Vol. 242, No. 2, 2009

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Nonarchimedean equidistribution on elliptic curves with global applications

Clayton Petsche

Vol. 242 (2009), No. 2, 345–375

Let 𝕂 be an algebraically closed, complete, nonarchimedean field, let E∕𝕂 be an elliptic curve, and let E denote the Berkovich analytic space associated to E∕𝕂. We study the μ-equidistribution of finite subsets of E(𝕂), where μ is a certain canonical unit Borel measure on E. Our main result is an inequality bounding the error term when testing against a certain class of continuous functions on E. We then give two applications to elliptic curves over global function fields: We prove a function field analogue of the Szpiro–Ullmo–Zhang equidistribution theorem for small points, and a function field analogue of a result of Baker, Ih, and Rumely on the finiteness of S-integral torsion points. Both applications are given in explicit quantitative form.

elliptic curves, nonarchimedean fields, function fields, Berkovich analytic spaces, equidistribution, integral points
Mathematical Subject Classification 2000
Primary: 11G05, 11G07, 11G50
Received: 20 December 2008
Revised: 13 April 2009
Accepted: 14 April 2009
Published: 1 October 2009
Clayton Petsche
Department of Mathematics and Statistics
Hunter College
695 Park Avenue
New York, NY 10065
United States