One-level density of the family of twists of an elliptic curve over function fields

Abstract: We fix an elliptic curve $E/\mathbb{F}_q(t)$ and consider the family ${E\otimes\chi_D}$ of $E$ twisted by quadratic Dirichlet characters. The one-level density of their $L$-functions is shown to follow orthogonal symmetry for test functions with Fourier transform supported inside $(-1,1)$. As an application, we obtain an upper bound of 3/2 on the average analytic rank. By splitting the family according to the sign of the functional equation, we obtain that at least $12.5\%$ of the family have rank zero, and at least $37.5\%$ have rank one. The Katz and Sarnak philosophy predicts that those percentages should both be $50\%$ and that the average analytic rank should be $1/2$. We finish by computing the one-level density of $E$ twisted by Dirichlet characters of order $\ell\neq 2$ coprime to $q$. We obtain a restriction of $(-1/2,1/2)$ on the support with a unitary symmetry.