New Geometric Methods in Number Theory and Automorphic Forms

Question

The MSRI is organising a programme with the above title from Aug 11, 2014 to Dec 12, 2014. Here is a short description from their website :

The branches of number theory most directly related to the arithmetic of automorphic forms have seen much recent progress, with the resolution of many longstanding conjectures. These breakthroughs have largely been achieved by the discovery of new geometric techniques and insights. The goal of this program is to highlight new geometric structures and new questions of a geometric nature which seem most crucial for further development. In particular, the program will emphasize geometric questions arising in the study of Shimura varieties, the $p$-adic Langlands program, and periods of automorphic forms.

Question Which new geometric structures, techniques and insights have been crucial for this recent progress ?

Answer 1

Knowing the organizers well and working in the field, I can try an answer, but this is nothing more than an educated guess.

First, the breakthroughs in question include

(i) The construction and study of Galois representations attached to self-dual cohomological automorphic forms for $Gl_n$ (satisfying local-global compatibility, etc.) This is the work of many people, based on the fundamental work of Arthur and Ngo, including Shin, Morel, Harris, Clozel, Labesse, and many many others. This can be considered as done, even if the four-volume Paris' book edited by Harris that should contain every detail is not completely ready.

(ii) The construction and study of Galois representations attached to not necessary self-dual cohomological automorphic forms for $Gl_n$, announced last year by Lan, Harris, Taylor and Thorne (the preprint has yet to be released).

(iii) The proof by Kisin and also by Emerton of large parts of the Fontaine-Mazur conjecture for $Gl_2$.

(iv) The proof of Sato-Tate by many people with various multiplicity, the two highest being Taylor and Harris.

(v) The progresses on the p-adic Langlands program, especially on the Breuil-Mezard conjecture.

(vi) The progress on the theory of Shimura varieties, including the proof of two major conjectures of the subject by Kisin (one has an older, not universally accepted, proof by Vas): the conjectures of Milne and of Langlands-Rapoport.

At first I thought I should include (0) the proof of the fundamental lemma by Ngo, but since none of the organizers is a specialist of this area, I am not sure.

Now why the emphasis on the "geometric methods", and what are those? Well, there is a même saying that along the traditional tripartite division of mathematicians as "algebraists", "analysts", and "geometers" (see e.g. Recoltes et Semailles), while people like Breuil (or Fontaine) are more on the algebraic side, and perhaps Colmez on the analytic side, people like Kisin and Emerton are really on the geometric side, and that their geometric intuition played a crucial role in their recent successes. Whatever you think of this même (or even of the tripartite classification) it is quite possible that it made its way to the mind of one or more of the organizer. The geometric insights and methods include

(a) the use of "eigenvarieties": families of automorphic or/and Galois representations that have a geometric structure, and whose geometric properties, local and global illuminate the properties of the individual objects that compose them. For example, this plays a crucial role in Emerton's proof of Fonntaine-Mazur's conjecture (iii), in constructing Galois representations by "passage to the limit" (i) and (ii), and also in recent progress toward Bloch-Kato and Birch-Swinnerton-Dyer conjecture (work of Chenevier and myself, Urban and Skinner), and also in the work on the Breuil-Mezard conjecture (Kisin first, then others)

(b) The better understanding of certain Shimura varieties, in particular the ones attached to unitary groups, in particular in connection with Rapoport-Zink spaces etc. For example one can cite the thesis work of Mantovan, which is used in Shin's subsequent work on (i). Also whatever Kisin uses to prove the conjecture about Shimura variety (at this point I don't know what it is, but I am organizing a seminar at Yale to learn this eventually)

(c) Also, the better understanding and the use of the boundary components of non-compact Shimura varieties, including in cases (this is mainly speculative so far) where this components has only the structure of a differentiable manifold, not of an algebraic variety. I am not sure, but idea like that plays a role in (iv).

(d) Study of cycle on Shimura's varieties, in particular in connection to periods and p-adic L-function (how to define them in higher ranks? that is very hard and important).

(e) if (0) is included (which as I have said, I am not sure of), the geometric methods of Ngo (and before him Laumon, Goreski, MacPherson: balloons, Hitchin's vibrations, etc.) used in proving the fundamental lemma, and perhaps also the ones of Laurent Lafforgue. But I think this might be the subject of another conference.

I hope that helps... Sorry to anyone I forgot to mention, my list of people having a part in the recent breakthroughs is far from complete.

