Mathematics – Number Theory
Scientific paper
2011-06-15
Mathematics
Number Theory
26 pages, 8 figures, 11 tables, submitted to Experimental Mathematics
Scientific paper
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.
Baig Salman
Hall Chris
No associations
LandOfFree
Experimental Data for Goldfeld's Conjecture over Function Fields does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Experimental Data for Goldfeld's Conjecture over Function Fields, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Experimental Data for Goldfeld's Conjecture over Function Fields will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-188933