About five years ago, Cheewhye Chin gave a great year-long seminar on Langlands correspondence for over function fields, building up to a description of Lafforgue’s proof of the theorem. In the beginning, he drew a diagram that captured the general architecture of the proof, and I liked it so much that I stole it for a talk I gave at Talbot in 2005. It seemed to get a good reception, and Mark Behrens pointed out that the Eichler-Shimura correspondence also fits into the picture with minimal alteration.
If we remove all of the explanatory text, the diagram looks like this:
I was a bit hesitant to draw this, because my advisor once told me, “If you ever find yourself drawing one of those meaningless diagrams with arrows connecting different areas of mathematics, it’s a good sign that you’re going senile.” Anyway, I’ll explain roughly how it works.
Langlands correspondence is a “bridge between two worlds,” or more specifically, an assertion of a bijection between certain Galois representations (algebraic world) and certain automorphic representations (spectral world). To build this bridge, we pass through the analytic world of L-functions and -factors, and the geometric world of moduli stacks and sheaves. Underlying this is the big open question of naturality, i.e., what happens as we vary our fields and groups, and is there some functor that produces these bijections? The geometric Langlands program was initially an attack on this question through categorification (although that word wasn’t in use at the time), but it seems to have acquired its own life force.
I’ll start by introducing the objects of interest, then the arrows. Some seem to be more manifestly natural than others; L-functions never appeared very functorial to me, but I imagine there is some engine behind them.
The objects in the algebraic world are representations that satisfy some conditions (irreducibility, continuity, finite-order determinant, almost everywhere unramified). F is the field of “meromorphic” functions on a smooth curve over a finite field (i.e., a finite extension of ), is a separable closure of it, and Galois representations are just homomorphisms from the group of automorphisms of the separable closure that fix F to invertible matrices with entries in . I don’t have a really good reason for the use of instead of , except that it yields the right answer. Totally disconnected fields give more continuous representations than the complex numbers, and this is in line with some philosophy involving the motivic Galois group that I never understood. The representations are characterized by their Frobenius eigenvalues, which form an unordered r-tuple of elements of for each unramified point of the curve, and Chebotarev density makes this characterization unique.
The objects in the analytic world are L-functions and epsilon factors. In the function field case, the L-functions are formal power series, which can be shown to be rational functions that satisfy functional equations with epsilon factors as correction terms. I’d be interested to hear if there is an easy way to look at them, because they usually give me a headache. For number fields, L-functions are just complex-analytic functions, and showing that they are well-behaved is often very difficult. Presumably, this difficulty has to do with geometry over the field with one element being a nasty, degenerate thing that possibly doesn’t exist.
The objects in the spectral world are certain infinite dimensional complex representations of , which is a really huge group. is the ring of adèles, defined as a restricted product of the completions of F with respect to all possible norms, and we are looking at representations of the group of invertible matrices with entries in this ring that have strong invariance properties. They are characterized by their Satake parameters, also known as Hecke eigenvalues, which are given by an unordered r-tuple of complex numbers assigned to almost all points of the curve. This characterization is unique, by a strong multiplicity one theorem due to Jacquet, Piatetskii-Shapiro, and Shalika.
The objects in the geometric world are moduli stacks of shtukas. Shtukas are vector bundles on a curve with some extra structure coming from Frobenius and Hecke actions. Vector bundles tend to be parametrized by rather badly behaved moduli spaces, and these are no exception, since they can exhibit arbitrarily bad singularities. The idea behind using these spaces is that they live over the curve, and admit actions of . Pushing the constant sheaf down to the curve and pulling back to the generic point gives an étale sheaf whose cohomology has commuting actions of both the Galois group and the adèlic group, and this produces a correspondence on representations.
Now I will describe the arrows. The diagram is a bit of a stretch here, because the actual induction in Lafforgue’s proof is somewhat more subtle than just running around a circle.
Given a moduli space, we get a Galois representation by taking l-adic étale cohomology. We obtain information about the representation by the Grothendieck(-Lefschetz-Verdier) trace formula and purity theorems.
From a Galois representation, we get an Artin L-function by forming an Euler product over points of the curve, with local factors determined by the action of Frobenius. Since our Galois representations come from geometry, we can say that the L-functions satisfy functional equations with known epsilon factors by Deligne’s Weil II and Laumon’s sheafy Fourier transform work.
We can also get nice L-functions from automorphic representations by a similar Euler product formalism. The arrow on the bottom left requires a converse theorem (due to Cogdell and Piatetskii-Shapiro), asserting that a sufficiently well-behaved L-function comes from an automorphic representation.
The arrow on the top left is Lafforgue’s arrow. The moduli stacks of shtukas encode all of the necessary representations, but they are jumbled together, so he made a rather complicated geometric construction, involving chopping up the stacks into stratifications by Harder-Narasimhan polygon and degree, quotienting by a free action of , and compactifying. The middle l-adic cohomology of the resulting spaces admits a filtration whose quotients have a commuting Galois and Hecke action. Taking a sum and semisimplifying yielded a correspondence (after several hundred pages of work – someone told me that Lafforgue proved it “by staring at the Arthur-Selberg trace formula for six years”).
[Does anyone know how to make a diagram in wordpress? My array got rejected, so I made a hack with periods.]