Simple algebraic groups with the same maximal tori, weakly commensurable Zariski-dense subgroups, and good reduction

Abstract

We provide a new characterization of the existence of good reduction for an absolutely almost simple algebraic group with respect to a discrete valuation of the base field that is formulated in terms of the existence of maximal tori with special properties. This characterization, in particular, shows that the Finiteness Conjecture for forms of an absolutely almost simple algebraic group over a finitely generated field that have good reduction at a divisorial set of places of the field would imply the finiteness of the genus of the group at hand. It also leads to a new phenomenon that we refer to as "killing of the genus by a purely transcendental extension." Yet another application deals with the investigation of "eigenvalue rigidity" of Zariski-dense subgroups, which in turn is related to the analysis of length-commensurable Riemann surfaces and general locally symmetric spaces. Finally, we analyze the genus problem for simple algebraic groups of type $\textsf{F}_4$.

Locations

• arXiv (Cornell University) - View - PDF