Low-lying zeros in families of elliptic curve L-functions over function fields
Journal article, 2022

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.

Function fields over finite fields

One-level density

Quadratic and cubic twists

Elliptic curves

L-functions

Frobenius class

Author

Patrick Meisner

Royal Institute of Technology (KTH)

Anders Södergren

Chalmers, Mathematical Sciences, Algebra and geometry

Finite Fields and their Applications

1071-5797 (ISSN) 1090-2465 (eISSN)

Vol. 84 102096

Värdefördelning för L-funktioner och zetafunktioner

Swedish Research Council (VR) (2016-03759), 2017-01-01 -- 2020-12-31.

Subject Categories

Algebra and Logic

Geometry

Mathematical Analysis

DOI

10.1016/j.ffa.2022.102096

More information

Latest update

9/1/2022 1