Shimura varieties at level and Galois representations

Abstract

We show that the compactly supported cohomology of certain $\text{U}(n,n)$- or $\text{Sp}(2n)$-Shimura varieties with $\unicode[STIX]{x1D6E4}_{1}(p^{\infty })$-level vanishes above the middle degree. The only assumption is that we work over a CM field $F$ in which the prime $p$ splits completely. We also give an application to Galois representations for torsion in the cohomology of the locally symmetric spaces for $\text{GL}_{n}/F$. More precisely, we use the vanishing result for Shimura varieties to eliminate the nilpotent ideal in the construction of these Galois representations. This strengthens recent results of Scholze [On torsion in the cohomology of locally symmetric varieties, Ann. of Math. (2) 182 (2015), 945–1066; MR 3418533] and Newton–Thorne [Torsion Galois representations over CM fields and Hecke algebras in the derived category, Forum Math. Sigma 4 (2016), e21; MR 3528275].

Patching and the Completed Homology of Locally Symmetric Spaces

“…It also follows from Scholze's work (again with the assumption that F is CM or totally real) that there is a lifting of ρ m valued in T S (U p ) m /I for some nilpotent ideal I ⊂ T S (U p ) m , and in fact we may assume that I 4 = 0 by [48, Theorem 1.3]. Moreover, the nilpotent ideal has been eliminated entirely when F is CM and p splits completely in F [22]. Definition 3.3.5.…”

Derived Hecke Algebra and Cohomology of Arithmetic Groups

“…In particular, the paper [1,Theorem 3.11] of Allen, Calegari, Caraiani, Gee, Helm, Le Hung, Newton, Scholze, Taylor and Thorne establishes, among many other results, the key local-global compatibility needed for Taylor-Wiles patching. Moreover, the paper [11] of Caraiani, Gulotta, Hsu, Johansson, Mocz, Reinecke, and Shih eliminates the use of nilpotent ideals in the Galois representations originally constructed by Scholze. While these do not precisely match with the inputs needed for our setup of the argument, they appear to address the key issues, and so to me it seems very likely that one could produce an unconditional version of our analysis in the near future.…”

“…the Q-linear dual L ˚of L carries H ˚pY pKq, Qq χ to itself. In particular, this means that (11) There is a natural graded action of ^˚L ˚on H ˚pY pKq, Qq χ .…”

A quotient of the Lubin–Tate tower II

“…A natural question is whether the analogue of Theorem 5.3.1 holds for other maximal (standard) parabolics Q = P. By Theorem A.1.5, the quotient M Q(K ) is not a perfectoid space, and so the method for proving Theorem A.1.5 breaks down. One could ask whether the vanishing theorem could still be salvaged by geometric methods (such as in [5], where a vanishing result is proven in a situation where the space in question is not perfectoid), but we currently see no way of doing this (in particular, we see no way of adapting the method of [5]). (3) It is also natural to ask about vanishing below the middle degree, but here things seem to be much more unclear.…”

“…C.J. also wishes to thank Daniel Gulotta, Chi-Yun Hsu, Lucia Mocz, Emanuel Reinecke, Sheng-Chi Shih and especially Ana Caraiani for all discussions relating to [5], which have had a large influence on this paper. C.J.…”