These are expanded notes prepared for a talk in a learning seminar on Caraiani-Scholze's paper On the generic part of the cohomology of compact unitary Shimura varieties, Spring 2016 at Columbia. We summarize the major ingredients of the proof, explain the preservation of perversity under the Hodge-Tate period map and deduce the main theorems: 1) the existence of Galois representations associated to the torsion classes in betti cohomology of certain compact unitary Shimura varieties; 2) after localized at a maximal ideal of the Hecke algebra satisfying a genericity assumption, the -cohomology is concentrated in the middle degree and torsion-free.
Recall our set-up. Let be a PEL datum of type A:
The PEL datum models the first cohomology of a polarized abelian variety of dimension with endomorphism by . Let be the group of the automorphisms of (as a -module) that preserves up to a similitude factor. Let be the associated unitary Shimura variety, which is a moduli space of such abelian varieties with -level structure.
Assume we are in one of the following two extremal cases:
Assume , , are unramified outside a finite set of primes (and in the second case). Let be the unramified Hecke algebra. Let be a maximal ideal. The first main result constructs an associated torsion Galois representation.
The proof requires three major ingredients:
(1) Sug Woo Shin has constructed Galois representations attached to the system of (characteristic 0) Hecke eigenvalues appearing in the cohomology of the Shimura varieties and Igusa varieties (in the above two extremal cases), by stable trace formula. In particular, there exists a Galois representation associated to system of Hecke eigenvalues in .
(2) The construction of the (Hecke equivalent) Hodge-Tate period map from the infinite level Shimura variety to the flag variety (for any Shimura varieties of Hodge type), whose fibers are related to the Igusa varieties (for any Shimura varieties of PEL type). In particular,the main result of Chap. 4 shows that for any geometric point , the fiber above ,
(3) The perversity of (for any compact Shimura variety of PEL type), which has the following consequence:
Using these 3 ingredients, now we can finish the proof of Theorem 1.
Now let us come to the second main result, which asserts the "generic part" of the cohomology of our compact unitary Shimura varieties vanishes outside the middle degree.
Generic principal series are mapped to zero under the Jacquet-Langlands correspondence to any group that is not quasi-split. As we saw last time, using this one deduces that only contributes to the "most ordinary part" of the Igusa variety:
Theorem 3 now follows easily from the genericity and the three main ingredients.
Finally, let us explain the proof of the perversity result. Let be any compact Shimura variety of PEL type with hyperspecial level at .
To motivate, recall two useful results for perverse sheaves in algebraic geometry:
By the minimality, we know that has support on the union So it has support in a closed subset of dimension equal to . The result then would follow if is "perverse". Why should it be? The intuition is that is an affine and partially proper (i.e., satisfies valuative criterion in the category of adic spaces). If we were working with schemes, this would mean is affine and proper, hence finite, and finite morphisms preserve perversity. Of course all the beauty of lies in its very non finite-type behavior, so we cannot literally say this. On the other hand, because of Ingredient (2), we only need to show the sheaf is perverse when restricted on an affinoid etale neighborhood of . Then we can pass to the special fiber (by the perfectoidness), where indeed becomes a finite map between affine schemes of finite type over the residual field.
The other issue is that admits the action of and is infinite dimensional. So it can only be perverse (or just constructible) after taking -invariants for open compact. So the idea for proving the perversity Theorem 2 is then to pass to finite levels and special fibers. Let us be more precise.
We choose such sufficiently small for each so that 's shrink to 1. Then we know that the fiber at can be computed using the cohomology of the special fiber using the right upper corner of the following diagram, When shrinking , we obtain Here is the specialization of . Now we can compute using the left lower corner as well, namely, first specialize, then apply (which now becomes a finite map between schemes of finite type). Because the general fact that specialization (aka, nearby cycle) preserves perversity, we know that each individual term in the direct limit is concentrated in degree . Hence the direct limit itself is also concentrated in degree . ¡õ