Number/Representation Theory

(This talk will take place in KP 113)

Hecke operators play a fundamental role in understanding the arithmetic properties of modular and automorphic forms. Since the advent of the original Eichler-Shimura relation, it has been clear that the mod-$p$ behavior of Hecke correspondences is crucial for such applications. However, one could argue a truly robust theory of such correspondences yielding convenient access to their mod-$p$ reductions has so far been elusive, especially when dealing with higher rank groups.

In this talk, I will present a new approach to these matters, using recent advances in $p$-adic geometry and $p$-adic cohomology, building on work of Drinfeld and Bhatt-Lurie, and combining them with a tool familiar to the geometric Langlands and representation theory community: the Vinberg monoid. In particular, this approach yields direct access to geometric incarnations of the 'standard' basis elements of the spherical Hecke algebra.

For another application, this approach also gives the first general construction of Rapoport–Zink spaces associated with exceptional groups.