The Hasse-Weil zeta functions of orthogonal Shimura varieties
Initiated by Langlands, the problem of comparing the Hasse-Weil zeta functions of Shimura varieties with automorphic L-functions has received continual study. The strategy proposed by Langlands, later made more precise by Kottwitz, is to compare the Grothendieck-Lefschetz trace formula for Shimura varieties with the trace formula for automorphic forms. Recently the program has been extended to some Shimura varieties not treated before. In the particular case of orthogonal Shimura varieties, we discuss the proof of Kottwitz's conjectural comparison (between the intersection cohomology of their minimal compactifications and the stable trace formulas). Key ingredients include point counting on these Shimura varieties, Morel's theorem on intersection cohomology, and explicit computation in representation theory mostly for real Lie groups.