On the prime Selmer ranks of cyclic prime twist families of elliptic curves over global function fields

TL;DR

This paper proves that the distribution of prime Selmer group sizes in certain elliptic curve families over function fields matches a conjectured distribution, using advanced number theory and probabilistic tools.

Contribution

It establishes the distribution of prime Selmer ranks for cyclic prime twists of elliptic curves over function fields, confirming a conjecture with explicit error bounds.

Findings

Distribution matches the conjectured model by Bhargava et al.

Explicit error bounds are provided for the distribution.

Uses tools like the Riemann hypothesis over function fields and Markov chain ergodicity.

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 Erd\"os-Kac theorem, and the geometric ergodicity of Markov chains.