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

Abstract

Runge's method is a tool to figure out integral points on algebraic 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 an explicit finiteness result for integral points on the Siegel modular variety A2(2).

Sur la méthode de Runge et les points entiers de certaines variétés modulaires de Siegel

“…Ainsi, le but de [LF16] est d'adapter la méthode de Runge en dimension supérieure, en particulier pour traiter non plus de courbes modulaires mais de variétés modulaires (par exemple celles de Siegel). Cette note s'articule autour des trois résultats principaux de l'article : le premier donne une généralisation de la méthode de Runge en dimension supérieure (partie 1), conçue pour une certaine souplesse d'utilisation.…”

“…Dans cette note, nous décrivons les principaux résultats de [LF16] qui visent à établir de nouveaux éclairages sur le comportement des points entiers de variétés modulaires en dimension supérieure.…”

A criterion for nondensity of integral points

“…Before Siegel's theorem on integral points of curves, Runge [18] proved some special cases over $$\mathbb {Q}$$ under the assumption that the divisor $$D$$ on the curve consisted of at least two components defined over $$\mathbb {Q}$$. Runge's method has been extended to the higher dimensional setting by Levin [13, 15] (see also Le Fourn's extension [12]), and over $$\mathbb {Q}$$, the main technical assumption is that the divisor $$D$$ on the variety consists of several components $$D_j$$ defined over $$\mathbb {Q}$$ having empty total intersection $$\cap _jD_j$$. This last assumption has been relaxed by Levin and Wang [16], by requiring that $$\cap _jD_j$$ is finite and its geometric points are, in fact, rational over $$\mathbb {Q}$$, among other technical assumptions on the geometry of the divisors $$D_j$$.…”