A tubular variant of Runge’s methodin all dimensions, with applications to integral points on Siegel modularvarieties

2016/11/30 by Samuel Le Fourn
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebra over a field #Algebraic Geometry and Number Theory #Modular design #Modular form #Siegel modular form #Variety (cybernetics) #math.NT #msc:11G10 #msc:11G18 #msc:14K25

paper · pdf · doi:10.2140/ant.2019.13.159

published as Alg. Number Th. 13 (2019) 159-209 · 46 pages, minor changes of notations

openalex created_date 2016/12/08 · arxiv created 2017/08/06 · openalex publication_date 2019/02/13 · arxiv updated 2019/03/06 · openalex updated_date 2026/08/05

Abstract

Runge’s method is a tool to figure out integral points on algebraic curves effectively in terms of height. This method has been generalized to varieties of any dimension, but unfortunately the conditions needed to apply it are often too restrictive. We provide a further generalization intended to be more flexible while still effective, and exemplify its applicability by giving finiteness results for integral points on some Siegel modular varieties. As a special case, we obtain an explicit finiteness result for integral points on the Siegel modular variety [math] .

Citations

• Toroidal Compactification of Siegel Spaces
• Diophantine Approximations and Value Distribution Theory
• Towards an Enumerative Geometry of the Moduli Space of Curves
• Bornes pour la torsion des courbes elliptiques sur les corps de nombres
• Algebraic Geometry