Vol. 14, No. 7, 2020

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Moments of quadratic twists of elliptic curve $L$-functions over function fields

Hung M. Bui, Alexandra Florea, Jonathan P. Keating and Edva Roditty-Gershon

Vol. 14 (2020), No. 7, 1853–1893
DOI: 10.2140/ant.2020.14.1853
Abstract

We calculate the first and second moments of L-functions in the family of quadratic twists of a fixed elliptic curve E over 𝔽q[x], asymptotically in the limit as the degree of the twists tends to infinity. We also compute moments involving derivatives of L-functions over quadratic twists, enabling us to deduce lower bounds on the correlations between the analytic ranks of the twists of two distinct curves.

Keywords
elliptic curve, L-function, rank, finite field, correlation
Mathematical Subject Classification 2010
Primary: 11M06
Secondary: 11M38
Milestones
Received: 1 February 2019
Revised: 24 June 2019
Accepted: 2 September 2019
Published: 18 August 2020
Authors
Hung M. Bui
Department of Mathematics
University of Manchester
United Kingdom
Alexandra Florea
Department of Mathematics
Columbia University
New York, NY
United States
Jonathan P. Keating
Mathematical Institute
University of Oxford
United Kingdom
Edva Roditty-Gershon
Department of Applied Mathematics
Holon Institute of Technology
Holon
Israel