The Cheewhye Diagram

About five years ago, Cheewhye Chin gave a great year-long seminar on Langlands correspondence for GL_r 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:

. \qquad \quad \text{Geometric}
. \qquad \nearrow \qquad \qquad \searrow
\text{Spectral} \qquad \qquad \text{Algebraic}
. \qquad \nwarrow \qquad \qquad \swarrow
. \qquad \quad \text{Analytic}

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 \epsilon-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 \text{Gal}(\overline{F}/F) \to GL_r(\overline{\mathbb{Q}_\ell}) 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 \mathbb{F}_p(t)), \overline{F} 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 \overline{\mathbb{Q}_\ell}. I don’t have a really good reason for the use of \overline{\mathbb{Q}_\ell} instead of \mathbb{C}, 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 \overline{\mathbb{Q}_\ell} 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 GL_r(\mathbb{A}_F), which is a really huge group. \mathbb{A}_F 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 r \times r 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 GL_r(\mathbb{A}). 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 \mathbb{Z}, 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.]

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19 thoughts on “The Cheewhye Diagram

  1. one of those meaningless diagrams with arrows connecting different areas of mathematics

    You mean, “a functor”? And to think I’d thought highly of algebraic geometry…

  2. John,
    I think he was referring to diagrams (like this one) whose arrows do not refer to functors. As far as I know, there is no interesting “category of L-functions” so the process of finding the L-function of a Galois or automorphic representation is not a functor in any reasonable sense.

    One might argue that L-functions should be categorified, and that the arrows could then be promoted to functors, but I haven’t heard of any good candidate categories. The geometric Langlands program does not seem to involve this sort of analytic data at all.

  3. Even if there’s a distinction to be drawn between “meaningless” and “meaningful” diagrams, it’s a better mathematician than I who can come up with a meaningful diagram without a meaningless diagram to tell them there should be a functor there.

  4. I don’t think he was completely serious when he made that comment, and at the time, I was holding a book (which shall remain unnamed) open to a page containing a particularly egregious example.

  5. About why $\bar{\mathbf{Q}}_{\ell}$ is better than $\mathbf{C}$ to get the algebraic objects: the topology would impose maps into $GL(r,\mathbf{C})$ to have finite image. In particular the eigenvalues of Frobenius would all be roots of unity. But we know many interesting Galois representations with eigenvalues which are not roots of unity, so it’s important to be able to deal with them.
    Also, in the number field case (I don’t know much about modular forms in the function field case, but I suspect it is just the same), the analogue of representations with finite image are well-known to pose many problems on the analytic side because they are spectrally “incomplete”, which means in particular that they are hard to detect/isolate using the Trace Formula or other tools. [Here is a sample question to which this is related: how many Galois extensions $K/\mathbf{Q}$ are there with discriminant $|D|\leq X$ and Galois group $A_5$, asymptotically as a function of $X$?]

    One feature of the function field case which certainly helps explain why it’s better understood at the moment is that the “algebraic” side here is very close to geometric already: the Galois representation amounts to a sheaf on a curve with function field $F$.

  6. Indeed, I think it is Grothendieck who said that the absolute Galois group being profinite, its reasonnable representations should have profinite coefficients. It always seemed to me to be a natural motivation for p-adic integers.

  7. Scott C. wrote:

    One might argue that L-functions should be categorified, and that the arrows could then be promoted to functors, but I haven’t heard of any good candidate categories.

    I think L-functions should be categorified. “Of course,” you say, “you would say that, since you think everything should be categorified.” True, but in this case it’s very clear that an L-function is just some sort of decategorified “distillation” of an object – we take the L-function of some sort of object to extract information about it. It’s just a fancier version of how we take the cardinality of a finite set or the generating function of a species.

    However, I don’t yet know what are the best categories of gadgets that have L-functions. There are a bunch, right? So then the job is to see if they can be unified. Ideally, taking an L-function always consists of first turning your object into some object in a fixed category C and then taking the L-function of that.

    I’m a bit further along when it comes to zeta functions. There are lots of things that have zeta functions, but for a bunch, we first construct a set with an action of the group Z, and then take the zeta function of that Z-set.

    This does not cover all examples of things that have zeta functions, but that’s not surprising: people keep extending the term “zeta function” in undisciplined ways, so it would be amazing if all zeta functions fit into a single framework. It’s good enough if a lot do.

    Same for L-functions. Anyone willing to make a guess?

    Is there such a thing as the L-function of a motive, or do we need a motive together with some representation of something?

  8. Anyone willing to make a guess?

    The category of motive over Q is reasonnable. A motive over Q comes with a representation via its étale realization.

  9. There’s a lot of discussion about what “is” an L-function (and corresponding definitions…) A fairly common point of view is that the right notion has to be “some Langlands L-function attached to an automorphic representation of a global field”. This has the advantage of being well-defined, and the inconvenient that the definition requires a lot of material. One common feature of all such L-functions, which is widely seen as a crucial component of the definition, is the existence of an Euler product. Another one is some kind of functional equation. (Gamma factors, which people are also used to seeing, do not count as common feature since they don’t exist over finite fields).

    There is supposed to be such a thing as the L-function of a motive over a global field (because a motive should automatically come with a Galois representation from which an L-function can be built), but not all (in fact, probably very few, in some sense) L-functions come from motives.

    And that an L-function coming from a motive is a Langlands L-function is ultimately one of the biggest question in the Langlands program. (The Shimura-Taniyama-Weil theorem being just one example of this type of statement).

