Average size of Selmer group in large q limit

TL;DR

This paper proves a function field analogue of Poonen-Rains heuristics, showing that the average size of p-Selmer groups over quadratic twists of elliptic curves over finite fields approaches p+1 as both the twist degree and field size grow large.

Contribution

It establishes the average size of p-Selmer groups over quadratic twists in the function field setting, confirming the heuristic prediction in the large q and degree limit.

Findings

Average size of p-Selmer groups approaches p+1 for large q and twist degree n.

The result holds for all but finitely many primes p.

The average rank of p-Selmer groups converges to p+1.

Abstract

In this paper, we prove a function field-analogue of Poonen-Rains heuristics on the average size of $p$-Selmer group. Let $E$ be an elliptic curve defined over $\mathbb{Z}[t]$. Then $E$ is also defined over $\mathbb{F}_q$ for any $q$ of prime power. We show that for large enough $q$, the average size of the $p$-Selmer groups over the family of quadratic twists of $E$ over $\mathbb{F}_q[t]$ is equal to $p+1$ for all but finitely many primes $p$. Namely, if we twist the curve in $\mathbb{F}_q[t]$ by polynomials of fixed degree $n$ and let both $n$ and $q$ approach to infinity, then the average rank of $p$-Selmer group converges to $p+1$.