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

2021/09/30 by Patrick Meisner, Meisner, Patrick, Anders Södergren +1 · 2 citations
Arts and Humanities · Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #French Historical and Cultural Studies #Historical Studies and Socio-cultural Analysis #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2110.00102

openalex publication_date 2021/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate the low-lying zeros in families of L-functions attached to quadratic and cubic twists of elliptic curves defined over \mathbbF_q(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.

Cited by

• Linear Combinations of Logarithms of L-functions over Function Fields at Microscopic Shifts and Beyond
• Average rank of elliptic curves over function fields

Related

• One-level density of the family of twists of an elliptic curve over function fields
• Some Results in the Theory of Low-lying Zeros: Determining the 1-level density, identifying the group symmetry and the arithmetic of moments of Satake parameters
• Conditional lower bounds on the distribution of central values in families of L-functions
• One-level density of families of elliptic curves and the Ratios Conjectures
• One-level densities in families of Grössencharakters associated to CM elliptic curves