Abelian Surfaces are Potentially Modular

Today I wanted (in the spirit of this post) to report on some new work in progress with George Boxer, Toby Gee, and Vincent Pilloni. Edit: The paper is now available here.

Recal that, for a smooth projective variety X over a number field F unramified outside a finite set of primes S, one may write down a global Hasse-Weil zeta function:

\(\displaystyle{ \zeta_{X,S}(s) = \prod \frac{1}{1 – N(x)^{-s}}}\)

where the product runs over closed points of a smooth integral model. From the Weil conjectures, the function \(\zeta_{X,S}(s)\) is absolutely convergent for s with real part at least \(1+m/2,\) where \(m = \mathrm{dim}(X).\) One has the following well-known conjecture:

Hasse–Weil Conjecture: The function \(\zeta_{X,S}(s)\) extends to a meromorphic function on the complex plane. Moreover, there exists a rational number A, a collection of polynomials \(P_v(T)\) for v dividing S, and infinite Gamma factors \(\Gamma_v(s)\) such that

\(\displaystyle{ \xi_{X}(s) = \zeta_{X,S}(s) \cdot A^{s/2} \cdot \prod_{v|\infty} \Gamma_v(s) \cdot \prod_{v|S} \frac{1}{P_v(N(v)^{-s})}}\)

satisfies the functional equation \(\xi_X(s) = w \cdot \xi_X(m+1-s)\) with \(w = \pm 1.\)

Naturally, one can be more precise about the conductor and the factors at the bad primes. In the special case when F = Q and X is a point, then \(\zeta_{X,S}(s)\) is essentially the Riemann zeta function, and the conjecture follows from Riemann’s proof of the functional equation. If F is a general number field but X is still a point, then \(\zeta_{X,S}(s)\) is (up to some missing Euler factors at S) the Dedekind zeta function \(\zeta_F(s)\) of F, and the conjecture is a theorem of Hecke. If X is a curve of genus zero over F, then \(\zeta_{X,S}(s)\) is \(\zeta_F(s) \zeta_F(s-1),\) and one can reduce to the previous case. More generally, by combining Hecke’s results with an argument of Artin and Brauer about writing a representation as a virtual sum of induced characters from solvable (Brauer elementary) subgroups, one can prove the result for any X for which the l-adic cohomology groups are potentially abelian. This class of varieties includes those for which all the cohomology of X is generated by algebraic cycles.

For a long time, not much was known beyond these special cases. But that is not to say there was not a lot of progress, particularly in the conjectural understanding of what this conjecture really was about. The first huge step was the discovery and formulation of the Taniyama-Shimura conjecture, and the related converse theorems of Weil. The second was the fundamental work of Langlands which cast the entire problem in the (correct) setting of automorphic forms. In this context, the Hasse-Weil zeta functions of modular curves were directly lined to the L-functions of classical weight 2 modular curves. More generally, the Hasse-Weil zeta functions of all Shimura varieties (such as Picard modular surfaces) should be linked (via the trace formula and conjectures of Langlands and Kottwitz) to the L-functions of automorphic representations. On the other hand, these examples are directly linked to the theory of automorphic forms, so the fact that their Hasse-Weil zeta functions are automorphic, while still very important, is not necessarily evidence for the general case. In particular, there was no real strategy for taking a variety that occurred “in nature” and saying anything non-trivial about the Hasse-Weil zeta function beyond the fact it converged for real part greater than \(1 + m/2,\) which itself requires the full strength of the Weil conjectures.

