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

TL;DR

This paper introduces a more flexible generalization of Runge's method for finding integral points on higher-dimensional varieties, demonstrated through finiteness results on Siegel modular varieties including an explicit case for A_2(2).

Contribution

It presents a new, more adaptable version of Runge's method applicable to all dimensions, expanding its utility in arithmetic geometry.

Findings

Finiteness results for integral points on certain Siegel modular varieties.

Explicit finiteness result for integral points on A_2(2).

Enhanced applicability of Runge's method in higher dimensions.

Abstract

Runge's method is a tool to figure out integral points on curves effectively in terms of height. This method has been generalised to varieties of any dimension, unfortunately its conditions of application are often too restrictive. In this paper, we provide a further generalisation 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 a totally explicit finiteness result for integral points on the Siegel modular variety $A_2(2)$.