Experimental Data for Goldfeld’s Conjecture over Function Fields

Abstract

This paper presents empirical evidence supporting Goldfeld’s conjecture on the average analytic rank of a family of quadratic twists of a fixed elliptic curve in the function field setting. In particular, we consider representatives of the four classes of nonisogenous elliptic curves over $\mathbb{F}_q(t)$ with $(q, 6) = 1$ possessing two places of multiplicative reduction and one place of additive reduction. The case of $q = 5$ provides the largest data set as well as the most convincing evidence that the average analytic rank converges to 1/2, which we also show is a lower bound following an argument of Kowalski. The data were generated via explicit computation of the $L$-function of these elliptic curves, and we present the key results necessary to implement an algorithm to efficiently compute the $L$-function of nonisotrivial elliptic curves over $\mathbb{F}_q(t)$ by realizing such a curve as a quadratic twist of a pullback of a "versal" elliptic curve. We also provide a reference for our open-source library ELLFF, which provides all the necessary functionality to compute such $L$-functions, and additional data on analytic rank distributions as they pertain to the density conjecture.