The first genuinely new example arrived in the work of Wiles (extended by others, including Breuil-Conrad-Diamond-Taylor), who proved that elliptic curves E/Q were modular. An immediate consequence of this theorem is that Hasse-Weil conjecture holds for elliptic curves over Q. Taylor’s subsequent work on potentially modularity, while not enough to prove modularity of all elliptic curves over all totally real fields, was still strong enough to allow him to deduce the Hasse-Weil conjecture for any elliptic curve over a totally real field. You might ask what have been the developments since these results. After all, the methods of modularity have been a very intense subject of study over the past 25 years. One problem is that these methods have been extremely reliant on a regularity assumption on the corresponding motives. One nice example of a regular motive is the symmetric power of any elliptic curve. On the other hand, if one takes a curve X over a number field, then h^{1,0} = h^{0,1} = g, and the corresponding motive is regular only for g = 0 or 1. The biggest progress in automorphy of non-regular motives has actually come in the form of new cases of the Artin conjecture — first by Buzzard-Taylor and Buzzard, then in the proof of Serre’s conjecture by Khare-Wintenberger over Q, and more recently in subsequent results by a number of people (Kassaei, Sasaki, Pilloni, Stroh, Tian) over totally real fields. But these results provide no new cases of the Hasse-Weil conjecture, since the Artin cases were already known in this setting by Brauer. (It should be said, however, that the generalized modularity conjecture is now considered more fundamental than the Hasse-Weil conjecture.) There are a few other examples of Hasse-Weil one can prove by using various forms of functoriality to get non-regular motives from regular ones, for example, by using the Arthur-Clozel theory of base change, or by Rankin-Selberg. We succeed, however, in establishing the conjecture for a class of motives which is non-regular in an essential way. The first corollary of our main result is as follows:

Theorem [Boxer,C,Gee,Pilloni] Let X be a genus two curve over a totally real field. The the Hasse-Weil conjecture holds for X.

It will be no surprise to the experts that we deduce the theorem above from the following:

Theorem [BCGP] Let A be an abelian surface over a totally real field F. Then A is potentially modular.

In the case when A has trivial endomorphisms (the most interesting case), this theorem was only known for a finite number of examples over \(\mathbf{Q}.\) In each of those cases, the stronger statement that A is modular was proved by first explicitly computing the corresponding low weight Siegel modular form. For example, the team of Brumer-Pacetti-Tornaría-Poor-Voight-Yuen prove that the abelian surfaces of conductors 277, 353, and 587 are all modular, using (on the Galois side) the Faltings-Serre method, and (on the automorphic side) some really quite subtle computational methods developed by Poor and Yuen. A paper of Berger-Klosin handles a case of conductor 731 by a related method that replaces the Falting-Serre argument by an analysis of certain reducible deformation rings.

The arguments of our paper are a little difficult to summarize for the non-expert. But George Boxer did a very nice job presenting an overview of the main ideas, and you can watch his lecture online (posted below, together with Vincent’s lecture on higher Hida theory). The three sentence version of our approach is as follows. There was a program initiated by Tilouine to generalize the Buzzard-Taylor method to GSp(4), which ran into technical problems related to the fact that Siegel modular forms are not directly reconstructible from their Hecke eigenvalues. There was a second approach coming from my work with David Geraghty, which used instead a variation of the Taylor-Wiles method; this ran into technical problems related to the difficulty of studying torsion in the higher coherent cohomology of Shimura varieties. Our method is a synthesis of these two approaches using Higher Hida theory as recently developed by Pilloni. Let me instead address one or two questions here that GB did not get around to in his talk:

What is the overlap of this result with [ACCGHLNSTT]? Perhaps surprisingly, not so much. For example, our results are independent of the arguments of Scholze (and now Caraiani-Scholze) on constructing Galois representations to torsion classes in Betti cohomology. We do give a new proof of the result that elliptic curves over CM fields are potentially modular, but that is the maximal point of intersection. In contrast, we don’t prove that higher symmetric powers of elliptic curves are modular. We do, however, prove potentially modularity of all elliptic curves over all quadratic extensions of totally real fields with mixed signature, like \(\mathbf{Q}(2^{1/4}).\) The common theme is (not surprisingly) the Taylor-Wiles method (modified using the ideas in my paper with David Geraghty).

What’s new in this paper which allows you to make progress on this problem? George explains this well in his lecture. But let me at least stress this point: Vincent Pilloni’s recent work on higher Hida Theory was absolutely crucial. Boxer, Gee, and I were working on questions related to modularity in the symplectic case, but when Pilloni’s paper first came out, we immediately dropped what we were doing and started working (very soon with Pilloni) on this problem. If you have read the Calegari-Geraghty paper on GSp(4) and are not an author of the current paper (hi David!), and you look through our manuscript (currently a little over 200 pages and [optimistically?!] ready by the end of the year), then you also recognize other key technical points, including a more philosophically satisfactory doubling argument and Ihara avoidance in the symplectic case, amongst other things.

