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
References in zbMATH (referenced in 4 articles , 1 standard article )
Showing results 1 to 4 of 4.
- Bui, Hung M.; Florea, Alexandra: Moments of Dirichlet (L)-functions with prime conductors over function fields (2020)
- Griffon, Richard: Bounds on special values of (L)-functions of elliptic curves in an Artin-Schreier family (2019)
- Cha, Byungchul; Fiorilli, Daniel; Jouve, Florent: Prime number races for elliptic curves over function fields (2016)
- Baig, Salman; Hall, Chris: Experimental data for Goldfeld’s conjecture over function fields (2012)