The Manin constant of elliptic curves over function fields

Pál, Ambrus

doi:10.2140/ant.2010.4.509

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The Manin constant of elliptic curves over function fields

Ambrus Pál

Vol. 4 (2010), No. 5, 509–545

DOI: 10.2140/ant.2010.4.509

Abstract

We study the $p$ -adic valuation of the values of normalised Hecke eigenforms attached to nonisotrivial elliptic curves defined over function fields of transcendence degree one over finite fields of characteristic $p$ . We derive upper bounds on the smallest attained valuation in terms of the minimal discriminant under a certain assumption on the function field, and provide examples to show that our estimates are optimal. As an application of our results, we prove the analogue of the degree conjecture unconditionally for strong Weil curves with square-free conductor defined over function fields satisfying the assumption mentioned above.

Keywords

elliptic curves, Hecke eigenforms, degree conjecture

Mathematical Subject Classification 2000

Primary: 11G05

Secondary: 11G40, 14F30

Milestones

Received: 31 March 2009

Revised: 2 December 2009

Accepted: 31 December 2009

Published: 10 July 2010

Authors

Ambrus Pál
Department of Mathematics Imperial College 180 Queen’s Gate London, SW7 2AZ United Kingdom

Vol. 4, No. 5, 2010