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Mathematische Annalen

, Volume 341, Issue 4, pp 789–824 | Cite as

Courbes algébriques et équations multiplicatives

  • Guillaume Maurin
Article

Abstract

We study the intersection of an algebraic curve C lying in a multiplicative torus over \({\bar{\mathbb{Q}}}\) with the union of all algebraic subgroups of codimension 2. Finiteness of this set has already been proved by Bombieri, Masser and Zannier under the assumption that C is not contained in a translate of a proper subtorus. Following this result, the question of the minimal hypothesis implying finiteness has been raised by these authors, giving rise to the conjecture~: finiteness holds precisely when C is not contained in a proper subgroup. We prove here this statement which is also a special case of more general conjectures stated independently by Zilber and Pink. Our proof takes its inspiration from an article by Rémond and Viada concerning the Zilber-Pink conjecture for curves lying in a power of an elliptic curve. Hence, it relies on a uniform version of the Vojta inequality proven via the generalized Vojta inequality of Rémond. The main task is to establish a lower bound for some intersection numbers, here on a whole family of surfaces obtained by blowing up a compactification of C ×  C.

Keywords

Algebraic Subgroup Nous Allons Nous Obtenons Nous Montrons Nous Rappelons 
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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Institut Fourier, UMR 5582Saint-Martin-d’Hères CedexFrance

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