3:00–4:00 pm Eckhart Hall Room 206
Higher Traces in Arithmetic Geometry
A central theme in arithmetic geometry is the passage from geometric invariants to arithmetic information by taking the trace of Frobenius. I will describe a higher categorical version of this procedure with a particular focus on applications to the Langlands correspondence over function fields. In the global unramified case, we obtain that the space of automorphic forms is the categorical trace (aka Hochschild homology) of Frobenius acting on (an appropriate version of) the automorphic category. This, in particular, leads to an enhancement of V. Lafforgue’s spectral decomposition of the space of automorphic forms. In the local case, we obtain an enhancement of the local Langlands conjecture of Fargues-Scholze in the case of function fields. This is based on joint works with Arinkin, Gaitsgory, Kazhdan, Raskin, and Varshavsky.