Groups with good reduction and buildings

in MATH 126

Over the last few years, the analysis of algebraic groups with good reduction has come to the forefront in the emerging arithmetic theory of algebraic groups over higher-dimensional fields. Current efforts are focused on finiteness conjectures for forms of reductive algebraic groups with good reduction that share some similarities with the famous Shafarevich Conjecture in the study of abelian varieties. Most results on these conjectures obtained so far have ultimately relied on finiteness properties of appropriate unramified cohomology groups. However, quite recently, methods based on building-theoretic techniques have emerged as a promising alternative approach. I will showcase some of these developments by sketching a new proof of a theorem of Raghunathan-Ramanathan concerning torsors over the affine line.