Determining large groups and varieties from small subgroups and subvarieties

This course will consist of two parts with the common theme of analyzing a geometric object via geometric sub-objects which are ‘small’ in an appropriate sense. In the first half of the course, we will begin by describing the work on H. W. Lenstra, Jr., on finding generators for the unit groups of subrings of number fields. Lenstra discovered that from the point of view of computational complexity, it is advantageous to first consider the units of the ring of S-integers of L for some moderately large finite set of places S. This amounts to allowing denominators which only involve a prescribed finite set of primes. We will develop a generalization of Lenstra’s method which applies to find generators of small height for the S-integral points of certain algebraic groups G defined over number fields. We will focus on G which are compact forms of GLd for d ≥ 1. In the second half of the course, we will turn to smooth projective surfaces defined by arithmetic lattices. These are Shimura varieties of complex dimension 2. We will try to generate subgroups of finite index in the fundamental groups of these surfaces by the fundamental groups of finite unions of totally geodesic projective curves on them, that is, via immersed Shimura curves. In both settings, the groups we will study are S-arithmetic lattices in a product of Lie groups over local fields. We will use the structure of our generating sets to consider several open problems about the geometry, group theory, and arithmetic of these lattices. In particular, we will consider the structure of the cohomology of S-arithmetic groups and characteristic p analogues. We will also discuss connections with the congruence subgroup problem, which is open in many of the cases under consideration in this project.