So what about modularity? Of course, we deduce our potential modularity result from a modularity lifting theorem. The reason we cannot deduce that Abelian surfaces are all modular, even assuming for example that they are ordinary at 3 with big residual image, is that Serre’s conjecture is not so easy. Not only is \(\mathrm{GSp}_4(\mathbf{F}_3)\) not a solvable group, but — and this is more problematic — Artin representations do not contribute to the coherent cohomology of Shimura varieties in any setting other than holomorphic modular forms of weight one. Still, there are some sources of residually modular representations, including the representations induced from totally real quadratic extensions (for small primes, at least). We do, however, prove the following (which GB forgot to mention in his talk, so I bring up here):

Proposition [BCGP]: There exist infinitely modular abelian surfaces (up to twist) over Q with End_C(A) = Z.

This is proved in an amusing way. It suffices to show that, given a residual representation

\(\overline{\rho}: G_{\mathbf{Q}} \rightarrow \mathrm{GSp}_4(\mathbf{F}_3)\)

with cyclotomic similitude character (or rather inverse cyclotomic character with our cohomological normalizations) which has big enough image and is modular (plus some other technical conditions, including ordinary and p-distinguished) that it comes from infinitely many abelian surfaces over Q, and then to prove the modularity of those surfaces using the residual modularity of \(\overline{\rho}.\) This immediately reduces to the question of finding rational points on some twist of the moduli space \(\mathcal{A}_2(3).\) And this space is rational! Moreover, it turns out to be a very famous hypersurface much studied in the literature — it is the Burkhardt Quartic. Now unfortunately — unlike for curves — it’s not so obvious to determine whether a twist of a higher dimensional rational variety is rational or not. The problem is that the twisting is coming from an action by \(\mathrm{Sp}_4(\mathbf{F}_3),\) and that action is not compatible with the birational map to \(\mathbf{P}^3,\) so the resulting twist is not a priori a Severi-Brauer variety. However, something quite pleasant happens — there is a degree six cover

\(\mathcal{A}^{-}_2(3) \stackrel{6:1}{\rightarrow} \mathcal{A}_2(3)\)

(coming from a choice of odd theta characteristic) which is not only still rational, but now rational in an equivariant way. So now one can proceed following the argument of Shepherd-Barron and Taylor in their earlier paper on mod-2 and mod-5 Galois representations.

What about curves of genus g > 2?: Over \(\mathbf{Q},\) there is a tetrachotomy corresponding to the cases g = 0, g = 1, g = 2, and g > 2. The g = 0 case goes back to the work of Riemann. The key point in the g = 1 case (where the relevant objects are modular forms of weight two) is that there are two very natural ways to study these objects. The first (and more classical) way to think about a modular form is as a holomorphic function on the upper half plane which satisfies specific transformation properties under the action of a finite index subgroup of \(\mathrm{SL}_2(\mathbf{Z}).\) This gives a direct relationship between modular forms and the coherent cohomology of modular curves; namely, cuspidal modular forms of weight two and level \(\Gamma_0(N)\) are exactly holomorphic differentials on the modular curve \(X_0(N).\) On the other hand, there is a second interpretation of modular forms of weight two in terms of the Betti (or etale or de Rham) cohomology of the modular curve. A direct way to see this is that holomorphic differentials can be thought of as smooth differentials, and these satisfy a duality with the homology group \(H_1(X_0(N),\mathbf{R})\) by integrating a differential along a loop. And it is the second description (in terms of etale cohomology) which is vital for studying the arithmetic of modular forms. When g = 2, there is still a description of the relevant forms in terms of coherent cohomology of Shimura varieties (now Siegel 3-folds), but there is no longer any direct link between these coherent cohomology groups and etale cohomology. Finally, when g > 2, even the relationship with coherent cohomology disappears — the relevant automorphic objects have some description in terms of differential equations on locally symmetric spaces, but there is no longer any way to get a handle on these spaces. For those that know about Maass forms, the situation for g > 2 is at least as hard (probably much harder) than the notorious open problem of constructing Galois representations associated to Maass forms of eigenvalue 1/4. In other words, it’s probably very hard! (Of course, there are special cases in higher genus when the Jacobian of the curve admits extra endomorphisms which can be handled by current methods.)

Finally, as promised, here are the videos: