Average size of Selmer group in large q limit

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 $p+1$.