Number Theory Seminar

We will discuss joint work with Si-Yingg Lee on generalizing the torsion vanishing results of Caraiani-Scholze and Koshikawa. Our results apply to the cohomology of general Shimura varieties (G,X) of PEL type A or C, localized at a suitably generic maximal ideal in the spherical Hecke algebra at primes p such that the local group at p is a group for which we know the Fargues-Scholze local Langlands correspondence is the semi-simplification of a suitably nice local Langlands correspondence. This is accomplished by combining Koshikawa's technique, the theory of geometric Eisenstein series over the Fargues-Fontaine curve, the work of Santos describing the structure of the fibers of the minimally and toroidally compactified Hodge-Tate period morphism for general PEL type Shimura varieties of type A or C, and ideas developed by Zhang on comparing Hecke correspondences on the moduli stack of G-bundles of the Fargues-Fontaine curve with the cohomology of Shimura varieties. In the process, we also establish a description of the generic part of the cohomology that bears resemblance to the work of Xiao-Zhu, but on the generic fiber. Moreover, we also construct a filtration on the compactly supported cohomology that differs from Manotovan's filtration in the case that the Shimura variety is non-compact. Our method showcases a very general paradigm; namely, that the behavior of the torsion cohomology localized at a semi-simple L-parameter under the action of the spectral Bernstein center constructed by Fargues and Scholze is related to the perversity of Hecke eigensheaves with Hecke eigenvalue given by that parameter. This allows us to formulate several new conjectures on the structure of the torsion cohomology, and, time permitting, we will discuss this.