Algebraic points on meromorphic curves

2012/04/27 by Mathilde Herblot, Herblot, Mathilde
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1204.6336

arxiv created 2012/04/27 · arxiv updated 2012/05/01

Abstract

The classic Schneider-Lang theorem in transcendence theory asserts that there are only finitely many points at which algebraically independent complex meromorphic functions of finite order of growth can simultaneously take values in a number field, when satisfying a polynomial differential equation with coefficients in this given number field. In this article, we are interested in generalizing this theorem in two directions. First, instead of considering meromorphic functions on C we consider holomorphic maps on an affine curve over the field C or C_p. This extends a statement of D. Bertrand, which applies to meromorphic functions on P¹(C) or P¹(C_p) minus a finite subset of points. Secondly, we deal with algebraic values taken by the functions, instead of rational values as in the classic setting, inspired by a work of D. Bertrand. We prove a geometric statement extending those two results, using the slopes method, written in the language of Arakelov geometry. In the complex case, we recover a special case of a result by C. Gasbarri.

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