Independence of the Zeros of Elliptic Curve L-Functions over Function Fields

The linear independence (LI) hypothesis, which states roughly that the imaginary parts of the critical zeros of Dirichlet L-functions are linearly independent over the rationals, is known to have interesting consequences in the study of prime number races, as pointed out by Rubinstein and Sarnak. In this article, we prove that a function field analog of LI holds generically within certain families of elliptic curve L-functions and their symmetric powers. More precisely, for certain algebro-geometric families of elliptic curves defined over the function field of a fixed curve over a finite field, we give strong quantitative bounds for the number of elements in the family for which the relevant L-functions have their zeros as linearly independent over the rationals as possible.