Arithmetic Questions in the Theory of Linear Algebraic Groups

Michigan State University, East Lansing MI

Investigators

Abstract

Linear algebraic groups are groups of matrices that are described by polynomial equations. Such groups arise as groups of symmetries of various objects and are ubiquitous across many areas of mathematics, including algebraic geometry, number theory, and mathematical physics. In the arithmetic context, work done over the last six decades has resulted in a well-developed theory of linear algebraic groups over the rational numbers and other similar fields. While activity in this area is still ongoing, over the last ten years various problems in Lie group theory, arithmetic geometry, and other subjects have led to significant interest in the properties of algebraic groups over fields of geometric origin. Building on previous work, the research program will investigate the arithmetic, geometric, and structural aspects of algebraic groups over such higher-dimensional fields, with a particular focus on various finiteness properties. Mentoring graduate students and developing courses at the undergraduate and graduate levels will be an integral part of this work. In addition, a book project will be undertaken to open up recent developments in the emerging arithmetic theory of algebraic groups over higher-dimensional fields to a broader audience. The project is a multi-faceted research program in the study of algebraic groups over higher-dimensional fields. The work will focus on the following three directions: the analysis of algebraic groups with good reduction and applications to local-global principles, the study of finiteness properties of unramified cohomology, and the investigation of rigidity phenomena for abstract homomorphisms of algebraic groups. A major goal in the study of groups with good reduction will be to make progress on a finiteness conjecture for forms of reductive algebraic groups over finitely generated fields having good reduction with respect to divisorial sets of discrete valuations. This work will significantly expand the scope of previous results, which dealt mainly with groups over fraction fields of Dedekind rings, and will also have important consequences for the properness of the global-to-local map in the Galois cohomology of algebraic groups. It turns out that, for certain types of groups, this finiteness conjecture is closely related to finiteness properties of unramified cohomology. As a result, one of the objectives will be to establish the expected finiteness of unramified cohomology in degree three for surfaces and certain higher-dimensional varieties over global fields. Concerning abstract homomorphisms, the goal will be to develop a substantial generalization of methods introduced in previous work to resolve a longstanding conjecture of Borel and Tits for all absolutely almost simple groups over infinite fields of relative rank at least two. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.