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

We fix an elliptic curve E / F q (t) and consider the family { E ⊗ χ 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 ℓ ≠ 2 coprime to q. We obtain a restriction of (− 1 / 2 , 1 / 2) on the support with a unitary symmetry. [ABSTRACT FROM AUTHOR]