New Results In Modularity, Part I

I usually refrain from talking directly about my papers, and this reticence stems from wishing to avoid any appearance of tooting my own horn. On the other hand, nobody else seems to be talking about them either. Moreover, I have been involved recently in a number of collaborations with multiple authors, thus sufficiently diluting my own contribution enough to the point where I am now happy to talk about them.

The first such paper I want to discuss has 9(!) co-authors, namely Patrick Allen, Ana Caraiani, Toby Gee, David Helm, Bao Le Hung, James Newton, Peter Scholze, Richard Taylor, and Jack Thorne. The reason for such a large collaboration is a story of itself which I will explain at the end of the second post. But for now, you can think of it as a polymath project, except done in a style more suited to algebraic number theorists (by invitation only).

In this first post, I will start by giving a brief introduction to the problem. Then I will state one of the main theorems and give some (I hope) interesting consequences. In the next post, I will be a little bit more precise about the details, and explain more precisely what the new ingredients are.

Like all talks in the arithmetic of the Langlands program, we start with:

The Triangle:

Let F be a number field, let p be a prime, and let S be a finite set of places containing all the infinite places and all the primes above p. Let G_S denote the absolute Galois group of the maximal extension of F unramified outside S. In many talks in the Langlands program, one encounters the triangle, which is a conjectural correspondence between the following three objects:

• A: Irreducible pure motives M/F (with coefficients) of dimension n.
• B: Continuous irreducible n-dimensional p-adic representations of G_S (for some S) which are de Rham at the places above p.
• C: Cuspidal algebraic automorphic representations \(\pi\) of \(\mathrm{GL}(n)/F.\)

In general, one would like to construct a map between any two of these objects, leading to six possible (conjectural) maps, which we can describe as follows:

• A->B: This is really the only map we understand, namely, etale cohomology. (I’m being deliberately vague here about what a motive actually is, but whatever.)
• B->A: This is the Fontaine-Mazur conjecture, and maybe some parts of the standard conjectures as well, depending on exactly what a motive is.
• B->C: This is “modularity.”
• C->B: This is the existence of Galois representations associated to automorphic forms.
• A->C: We really think of this as A->B->C and also call this modularity.
• C->A: Again, this is a souped up version of C->B. But note, we still don’t understand how to do this even in cases where C->B is very well understood. For example, suppose that \(\pi\) comes from a Hilbert modular form with integer coefficients of trivial level over a totally real field F of even degree. We certainly have an associated compatible family of Galois representations, and we even know that its symmetric square is geometric. But it should come from an elliptic curve, and we don’t know how to prove this. The general problem is still completely open (think Maass forms). On the other hand, often by looking in the cohomology of Shimura varieties, one proves C->A and uses this to deduce that C->B.

This triangle is also sometimes known as “reciprocity.” The other central tenet of the Langlands program, namely functoriality, also has implications for this diagram. Namely, there are natural operations which one can easily do in case B which should then have analogs in C which are very mysterious.

Weight Zero: For all future discussions, I want to specialize to the case of “weight zero.” On the motivic/Galois side, this corresponds to asking that the representations are regular and which Hodge-Tate weights which are distinct and consecutive, namely, [0,1,2,…,n-1]. The hypotheses that the weights are distinct is a restrictive but crucial one — already the case when F = Q and the Hodge-Tate weights are [0,0] is still very much open (specifically, the case of even icosahedral representations). On the automorphic side, the weight zero assumption corresponds to demanding that the \(\pi\) in question contribute to the cohomology of the associated locally symmetric space with constant coefficients.

For example, if n=2, then we are precisely looking at abelian varieties of GL(2) type over F (e.g. elliptic curves). This is an interesting case! We know they are modular if F is Q, or even a quadratic extension of Q. More generally, we know that if F is totally real, then such representations are at least potentially modular, that is, their restriction to some finite extension \(F’/F\) is modular. This is often good enough for many purposes. For example, it is enough to prove many cases of (some version of) B->A. In this case, we have quite complete results, although still short of the optimal conjectures, especially in the case when the residual representation is reducible.

There are many other modularity lifting results generalizing those for n=2, but they really involve Galois representations whose images have extra symmetry properties. In particular, they are either restricted to representations which preserve (up to scalar) some orthogonal or symplectic form, or they remain unchanged if one conjugates the representation by an outer automorphism of G_F (for example when \(F/F^+\) is CM and one conjugates by complex conjugation). There were basically no unconditional results which applied either in the situation that n > 2 or that F was not completely real, and the representation did not otherwise have some restrictive condition on the global image. Our first main theorem is to prove such an unconditional result. Here is such a theorem (specialized to weight zero):

Theorem [ACCGHLNSTT]: Let F be either a CM or totally real number field, and p a prime which is unramified in F. Let

\(\rho: G_S \rightarrow \mathrm{GL}_n(\overline{\mathbf{Q}_p})\)

be a continuous irreducible representation which is crystalline at v|p with Hodge-Tate weights [0,1,..,n-1]. Suppose that

1. The residual representation \(\overline{\rho}\) has suitably big image.
2. The residual representation is “modular” in the sense that there exists an automorphic form \(\pi_0\) for \(\mathrm{GL}(n)/F\) of weight zero and level prime to p such that \(\overline{r}(\pi_0) = \overline{\rho}.\)

Then \(\rho\) is modular, that is, there exists an automorphic representation \(\pi\) of weight zero for \(\mathrm{GL}(n)/F\) which is associated to \(\rho.\)

One could be more precise about what it means to have big image. In fact, I can do this by saying that it has enormous image after restriction to the composite of the Galois closure of F with the pth roots of unity. Here enormous is a technical term, of course. There is also a version of this theorem with an ordinary (rather than Fontaine-Laffaille) hypothesis (more on this next time).

