Torsion Vanishing for Some Shimura Varieties
We generalize the torsion vanishing results of Caraiani-Scholze and Koshikawa. Our results apply to the cohomology of general Shimura varieties $(\mathbf{G},X)$ of PEL type $A$ or $C$, localized at a suitable maximal ideal $\mathfrak{m}$ in the spherical Hecke algebra at primes $p$ such that $\mathbf{G}_{\mathbb{Q}_{p}}$ 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 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. 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, allowing us to circumvent some of the circumlocutions taken by Cariani-Scholze. Our method showcases a very general strategy for proving such torsion vanishing results, and should bear even more fruit once the inputs are generalized. Motivated by this, we formulate an even more general torsion vanishing conjecture.