Experimental data for Goldfeld’s conjecture over function fields 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 $Bbb F_q(t)$ 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 $Bbb F_q(t)$ 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.

Keywords for this software

Anything in here will be replaced on browsers that support the canvas element