Trustees of Indiana University
Grant: $135,794 - National Science Foundation - Jun. 29, 2009
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Award Description: The principal investigator proposes to study moduli spaces of p-divisible groups and the cohomology groups of such spaces. These cohomology groups furnish representations of certain p-adic reductive groups and Galois groups. Moreover, the way these representations appear in the cohomology gives rise to relations between representations of different groups (at least conjecturally); these relations are the so-called Langlands correspondences. Depending on the type of cohomology theory one uses, one gets different types of correspondences. The project naturally comprises two sub-projects, depending on the type of cohomology theory: project A studies the de Rham cohomology, and project B the `-adic cohomology of certain moduli spaces. More specifically, project A is about relating certain 2-dimensional potentially crystalline Galois representations to Banach space representations of GL2. These Banach spaces shall be constructed using lattices in spaces of differential forms which are defined on coverings of the p-adic upper half plane. This is the main purpose of Project A which has, however, another two related sub-projects, namely a corresponding project for GLn and a project which is similar in spirit but concerns representations of the multiplicative group of a division algebra. In project B the PI proposes to investigate the classical smooth Langlands correspondence (for fields of positive characteristic) by purely local means, and suggests a line of research to compare local epsilon constants. The main innovative idea here is to use a formula for the Galois-epsilon constant, due to G. Laumon, and to relate it to a formula of Bushnell for the GLn-epsilon constant. The intellectual merit of our proposed activity is, mainly, to elucidate certain aspects of the geometry of deformation spaces of p-divisible groups. Project A will provide new examples of padic Langlands correspondences by methods which have not been explored so far. The program will therefore play a vital role in the general research activity connected to the p-adic Langlands Program. Project B proposes completely new ideas to study a known correspondence. The results will be of use for Representation Theorists and Number Theorists, and may well have also applications in other areas of Pure Mathematics like Topology (Homotopy Theory) and the Geometric Langlands Program. The PI is very much interested in getting students involved in activities which are in one way or another connected to the projects outlined above. The proposed projects are all based on geometric constructions and investigations which can be illustrated and therefore made comprehensible also for non-experts. It can thus be expected that the excitement of discovery which goes with the proposed research will enhance teaching on all levels. The p-adic Langlands Program is an area of research which started to develop less than ten years ago, and the PI is the only scholar at his organization working in this field. The proposed research will hence help to build up a research group and will connect the PI’s organization, and his students, to other researchers, universities and research institutions which previously had less contact to the PI’s organization. In this respect, it will help to build scientific and educational infrastructure.
Project Description: PI will study moduli spaces of p-divisible groups and the cohomology groups of such spaces. These cohomology groups furnish representations (reps) of certain p-adic reductive groups and Galois groups. The way these reps appear in the cohomology gives rise to relations between reps of different groups; these relations are Langlands correspondences. Depending on the cohomology theory one uses, one gets different types of correspondences. The project comprises two sub-projects: project A studies the de Rham cohomology, and project B the `-adic cohomology of certain moduli spaces. Project A is about relating certain 2-dimensional potentially crystalline Galois reps to Banach space reps of GL2. These Banach spaces shall be constructed using lattices in spaces of differential forms which are defined on coverings of the p-adic upper half plane. Project A has two related sub-projects: a corresponding project for GLn and a project which is similar in spirit but concerns reps of the multiplicative group of a division algebra. In project B the PI proposes to investigate the classical smooth Langlands correspondence (for fields of positive characteristic) by purely local means, and suggests a line of research to compare local epsilon constants. The idea is to use a formula for the Galois-epsilon constant, due to G. Laumon, and to relate it to a formula of Bushnell for the GLn-epsilon constant. Certain aspects of the geometry of deformation spaces of p-divisible groups will be elucidated. Project A will provide new examples of padic Langlands correspondences by methods which have not yet been explored. The program will play a vital role in the research activity connected to the p-adic Langlands Program. Project B proposes ideas to study a known correspondence. The results will be of use for Rep Theorists and Number Theorists and may have applications in other areas of Pure Math like Topology (Homotopy Theory) and the Geometric Langlands Program.
Jobs Summary: Additional Pay: Acad Services (Total jobs reported: 0)
Project Status: Less Than 50% Completed
This award's data was last updated on Jun. 29, 2009. Help expand these official descriptions using the wiki below.
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