DS10 - Défi de tous les savoirs

PERfectoids, Completed cohomology, LAnglands correspondence and TORsion in the cohomology – PerCoLaTor

Perfectoid, Langlands correspondance and Torsion in the Cohomology

The geometric realizations of the Langlands correspondance, both in its global incarnation with Shimura varieties, and local with Lubin-Tate spaces, enters now in a new cycle of development with in particular the works of Fargues, Fontaine and Scholze with the introduction of perfectoid spaces. Our project aim to develop the interaction between this new perfectoid world with the issues of the Langlands program.

Perfectoid approachs of Langlands program

Recent work of Scholze on perfectoids, infinte level Shimura varieties and finiteness of de Rham cohomology on adic spaces, have generated a lot of new approachs ont the Langlands program.<br /><br />Thanks to works of Berger, Breuil, Colmez , Emerton, Kisin and Paskunas, the theory of p-adic representations of GL_2(Q_p), in the context of p-adic Langlands program, is rather well understood. Going beyond this case, by considering for example a finite extension of Q_p, or more challenging <br />reductive groups other than GL_2, seems to call for completely new techniques and ideas.<br /><br />-Purely locally, we first expect to establish a theory of Lubin-Tate (phi,Gamma)-modules in order to be able to generalize the construction of Colmez's functor to other groups. <br /><br />-From a geometric viewpoint, a promising starting point towards the generalization of the correspondence to other groups would be a comparaison between Scholze' recent construction of a<br />partial correspondence using Drinfeld's and Lubin-Tate's towers and Colmez's construction for GL_2(Q_p).<br /><br />- Working more globally, we can use the Hodge-Tate map from the infinite level Harris-Taylor-Kottwitz Shimura varieties to the projective space,<br />seen as an adic space. Via the notion of Drinfeld sheaf, we should be able to obtain a new realization of the l-adic Langlands correspondence and hopefully in a second time a new tool for the p-adic version.<br /> <br />The heart of the Langlands program remains the construction of global Galois representations attached to algebraic automorphic representations of reductive groups. For a PEL reductive group and for cohomological automorphic representations, huge advances have been made. We will work on the following step : the construction of Galois representations for some algebraic non cohomological representations, with in mind applications to the modularity of motives with non-regular Hodge-Tate weights. In parallel, we will pursue a project on the search for unramified automorphic forms and motives.

1) Concerning p-adic representations of reductive groups, in order to go beyond the case of GL(2,Q_p), one of the projects consists in constructing a theory of overconvergent (phi,Gamma)-modules of Lubin-Tate type in many variables and then to link it with the p-adic Fourier theory of Schneider-Teitelbaum as well as with global geometric constructions such as Tsuji's reciprocity law.

2) We will work on controling torsion in the cohomology of Shimura varieties, either at infinite level for the completed cohomology or at a finite level. We plan to use the Harder-Narasimhan filtrations to construct a dévissage of the completed cohomology.

3) Concerning the study of p-adic families of modular forms, we plan to extend the construction by Andreatta-Iovita-Pilloni of eigenvarieties to PEL Shimura varieties with no ordinary locus. Here, the notion of mu-ordinarity will have to be used. It also seems possible to go beyond the PEL case, thanks to the recent work of Kisin and Madapusi-Pera.

4) Concerning the construction of Galois representations attached to algebraic automorphic forms, we hope to obtain Galois representations for some algebraic, non-cohomological Galois representations. We have some work in progress for representations which occur in the coherent cohomology. This paves the way for modularity theorems for non-regular Galois representations.

5) Concerning comparison theorems between p-adic etale and de Rham cohomology, the recent results of Scholze lead one to work on a cristalline version. Recently, Déglise and Niziol defined the notion of syntomic sheaf with coefficients as a sub-category of the category of p-adic constructible sheaves. There should exist a comparison theorem for these sheaves. After the work of Tsuji, we know that the p-adic sheaf of nearby cycles is syntomic (at least for large p) and we would like to study it in the case of Shimura varieties.

1) In the perfectoid field, with the help of diamonds theory of Scholze, Fargues formulated a new conjecture in the spirit of classical geometric Langlands program, with the curve X/F_q being replaced by Fargues-Fontaine curve.

2) Considering locally analytic vectors for the action of Gamma inside some big modules defined with the help of some period rings of Fontaine, Laurent Berger proposed a new approach of the classification of Galois representations in terms of (phi,Gamma)-modules, giving the overconvergent property in the cyclotomic case as well as in the Lubin-Tate case.

3) Studying torsion in the cohomology of Shimura varieties of Kottwitz-Harris-Taylor cases, Boyer obtained rather general cases of the Ihara's lemma as conjectured by Clozel-Harris-Taylo.