Let me now give a few nice theorems which can be deduced from the theorem above:

Theorem [ACCGHLNSTT]: Let E be an elliptic curve over a CM field F. Then E is potentially modular.

When I had a job interview at MIT in 2006, I was asked by Michael Sipser, the chair at the time, to come up with a theorem which (in a best case scenario) I would hope to prove in 10 years. I said that I wanted to prove that elliptic curves over imaginary quadratic fields were modular. (Reader, I got the job … then went to Northwestern.) It is very gratifying indeed that, roughly 10 years later, this result has actually been proved and that I have made some contribution towards its eventual resolution. (OK, we have potential modularity rather than modularity, but that is splitting hairs…). It is also amusing to note that a number of co-authors were still in high school at this time! (Fact Check: OK, just one…)

In fact, one can improve on the theorem above:

Theorem [ACCGHLNSTT]: Let E be an elliptic curve over a CM field F. Then Sym^n(E) is potentially modular for every n. In particular, the Sato-Tate conjecture holds for E.

Finally, for an application of a different type, suppose that \(\pi\) is a weight zero cuspidal algebraic automorphic representation for \(\mathrm{GL}(2)/F.\) For each prime v of good reduction, one can associate to \(\pi_v\) a pair of Satake parameters \(\{\alpha_v,\beta_v\}\) satisfying \(|\alpha_v \beta_v| = N(v).\) The Ramanujan conjecture says that one has

\(|\alpha_v| = |\beta_v| = N(v)^{1/2}.\)

An equivalent formulation is that the sum \(a_v\) of these two eigenvalues satisfies \(|a_v| \le 2 N(v)^{1/2}.\) We prove the following:

Theorem [ACCGHLNSTT]: Let F be a CM field, and let \(\pi\) be a weight zero cuspidal algebraic automorphic representation for \(\mathrm{GL}(2)/F.\) Then the Ramanujan conjecture holds for \(\pi.\)

If F is totally real, then the Ramanujan conjecture follows from Deligne’s theorem. One can associate to \(\pi\) a motive, whose Galois representation is either \(\rho = \rho(\pi)\) or \(\rho^{\otimes 2}.\) Then, by applying purity to these geometric representations, one deduces the result. (Of course, this was famously proved by Deligne himself in the case when F = Q. The case of a totally real field, especially in cases where one has to go via a motive assoicated to \(\rho^{\otimes 2},\) is due (I think) to Blasius.) This is decidedly not the way we prove this theorem. In fact, we do not know how to prove the Fontaine-Mazur conjecture for the representation \(\rho\) associated to \(\pi,\) even in the weak sense of showing that \(\rho\) or even \(\rho^{\otimes 2}\) appears inside the cohomology of some projective variety. Instead, we prove that \(\mathrm{Sym}^n \rho\) is potentially modular, then use the weaker convexity bound to prove the inequality:

\(|\alpha_v|^n \le N(v)^{n/2 + 1/2}.\)

Taking n sufficiently large, we deduce that \(|\alpha_v| \le N(v)^{1/2},\) which (by symmetry) proves the result. Experts will recognize this as precisely Langlands’ original strategy for proving Ramanujan using functoriality! In a certain sense, this is the first time that Ramanujan has been proved without a direct recourse to purity. I say “in some sense”, because there is also the ambiguous case of weight one modular forms. Here the Ramanujan conjecture (which is \(|a_p| \le 2\) in this case) was deduced by Deligne and Serre as a consequence of showing that \(\rho\) has finite image so that alpha_v and beta_v are roots of unity. On the other hand, that argument does also simultaneously imply that the representations are motivic. So our theorem produces, I believe, the only cuspidal automorphic representations for \(\mathrm{GL}(n)/F\) for which we know to be tempered everywhere and yet for which we do not know are directly associated in any way to geometry.

Question: Suppose I’m sitting in my club, and Tim Gowers asks me to say what is really new about this paper. What should I say?

Answer: The distinction (say) between elliptic curves over imaginary quadratic fields and real quadratic fields, while vast, is quite subtle to explain to someone who hasn’t thought about these questions. You could explain it, but the club is hardly a place to do so. Instead, go with this narrative: We generalize Wiles’ modularity results for 2-dimensional representations of Q to n-dimensional representations of Q. If you are pressed on previous generalizations, (especially those due to Clozel-Harris-Taylor), say that Wiles is the case GL(2), Clozel-Harris-Taylor is the case GSp(2n), and our result is the case GL(n).

If you had slightly more time, and the port has not yet arrived, you might also try to explain how the underlying geometric objects involved for GSp(2n) are all algebraic varieties (Shimura varieties), but for GL(n) they involve Riemannian manifolds which have no direct connection to algebraic geometry. Here is a good opportunity to name drop Peter Scholze, and explain how this is the first time that the methods of modularity have been combined with the new world of perfectoid spaces.

Click here for Part II