This is a sequel to this post, although as it turns out we still won’t actually get to anything substantial — or indeed anything beyond an introduction — in this post.
Let me begin with some overview. Suppose that is a locally symmetric space, where is a semi-simple group which does not admit discrete series. To be concrete, suppose that and that is an congruence subgroup of level in the Bianchi group — recall here that is an imaginary quadratic field. Since days of yore (Langlands, Clozel, Fontaine-Mazur, etc.), everyone has expected that there should exist a bijection between the following objects:
- Regular algebraic cuspidal autormophic representations for of level and weight zero (= the same infinitesimal character as the trivial representation).
- Cuspidal (= non-boundary) cohomology classes which are eigenforms for the ring of Hecke operators .
- Weakly compatible families of two-dimensional Galois representations of which are irreducible of level and Hodge-Tate weight .
- Irreducible semi-stable -adic Galois representations of weight and level (determined in the usual way) dividing .
- Abelian Varieties of -type over which don’t have CM by .
The equivalence between and follows from Matsushima/Franke. The (conjectural) relationship between and is the problem of reciprocity in the Langlands programme. It consists of two directions; existence (i.e. constructing Galois representations from automorphic forms) and modularity (i.e. showing nice Galois representations are automorphic). Both of these directions are difficult. The passage from is easy, whereas is basically the Fontaine-Mazur conjecture (not so easy). Actually even that last statement is not quite correct: if one knows that is pure of weight one with non-negative Hodge-Tate weights and arises up to twist in the cohomology of some smooth proper algebraic variety, then it should actually arise in cohomology without having to twist and hence come from an Abelian variety; proving this, however, is probably hard (as in Standard conjectures hard). Note that follows, in the case of classical modular forms, from a geometric construction of Shimura, but that idea doesn’t work here, and in fact this arrow is completely open and we shall say no more about it. In the particular case of imaginary quadratic fields, Harris, Soudry, and Taylor showed in 1992 that under certain favourable conditions. This case is slightly exceptional in this regard, since there exist functorial transfers of such to which do contribute to the -cohomology of Shimura varieties and hence can be directly related to the coherent cohomology of Shimura varieties (although not directly to Betti cohomology, because the corresponding weights are not regular.) As readers of this blog know, only very recently, Harris-Lan-Taylor-Thorne established the same result for over a CM field. (Small lie: not all the desired properties of the corresponding Galois representations, — i.e. local-global compatibility — have been established. I think Ila Varma is working this out for her thesis.)
It was observed early on that the cohomology groups of are not, in general, torsion free. So what then does a torsion class represent? Computations by Grunewald, Helling, and Mennicke in an 1978 paper suggested that torsion classes (specifically, two-torsion with Hecke eigenvalues in ) in these groups should be associated to Galois representations of the field . Apparently there are even some unpublished notes from Grunewald in 1972 doing similar things, although I have only ever heard rumors of their existence (to be fair, I heard those rumours from Grunewald, so they’re probably pretty reliable). So very early on there were hints that a further story was going on between torsion classes and Galois representations that wasn’t immediately related to the (conjectural) story coming from automorphic forms. The first general and precise conjecture along these lines were formulated by Ash in his 1992 Duke paper, with further refinements by Ash–Sinnott, and Ash–Doud–Pollack. Unlike the previous speculations of Grunewald, these conjectures were precisely formulated and falsifiable, and in the spirit of Serre’s original conjecture. Moreover, Herzig did actually come along and falsify them, by finding a more natural prediction for the set of possible Serre weights which turned out to be different from the formulation of Ash et. al., and Herzig’s formulation subsequently proved (numerically!) to give the right answer in these cases (Edit: see comments, this is not quite correct). At any rate, for quite some time, we have expected that mod-p torsion classes give rise to Galois representations, and following the conjectures of Serre, Ash, and others, one can be quite precise about exactly the local properties the corresponding Galois representations should have at primes of bad reduction. What is perhaps more recent is the idea that, especially for groups with no discrete series, that torsion is not merely a techincal nuisance, but rather is the source of “most” of the interesting Galois representations. In particular,
- The phenomenon whereby Galois representations coming from the countably many classical automorophic forms are dense in a suitable universal deformation ring (Böckle, Gouvêa-Mazur, Chenevier) will be totally false when does not have discrete series. On the other hand, the representations coming from torsion should cut out all of the universal deformation ring.
- That in order to answer the most pressing questions concerning reciprocity (even in characteristic zero!) one needs to understand torsion classes.
For one take on this, I might suggest reading Section 1.1, Speculations on
p-adic functorality, of this paper. For another interesting perspective, you should also read this, as well as the accompanying review.
So let us assume then that studying torsion representations and associating them to Galois representations is an Important Goal. How do we construct them? An observation also going back a long way (I believe to Harder??) is the following. Even though may not admit discrete series, there may exist a group containing a parabolic with as a Levi. If does admit discrete series, then there will exist a Shimura variety whose Borel-Serre compactification will have at least one boundary component which is a torus bundle over , and as a result one obtains a map (with some mixing of degrees) . Now one is theoretically in better shape, because this map should be compatible (in some sense) with Hecke operators, and the latter group has a chance to admit comparisons to étale cohomology groups which do come with Galois representations. There are three immediate problems:
- The compactification will be singular, except in the case of modular curves.
- Given a long exact sequence for (co)homology relative to a boundary divisor, it’s not clear whether the cohomology in the boundary ends up in or .
- Just because a class has an interesting Hecke eigensystem doesn’t mean that that étale cohomology sees an interesting Galois representation.
