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.
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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. Niziol succeeded to obtain remarkable results on this new cohomology, proving for example it's a absolute Hodge cohomology.
Then on could try to study the classical construction of the geometric Langlands theory to adapt them to the diamonds case. The approach of Boyer should be extended to this case. One question now is to understand the Galois representations attached to them: in particular their should be triangulines in some sense.
Gelbart : An elementary introduction to the Langlands program
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.
This subject is now at the heart of an increasing number of questions and projects. 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.
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They should also give us a window through which to later study the case of GL n. 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. 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. 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.
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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.webdisk.lauren.reclaim.hosting/20171.php
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The author of this summary is the project coordinator, who is responsible for the content of this summary. During the first week July 8th — 12th , we give an introduction to the fundamental concepts concerning automorphic forms and automorphic representations; for the second week July 15th — 19th , we introduce the main techniques in the Langlands Program, namely trace formula and automorphic L-functions. First week. Chenyan Wu.
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- Student Arithmetic Seminar?
Fangyang Tian. Kam Fai Tam. Second week.
Chung Pang Mok. Chen Wan. Yuanqing Cai. Schedule and venue.
Accommodation arrangements for students. For registered students with recommendation by supervisors : accommodation would be provided; non-registered students have to arrange for their own accommodation. We cordially welcome the participation of senior undergraduate students, graduate students, post-graduates, and other researchers.
School of Mathematical Sciences, Soochow University. June For the lecture time it is one hour for each lecture, but it is quite flexible. The lecture schedule of the first week:.