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

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

doi:10.2140/ant.2020.14.1853

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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 $𝔽_{q} [x]$ , 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

Vol. 14, No. 7, 2020