Vol. 4, No. 5, 2010

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The Manin constant of elliptic curves over function fields

Ambrus Pál

Vol. 4 (2010), No. 5, 509–545
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