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

Abstract

We fix an elliptic curve E/Fq(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 = 2 coprime to q. We obtain a restriction of (−1/2, 1/2) on the support with a unitary symmetry.

Low-lying zeros in families of elliptic curve $L$-functions over function fields

“…is not the same as the one stated here. However, one may easily deduce this result from that of [9].…”

“…Important results in this direction were proved by Rudnick [30] and Bui-Florea [4] who investigated the low-lying zeros of quadratic Dirichlet L-functions in the hyperelliptic ensemble. More recently, Comeau-Lapointe [9] investigated expected values of traces of high powers of the Frobenius class and the one-level density of families of quadratic twists of elliptic curves in this context and used the results to give upper bounds on the average rank in these families. In this paper, we refine results in [9] to isolate lower order terms and…”

“…More recently, Comeau-Lapointe [9] investigated expected values of traces of high powers of the Frobenius class and the one-level density of families of quadratic twists of elliptic curves in this context and used the results to give upper bounds on the average rank in these families. In this paper, we refine results in [9] to isolate lower order terms and…”

“…In recent years, there has been an increased interest in a variety of different aspects of higher order characters and twists; see, e.g., [1,7,8,9,10,11,12,13,27,28]. Motivated by this development, we investigate expected values of traces of high powers of the Frobenius class and the one-level density of families of cubic twists of elliptic curves of the form y 2 = x 3 + B defined over F q (t).…”

“…It was recently proven by Comeau-Lapointe [9] that, for ǫ > 0, n ∈ Z ≥1 and N > 4n + 16, the averages of traces of powers of Θ ED satisfy 2,3…”