On torsion in the cohomology of locally symmetric varieties

By Peter Scholze. 2013

294 citations recorded by the source corpus.

The main result of this paper is the existence of Galois representations associated with the mod $p$ (or mod $p^m$) cohomology of the locally symmetric spaces for $\GL_n$ over a totally real or CM field, proving conjectures of Ash and others. Following an old suggestion of Clozel, recently realized by Harris-Lan-Taylor-Thorne for characteristic 0 cohomology classes, one realizes the cohomology of the locally symmetric spaces for $\GL_n$ as a boundary contribution of the cohomology of symplectic or unitary Shimura varieties, so that the key problem is to understand torsion in the cohomology of Shimura varieties. Thus, we prove new results on the $p$-adic geometry of Shimura varieties (of Hodge type). Namely, the Shimura varieties become perfectoid when passing to the inverse limit over all levels at $p$, and a new period map towards the flag variety exists on them, called the Hodge-Tate period map. It is roughly analogous to the embedding of the hermitian symmetric domain (which is roughly the inverse limit over all levels of the complex points of the Shimura variety) into its compact dual. The Hodge-Tate period map has several favorable properties, the most important being that it commutes with the Hecke operators away from $p$ (for the trivial action of these Hecke operators on the flag variety), and that automorphic vector bundles come via pullback from the flag variety.

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Citations in the corpus

On the p-adic cohomology of the Lubin-Tate tower (2015, methodology): “As the Shimura curves under consideration are not of Hodge type, one cannot formally use the construction of a global Hodge-Tate period map in [28] to get one in this setup.”
Modularity lifting beyond the Taylor–Wiles method (2012, background): “ m. However, for historical reasons, we continue to work in the present setting, with the understanding that the real diﬃculty in Conjecture B 88 FRANK CALEGARI AND DAVID GERAGHTY lies (after Scholze [Sch15]) with proving the non-vanishing of (co-)homology groups in the required range after localiza”
Strata Hasse invariants, Hecke algebras and Galois representations (2015, result): “lar case [15]. Scholze achieved a breakthrough in the Betti case, by associating pseudo-representations to the cohomology of the locally symmetric spaces of GL(n) over a CMﬁeld, with mod pncoeﬃcients [94]. By contrast, much less was known prior to this work concerning the coherent cohomology of Shim”
A torsion Jacquet--Langlands correspondence (2012, background): “On the other hand, Scholze’s results [84] mentioned previously constructs Galois representations ρm as above without showing that they the appropriate Galois representation has the requisite local behavior at v|p.”
Geometrization of the local Langlands correspondence: an overview (2016, background): “ model. We note Sthis integral model, Sb η = ShG(Z p). Recall the following theorem of Scholze (this is a rewriting, Scholze theorem is more powerful but this is the only thing we need). Theorem 7.1 ([28]). The pro-´etale sheaf lim ←− Kp Sh⋄ Kp is representable by an E-perfectoid space Sh∞. Moreover”