Scholze on Torsion 0

This will be the first zeroth of a series of posts talking about Scholze’s recent preprint, available here. This is mathematics which will, no question, have more impact in number theory than any recent paper I can think of. The basic intent of this post is to commit to future posts in which I will discuss the details. I should remark that Scholze’s writing is pretty clear, so these posts will mainly be for my own benefit rather than yours.

Here are some of the specific points that I might cover:

Basics: The Hodge-Tate Period map, Perfectoid spaces, etc. To be honest, I will probably skip the details here to begin with, and discuss them only at points where they become fundamental for understanding.

Theorem IV.3.1: The action of Hecke on the completed cohomology groups \(\widetilde{H}^i(\mathbf{Z}/p^n \mathbf{Z})\) for Shimura varieties is detected by the action of Hecke on classical cuspidal automorphic forms. Although it may end up being no easier to consider, this result is already intersting in some quite degenerate cases. For example, this is new even for \(X = U(2,1)/\mathbf{Q}\) and \(i = 1\) (Gee and Emerton’s results, for example, are contingent on the relevant Galois representations being three dimensional — now one knows that they are!). A very similar example is the case of a compact inner form of \(U(2,1)\) (so called Rogawski lattices) or, more generally, the simple Shimura variety of Kottwitz-Harris-Taylor type. Can one show in those cases that \(\widetilde{H}^i\) vanishes outside degree zero and outside the middle dimension? A weaker question: can one compute the completed cohomology in degree one? Compare with the work of Pascal Boyer.

Local Global Compatibility: Suppose one is in the ordinary case. Then the HLTT approach (via congruences, discussed previously on this blog (here, here, and here) should allow one to establish some cases of local-global compatibility. At ramified primes \(\ell \ne p\), the HLTT approach should also work, especially if one is also willing to assume that the residual representation is absolutely irreducible (using base change arguments). What can one do in the torsion case?

The Nilpotent Ideal: Scholze ultimately constructs Galois representations over \(\mathbf{T}/I\) for an ideal \(I\) such that \(I^m = 0\). The necessity of this ideal arises from a spectral sequence argument. (The parameter \(m\) only depends on the degree of the field and \(n\).) The Calegari-Geraghty modularity lifting argument (in the minimal case) can still be made to apply even with the presence of this ideal if one is in the minimal case, but not in the non-minimal case which will require \(m = 1\) (the Taylor Ihara’s avoidance trick requires more precise control than the minimal case). Are there any circumstances (extra assumptions, etc.) in which one can prove that \(m =1\)?