Vol. 13, No. 1, 2019

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A tubular variant of Runge's method in all dimensions, with applications to integral points on Siegel modular varieties

Samuel Le Fourn

Vol. 13 (2019), No. 1, 159–209
DOI: 10.2140/ant.2019.13.159
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 ${A}_{2}\left(2\right)$.

Keywords
Runge's method, integral points on varieties, abelian varieties
Mathematical Subject Classification 2010
Primary: 11G35
Secondary: 11G10, 14G05, 14G35

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