    Categorifying L-functions might be nice, but one has to keep in mind that it’s often for analytic reasons that L-functions of the type above are useful. They are really interesting in regions where the series or product expansions do not converge (e.g., for the Riemann Hypothesis), and analytic continuation of holomorphic functions is not a formal process. On the formal level, there are already categories around, coming from representation theory, including the hypothetical Langlands group. Over finite fields, one can work with geometric categories of sheaves, and consider the L-function only when it’s useful (all the more so that L-functions typically do not distinguish between direct sums and non-trivial extensions).

    (Note that there are also things called p-adic L-functions; they are much more algebraic in nature and have a very sophisticated categorical background which I don’t understand).

  10. Since Greg changed the subject, I’d like to add another voice to a subject I am sure you have already gone through before. I think certain people may want to steadfastly AVOID the Math Jobs Wiki: namely people who might be sitting on hiring committees. I think it is a great idea for applicants to share information to give all applicants the best possible position in the job market. But, unfortunately, some hiring committees make hires based only on “buzz” and not on genuine evaluations of the applications. For those committees, reading the Wiki is only going to exacerbate a problem all of us keenly feel (and ultimately suffer from).

  11. Perhaps I should have replied to some of the comments before the conversation got derailed. I guess I hadn’t really thought about the social obligations incurred by blog posting.

    Emmanuel (if you’re still reading this), there was one sentence in particular that really stuck out at me, which was your reference to the trace formula. I’ve mostly seen it invoked as some kind of miraculous oracle (the trace formula predicts X,Y,Z), or sometimes used explicitly amidst an impenetrable forest of orbital integrals, but I never understood the yoga behind it. Do you know a good place to learn about that?

  12. My mention of the trace formula concerned the following type of applications : one wants to show that a certain irreducible representation (say $\pi$) occurs as a component (direct summand) of some other representation $\rho$, and to do this one can try to write down a linear projection on the space of $\pi$. In the cases of interest here, the representations are those of an adélic group $G(\mathbf{A})$ and $\pi$ corresponds to some generalizations of modular forms; it is described uniquely by local information (Hecke eigenvalues, weight or Laplace eigenvalue), while $\rho$ is the regular representation on the space of $L^2$ functions on $G(\mathbf{A})$ invariant under $G(\mathbf{Q})$.

    One source of difficulty is that if one chooses the local components to define $\pi$ “at random”, there’s no chance whatsoever it will occur (discreetly) in $\rho$. But often one can construct a likely candidate $\pi$ (likely because of the Langlands program) by taking a component of $\pi’$ corresponding to some other group, for which there is a local correspondance with $G$, and use the local data for $\pi’$ to construct some local data describing $\pi$. Or one can use motives/Galois representations to build local data, for instance as can be done concretely in the case of elliptic curves.

    To try to see if/how $\rho$ contains $\pi$, the trace formula is a tool that gives some kind of formula for the “trace” of convolution operators on the space of $\rho$, i.e., one may want to think of it as the best approximation that can exists for a formula for the character of $\rho$, which does not exist in general (infinite dimensional representation, existence of continuous spectrum). This trace is (more or less) a sum over all irreducible components of $\rho$, weighted by coefficients for which there is some freedom. Then one may try to use these coefficients in such a way that they are zero except for the desired component $\pi$ (i.e., except for subrepresentations of $\rho$ having the right local components) ; then the trace formula proves the existence of $\pi$ in $\rho$, provided the other side of the formula can be shown to be non-zero. This other side (the “geometric” side) is built out of expressions involving conjugacy classes of $G(\mathbf{Q})$, typically, and the orbital integrals you mention in particular.

    Unfortunately, this simple picture is really hard to translate because of lots of (intrinsic) analytic difficulties. Even from the most optimistic viewpoint, there is clearly going to be serious issues if the local parameters of interest are inside the known continuous spectrum, because the coefficients in the trace formula will not be able to separate the desired discrete component from those of the continuous spectrum (this is what occurs when trying to count $S_5$ extensions of $\mathbf{Q}$).

    For references, D. Bump explains a very basic example of the principle (where the analytic difficulties are very slight) in one of his chapters of the book “An introduction to the Langlands program”, edited by J. Bernstein and S. Gelbart ; in the Clay Math Institute Conference of 2005, there is an extensive introduction by Arthur of the general trace formula, with all difficulties explained, step by step.

  13. Emmanuel,
    Thank you very much. That was the best explanation I’ve seen. I had heard someone describe the trace formula as a generalization of Frobenius reciprocity, but I was lacking some context until now.

    It sounds like the biggest problem when working on the spectral side is the existence of a continuous part of the spectrum in L^2. I suppose excising this part is the reason why people put so much effort into looking for good equivariant truncations and compactifications of quotient spaces. Is this more or less correct?

  14. I dont know much about truncations, and rather less about compactifications, but it certainly seems like the truncations in particular are very crucial. Arthur’s notes describe in detail the truncation on the spectral side, with the example of GL(3) taken as an illustration of the complications that arise. Part of the difficulty is that, starting from GL(3), the continuous spectrum mirrors in a way the structure theory of the group involved (i.e., all lower-rank groups embedded as parabolic subgroups contribute), and that leads to some fairly extensive combinatorics also.

  15. For some truly meaningless diagrams related to category theory, see the oeuvre of Rudolf Kaehr (excerpted at Log24.net on Feb. 16, 2008). Kaehr is a good source of particularly sophisticated meaningless diagrams. See, for instance, his paper “Double Cross Playing Diamonds,” which cites the 1998 paper “Categorification” of Baez and Dolan.

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