The third issue is a genuine problem. If one has a Hecke eigenclass in the etale cohomology of a Shimura variety, even in characteristic zero, then all one can deduce is that the corresponding Galois representation is annihilated by the corresponding characteristic polynomial of Frobenius. But this is not always enough to get the correct Galois representation! It’s probably worthwhile to consider two examples.
First, a somewhat degenerate example. Let , , , and . Now is (for some level structure) a finite set of points, and is a modular curve with cusps. The boundary map realizes as a set of cusps in . The Hecke operators act on cusps by the degree map, i.e. . This coincides with the action of Hecke operators on . So, in the etale cohomology group we have a Hecke eigenclass which we imagine (looking at the Hecke operators) to be associated to the Galois representation where is the cyclotomic character. Yet is one dimensional, and so we only see half of the Galois representation, namely, the trivial character. Now as it turns out, the other piece of the Galois representation can be seen in , which is now mixed because is not projective (so the cuspidal part has motivic weight one, and this other piece of the Eisenstein series has weight two). So even in this trivial case, we see that a Hecke eigenclass in etale cohomology may have a less interesting Galois representation than the Hecke eigenvalues might suggest. From the Eicher-Shimura relation, we do get that the (trivial) Galois representation which does occur is annihilated by the characteristic polynomial of Frobenius , indeed it is annihilated by the first factor.
Second, let , , , and . Here is taken to split over . The cohomology of (a torus bundle over) the Bianchi group maps into the cohomology of . The characteristic polynomials of the Hecke operators are, morally, the following. If is the (conjectural) Galois representation associated to an eigenclass on the Bianchi group, then is a six dimensional (reducible) representation which is a direct sum for a four dimensional representation and a Grossencharacter and its conjugate (which are related to the central character of the original form and its conjugate). Now the characteristic polynomials of Frobenius on this Galois representation are, by Eichler–Shimura, the characteristic polynomials of Hecke on the image of this cohomology class in the cohomology . Without assuming one has , one can phrase the above purely in terms of Satake parameters, but this way of saying it makes clearer what is going on, even though we don’t know yet that actually exists. If one could find the Galois representation (and in particular ) inside the etale cohomology of one would (almost) be done, but instead, the classes which actually turn up in etale cohomology in these degrees are the reducible terms in corresponding to the Grossencharacters rather than to the interesting representation we are looking for. So as above, even in characteristic zero, one has the interesting Hecke eigenclass, but not the Galois representation.
These examples suggest that to understand what is going on we first need to get a better understanding of Shimura varieties. Most of the recent history of understanding Shimura varieties (and the Galois representations associated to automorphic forms) has concentrated on the cohomology arising from cuspidal automorphic representations. In this classical setting, the automorphic representations have a classical avatar as global sections of certain coherent bundles on . (For example, classical modular forms of weight are global sections of the line bundle .) If we want to restrict to cusp forms, we can also take the corresponding extension of these sheaves to minimal (or toroidal, doesn’t matter) compactifications which vanish appropriately at the boundary. If we denote these automorphic bundles by , then another way of saying this is that the action of Hecke operators on
is now understood if is, for example, a Shimura variety of unitary type over a totally real field. Even getting this far is a somewhat monumental task that required, amongst other things, Ngo’s work on the Fundamental Lemma, work of Kottwitz, Clozel, some large fraction of Jussieu, the work of Shin, and many more. In fact, as far as local-global compatibility goes, the ink is barely dry on the most recent work. Now we can at least state, in vague terms, the following:
Theorem [Scholze, IV.3.1]: For (many) Shimura varieties , the action of on torsion classes in Betti cohomology factors through the action on coherent cusp forms in characteristic zero.
Two examples: If is the modular curve, then this says that the action of Hecke operators on can be realized by the action on classical modular cuspidal eigenforms modulo powers of . Given how we think about modular forms, this is almost tautological, because, by Eichler-Shimura, we can pass between cohomology classes and classical modular forms (in this case, we can even do this via the Hodge decomposition of ). However, there is a little wrinkle: we do see Eisenstein classes in Betti cohomology, and this theorem says that we can realize these as coming from cusp forms, so this result also implies that there exist cusp forms which are congruent to Eisenstein classes modulo . Since we are ultimately interested in classes coming from the boundary of some compactification, we don’t want to ignore this case. Still, it’s not so difficult to prove.
If comes from (so it is a arithmetic complex hyperbolic manifold of real dimension , also known as a Picard modular surface), then we can look at the group . The characteristic zero classes here are known to correspond to endoscopic automorphic representations (and thus to not exist in the co-compact case) and are understood. However, unlike in the modular curve case, we no longer know that this group is torsion free, and in general, it may not be. So, a priori, all we know about the torsion classes and their Hecke operators is that there exists a Galois representation which is annihilated by the characteristic polynomial of , using Eichler-Shimura. These polynomials are all of fixed degree (three in this case), but that doesn’t give any lower bound on the dimension of this representation. This is an even more stark example of the well known phenomenon that Eichler-Shimura is pretty much useless for constructing Galois representations outside the case of dimension two where knowing both the trace and determinant tells you a lot. For example, suppose you have an irreducible representation of a finite group in characteristic zero such that all the elements of have a minimal polynomial of degree at most : then you can’t a priori bound the dimension of ! As an example, the extra-special -group of order has a representation of dimension all of whose elements have images satisfying the degree two polynomials or . So, before Scholze, we could not say anything about the dimensions of mod- Galois representations arising from torsion in the first homology of . However, using Scholze, we can now deduce that any such representation comes from a classical cusp form, and hence must (in this case) have dimension three!