arXiv subjects
Publications and source records attributed to Aaron Levin.
We study the S-integral points on the complement of a union of hyperplanes in projective space, where S is a finite set of places of a number field k. In the classical case where S consists of the set of archimedean places of k, we completely characterize, in terms of the hyperplanes and the field k, when the (S-)integral points are not Zariski-dense.
We prove new theorems which are higher-dimensional generalizations of the classical theorems of Siegel on integral points on affine curves and of Picard on holomorphic maps from $\mathbb{C}$ 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 some special cases we will be able to relate our conjectures to existing conjectures. In this respect, we are also led to formulate a new conjecture relating the absolute discriminant and height of an algebraic point on a projective variety over a number field.