Generalizations of Siegel’s and Picard’s theorems

Abstract

We prove new theorems that are higher-dimensional generalizations of the classical theorems of Siegel on integral points on affine curves and of Picard on holomorphic maps from C to affine curves. These include results on integral points over varying number fields of bounded degree and results on Kobayashi hyperbolicity. We give a number of new conjectures describing, from our point of view, how we expect Siegel's and Picard's theorems to optimally generalize to higher dimensions. In this article we prove new theorems that are higher-dimensional generaliza tions of the classical theorems of Siegel on integral points on affine curves and of Picard on holomorphic maps from C to affine curves. In Section 2, we will give the statements of Siegel's and Picard's theorems, and we will recall how these two theorems from such seemingly different areas of mathematics are related. We will then proceed to give a number of new conjectures describing, from our point of view, how we expect Siegel's and Picard's theorems to optimally generalize to higher dimensions. These include conjectures on integral points over varying number fields of bounded degree and conjectures addressing hyperbolic questions. These conjectures appear to be fundamentally new. We will then summarize our progress on these conjectures. We have been able to get results in all dimensions, with best-possible results in many cases for surfaces.