On the Langlands--Kottwitz--Scholze method

Abstract

Using recent advances in the integral canonical models of Shimura varieties using syntomic methods, we give an extension of the Langlands--Kottwitz--Scholze method from Scholze's paper on the Langlands--Kottwitz method for deformation spaces of $p$-divisible groups. Namely, we define analogues $\phi^{\mathcal{G},\mu}_{\tau,h}$ of the local test functions from op. cit. in full generality, show they satisfy reasonable harmonic-analytic properties, and give rise to trace formulae for the Galois-Hecke action on the cohomology of Shimura varieties of bad reduction. Additionally, we use these new local test functions and the work of Fargues--Scholze to state a more robust and unconditional version of the Scholze--Shin conjecture. Along the way we study the $(\mathcal{G},\mu)$-apertures of Drinfeld and Gardner--Madapusi, in particular showing that such $(\mathcal{G},\mu)$-apertures carry no more information than a Tannakian version of Fontaine--Laffaille theory, at least over nice bases, and give a very explicit description of the universal deformations of $(\mathcal{G},\mu)$-apertures.