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

Abstract

We investigate the low-lying zeros in families of L-functions attached to quadratic and cubic twists of elliptic curves defined over Fq(T ). In particular, we present precise expressions for the expected values of traces of high powers of the Frobenius class in these families with a focus on the lower order behavior. As an application we obtain results on one-level densities and we verify that these elliptic curve families have orthogonal symmetry type. In the quadratic twist families our results refine previous work of Comeau-Lapointe. Moreover, in this case we find a lower order term in the one-level density reminiscent of the deviation term found by Rudnick in the hyperelliptic ensemble. On the other hand, our investigation is the first to treat these questions in families of cubic twists of elliptic curves and in this case it turns out to be more complicated to isolate lower order terms due to a larger degree of cancellation among lower order contributions.

On the vanishing of twisted $L$-functions of elliptic curves over rational function fields

“…For quadratic twists, it is possible to prove results on the one-level density strong enough to deduce that a positive proportion of twists with even (respectively odd) analytic rank do no vanish (respectively vanish of order 1) at the central critical point [HB04,CL22]. The one-level density, or average rank, of higher order twists for elliptic curves L-functions was studied by [Cho] over number fields and [MS,CL22] over function fields. Quadratic twists of elliptic curve over functions fields were also studied by [BFKRG20] who obtained results on the correlation of the analytic ranks of two twisted elliptic curves.…”