Erdos-Kac theorem, and the geometric ergodicity of Markov chains.

Zoom link: https://uic.zoom.us/j/82357749325?pwd=L1JoRm9rUjZJbDNZeUJvdDljWGdhUT09