Answer 2

To complement Joel's wonderful and (as far as I understand) very much on point answer, let me quote from the proposal for the parallel program on Geometric Representation Theory, which touches on several related themes:

Representation theory is the study of the basic symmetries of mathematics and physics. The primary aim of the subject is to understand concrete linear models for abstract symmetry groups. A signature triumph of the past century is our understanding of compact Lie groups. At the foundation, there is Cartan's classification of Lie algebras and Borel-Weil-Bott's uniform construction of all representations in the cohomology of line bundles on flag varieties. Thus we have a list of every compact Lie group we could ever encounter and every way in which it could appear concretely as a matrix group. Furthermore, there is a deep combinatorial theory of other key structures such as the tensor product of representations. The ideas and results of this subject are the basic input to diverse areas from number theory to quantum field theory.

Though mysteries still remain, the theory of compact Lie groups is a representative model for what we would like to achieve with other symmetry groups. Because of their universal importance, symmetry groups come in many different flavors: finite groups, Lie groups, $p$-adic groups, loop groups, ad\'elic groups,... and the list will only increase with the discovery of new important structures. For the above examples, our understanding is still very coarse though the last decades have witnessed breathtaking advances. The Langlands program along with its geometric spinoffs provide a visionary roadmap for where the subject could go in the coming years. In particular, current developments such as the recent proof of the Fundamental Lemma create great optimism that geometric techniques will have a deep impact.

Geometric representation theory seeks to understand groups and representations as a consequence of more subtle but fundamental symmetries. A groundbreaking example of its success is Beilinson-Bernstein's uniform construction of all representations of Lie groups via the geometry of $\mathcal D$-modules on flag varieties. The result is not difficult to state and prove but has the Borel-Weil-Bott theorem and the Kazhdan-Lusztig multiplicity conjectures as immediate consequences. A reasonable reaction is to wonder what allows one to prove deep results about representations of Lie groups with so little effort. One answer is that the true focus of the Beilinson-Bernstein theory is not representations but rather the symmetries of flag varieties. The geometric notion of $\mathcal D$-module allows one to localize symmetries and to apply the sheaf-theoretic techniques of algebraic geometry. In this way, our understanding of representations of Lie groups follows from that of infinitesimal symmetries of algebraic varieties.

There are now many instances where difficult questions about representations can be translated into more tractable questions about geometry. Other famous examples include the Deligne-Lusztig theory of representations of finite groups of Lie type, the Springer theory of representations of Weyl groups, the Kazhdan-Lusztig theory of modules for Hecke algebras, and Lusztig and Nakajima's theory of representations of Kac-Moody algebras and quantum groups via quiver varieties. In the theory of automorphic forms, one of the fundamental tools is the realization of representations of ad`elic groups in the cohomology of Shimura varieties (in the case of number fields) and Drinfeld modular varieties (in the case of function fields). The recent proof of the Fundamental Lemma exploits the fact that computing orbital integrals in $p$-adic groups can be reduced to calculating cohomology of fibers of Hitchin's integrable system.

The twin modern goals of geometric representation theory are to explore the above dictionaries and to discover new unexpected ones. As an important consequence, the geometric realization of representations often reveals deeper layers of structure in the form of categorification. Categorification typically turns numbers (for example, the coefficients of Kazhdan-Lusztig polynomials) into the dimensions of vector spaces (in this case, the Ext groups of intersection cohomology sheaves). It is a primary explanation for miraculous integrality and positivity properties in algebraic combinatorics. At the center of geometric representation theory is Grothendieck's categorification of functions by $\ell$-adic sheaves. An important example is Lusztig's theory of character sheaves: it provides a uniform geometric source for the characters of all finite groups of Lie type. Another important example is the theory of canonical bases: here categorification replaces representation spaces with linear categories equipped with canonical generating objects. Another broad example is the geometric Langlands program: it provides a categorification of the Langlands program in the setting of function fields, providing new insights into many classical constructions.

Geometric representation theory has close and profound connections to many fields of mathematics, which we expect to play a significant role in the program. Perhaps the most significant are to number theory, via the theory of automorphic forms, L-functions and modularity. Much current activity in the field is motivated either directly by problems in number theory or by more tractable geometric analogues thereof. Another major influence on the subject comes from physics, in particular gauge theory, integrable systems, and recently topological string theory. There is a significant interaction with the theory of $C^*$-algebras through the Baum-Connes conjecture, an instance of which provides an organizing principle for representation theory of real and $p$-adic groups. Finally, geometric representation theory is closely entwined with very active areas in combinatorics such as Schubert calculus, its affine analogue, and the theory of Macdonald polynomials.

The proposed program has two primary goals:

1. To bring together researchers working in the arithmetic and geometric Langlands programs so as to discover new relations between the objects appearing in the two subjects.

2. To explore new principles and paradigms within geometric representation theory.

A particular emphasis will be placed on geometric methods in representation theory over local fields. This is an area which has advanced dramatically in the past decade. More importantly, it holds great promise for future discoveries of interest to a diverse collection of researchers.