Experimental Data for Goldfeld's Conjecture over Function Fields

TL;DR

This study provides empirical evidence supporting Goldfeld's conjecture in the function field setting by computing L-functions of quadratic twists of elliptic curves, showing the average analytic rank approaches 1/2.

Contribution

The paper introduces an efficient algorithm and open-source library for computing L-functions of elliptic curves over function fields, supporting the conjecture with extensive data.

Findings

Average analytic rank converges to 1/2 for the studied families.

The data supports the density conjecture related to analytic ranks.

An open-source tool for L-function computation is provided.

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 non-isogenous elliptic curves over 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 was 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 non-isotrivial elliptic curves over 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.