ELLFF

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.

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