Linear algebraic groups with good reduction and the genus problem

We will first discuss the notion of good reduction with respect to a discrete valuation for a reductive linear algebraic group, and then formulate a finiteness conjecture for the forms having good reduction at a divisorial set of places of a finitely generated field. This conjecture provides a uniform approach to several problems including the genus problem for division algebras and algebraic groups, the properness of the global-to-local map in Galois cohomology and the analysis of weakly commensurable Zariski-dense subgroups — the latter also has some geometric applications. We will present some of the available results on the finiteness conjecture — see the survey article A.R., I. Rapinchuk, “Linear algebraic groups with good reduction”, Res. Math. Sci. 7(2020) for more information.