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 ${\mu }_p$. Based on the works by Swinnerton-Dyer and Klagsbrun, Mazur, and Rubin, we prove that the probability distribution of 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 Erdös-Kac theorem, and the geometric ergodicity of Markov chains.

Version 4: Incorporated comments from anonymous referee. Three key updates made: (1) Updated Definition 4.7 on splitting partitions. (2) Updated the proof of Proposition 5.13. (3) Updated Theorem 6.6 and non-optimal bounds for geometric rate of convergence of Markov chains