Sal Baig
2010-12-03
Title
The Average Analytic Rank in a Family of Quadratic Twists of an Elliptic Curve over nt(2f)20101203/latex_78dda8521c4935de09a70acc1a2e3ecad9058f4f_p1.png "$F_q(t)$")
Abstract
The nt(2f)20101203/latex_ed45432016aa8f461445b268b596ea92edbcc79e_p1.png "$L$")
-function of a non-isotrivial elliptic curve over a function field of positive characteristic is known to be a polynomial with integer coefficients. Its analytic rank can be thus computed exactly, providing data to test the validity of Goldfeld's conjecture in the function field case. This conjecture claims that the average rank in a family of a quadratic twists of a fixed elliptic curve approaches 1/2 as the degree nt(2f)20101203/latex_2417891192eded7e4620724a80f9602630c0b1ae_p1.png "$d$")
of the twisting polynomials increase. We (for the most part) show that this average rank is at least 1/2 as nt(2f)20101203/latex_dc75bbf9b9c1ec2cffe8e65edb9e1e199783ac25_p1.png "$d\rightarrow\infty$")
.