4) About classicity questions, Bijakowski, Pilloni and Stroh succeeded to generalized Kassaei series using Fargues theory of degree, in a rather general context.

5) The syntonic cohomology was introduced to give a p-adic analoge of the Deligne-Beilinson cohomology. Niziol succeeded to obtain remarkable results on this new cohomology, proving for example it's a absolute Hodge cohomology.

1) To give a precise meaning of the Fargues conjecture, on has to develop a theory of perverse sheaves on diamonds. Then on could try to study the classical construction of the geometric Langlands theory to adapt them to the diamonds case.

2) Concerning problems R=T, on should have to be able ta tackle questions of torsion in the cohomology of non compact Shimura varieties. The approach of Boyer should be extended to this case.

3) Niziol propose to study the links of the syntonic cohomology with fiber bundles on the Fargues-Fontaine curve and their relation with Banach-Colmez spaces.

4) Andreatta, Iovita and Pilloni succeeded to construct overconvergent forms in caracteristic p, which wa conjectured by Colemean. One question now is to understand the Galois representations attached to them: in particular their should be triangulines in some sense.

5) Berger carry on studying his new approach of (phi,Gamma)-modules of Lubin-Tate spaces, with overconvergent questions.

14 articles and 35 preprints

Since its introduction in the 70's, the Langlands program has been the object of an incredible number of works by some of the world's most talented mathematicians, with striking arithmetic applications such as the proof of Fermat's Last theorem and the Sato-Tate conjecture.

In the past decade or two, beginning with the pioneering work of Breuil, a p-adic version of this program has emerged.
By the work of Colmez, the case of the group GL(2,Q_p) is now settled and furthermore Emerton established the compatibility of this p-adic correspondence with the global one. This subject is now at the heart of an increasing number of questions and projects.

During the last three years, very rich and deep mathematical objects were invented: the Fargues-Fontaine curve, the notion of perfectoid space, the pro-étale site, infinite level perfectoid Shimura varieties. We believe that these concepts will lead to a revolution in the field, just like deformation techniques for Galois representations and Taylor-Wiles systems did twenty years ago; the best proof of this affirmation is the great number of international conferences and workshops already organized on these topics, attracting an increasing number of mathematicians.

These emerging topics are extremely challenging and appeal to a large variety of competencies, which are amply covered by the various members of our group. We can organize the various directions of research into five broad themes:

1) p-adic representation theory of p-adic groups and p-adic Galois representations of p-adic local fields: we aim to go further than the case of the group GL(2,Q_p) settled by Colmez.

2) Infinite level Shimura varieties and Rapoport-Zink spaces: we believe that the Lubin-Tate spaces of infinite level will allow us to reprove Colmez's result for GL(2). They should also give us a window through which to later study the case of GL(n).

3) p-adic families of modular forms: we would like to unify the various points of views and to study some new situations like non-PEL Shimura varieties.

4) The global Langlands program for number fields : the study of coherent cohomology will allow us to explore the relations between non-cohomological algebraic automorphic forms and non regular Galois representations. We also aim to study the completed cohomology using the Hodge-Tate period map and the Harder-Narasimhan stratifications. On the automorphic side we will pursue our program on level 1 automorphic forms.

5) Comparaison theorems for cristalline and syntomic cohomology of adic spaces.

The French arithmetic school, since the introduction of the Langlands program and till its more recent developments, has always been at the avant-garde : we mention the proof of the local Langlands correspondance by Laumon-Rapoport-Stuhler, Harris-Taylor and Henniart, the global correspondence for function fields by L. Lafforgue, the proof of the fundamental lemma by Laumon-Ngo, Ngo and Chaudouard-Laumon, the introduction of the p-adic Langlands program by Breuil and Colmez's proof for GL(2,Q_p).The goal of the project is to continue this long tradition of French leadership, providing a necessary support to develop scientific activity inside the group or in cooperation with mathematicians abroad, in a very hot and promising topic.

Our group is composed of internationally recognized advanced researchers, bright early stage researchers and promising fresh post doctoral students. As we expect that the subject will develop quickly, the support of the ANR is necessary in order to organize regular workshops, mini-conferences and a big final conference, to travel to attend congresses, to invite world class experts for short stays, to hire post-doctoral students (36 months) and finally to convince the best students and postdocs to develop their projects in our universities.

Project coordination

Pascal Boyer (Laboratoire d'Analyse, Géométrie et Applications)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.


ENS de Lyon Unité de Mathématiques Pures et Appliquées
Université Paris 13 - LAGA Laboratoire d'Analyse, Géométrie et Applications

Help of the ANR 329,680 euros
Beginning and duration of the scientific project: September 2014 - 48 Months

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