On the Average of p-Selmer Ranks in Quadratic Twist Families of Elliptic Curves Over Global Function Fields

Abstract

Let $\mathbb{F}_{q}$ be a finite field whose characteristic is relatively prime to $2$ and $3$. Let $p$ be a prime number that is coprime to $q$. Let $E$ be an elliptic curve over the global function field $K = \mathbb{F}_{q}(t)$ such that $\textrm{Gal}(K(E[p])/K)$ contains the special linear group $\textrm{SL}_{2}(\mathbb{F}_{p})$. We show that if the quadratic twist family of $E$ has an element whose Néron model has a multiplicative reduction away from $\infty $, then the average $p$-Selmer rank is $p+1$ in large $q$-limit for almost all primes $p$.