Graduate Number Theory Seminar
Sun Woo Park
University of Wisconsin-Madison
On the prime Selmer ranks of cyclic prime twist families of elliptic curves over global function fields
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
Wednesday April 12, 2023 at 4:00 PM in Zoom