"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)."
The PI proposes to investigate representations of p-adic reductive groups (mainly general linear groups) and Galois groups which are (simultaneously) realized by the cohomology of deformation spaces of p-divisible groups. The proposed research naturally falls into two projects, according to the type of cohomology theory that is considered. The rst project is concerned with the de Rham cohomology of the p-adic symmetric spaces de ned by Drinfeld. These can be considered as the special bers of formal schemes, and these formal schemes represent deformation functors for certain p-divisible formal groups (with extra structures). It was shown by P. Schneider and U. Stuhler that the (compactly supported) de Rham cohomology groups of these spaces realize generalized Steinberg representations (of general linear groups). Torsion points on the universal formal group give rise to etale coverings of the symmetric spaces, and global arguments (relying on the use of Shimura varieties) show that their cohomology groups realize supercuspidal representations. Be- cause these spaces are Stein spaces, their cohomology groups can be computed by means of rigid-analytic di erential forms. Using the relation to di erential forms, the project aims at constructing lattices in these supercuspidal representations which then give rise to Banach space representations. Furthermore, the PI seeks to relate these Banach space representations to certain families of potentially crystalline Galois representations, at least when the spaces are one-dimensional (in which case the Galois representations are of di- mension two), and when the covering is 'tame'. The second project is concerned with the etale l-adic cohomology of deformation spaces of one-dimensional formal groups. These cohomology groups are known to realize the local Langlands correspondence between n- dimensional Weil group representations and representations of the p-adic GL(n). Again, the only known proofs for this fact, due to M. Harris and R. Taylor (respectively P. Boyer in the equal characteristic case) depend on global arguments. The PI proposes to investi- gate the equality between epsilon constants of representations which correspond to each other, in the case when the base eld is of positive characteristic, by using local methods from rigid-analytic geometry. For the Galois representation a formula due to G. Laumon describes the epsilon constant in terms of a local Fourier transform, and this Fourier transform in turn has a rigid-analytic description. Analyzing this description the PI seeks to relate Laumon's formula to a formula of Bushnell and Frohlich for the corresponding representation of GL(n).
The proposal is about investigating solutions of algebraic equations with special emphasis on methods involving congruences with respect to a xed prime number p. For instance, even though a polynomial in general may have no roots which are rational numbers, there may well be integers which are roots up to multiples of (arbitrary) high powers of p. This can be considered as an approximate solution with respect to p, and these approximate solutions give rise to a solution which is no longer an ordinary number, but a so-called p- adic number. The arithmetic of p-adic numbers is considerably easier than the arithmetic of rational numbers, yet complex enough to reveal lots of information about algebraic equations (with rational coefficients). Moreover, it is possible to do analysis with p-adic numbers. The local Langlands correspondence describes the arithmetic of p-adic numbers in terms of actions of certain groups of matrices on linear spaces. These linear spaces are derived from non-linear or 'curved' spaces by 'taking homology', which is a process of converting geometric information into linear spaces. The curved spaces in turn, which are like surfaces in spaces of p-adic numbers, are accessible to analytic methods, with functions depending on p-adic numbers as arguments. The PI proposes to use these available analytic methods to investigate the curved spaces (and subsequently the linear spaces they give rise to) with the goal of getting a better understanding of the local Langlands correspondence and extending it beyond its classical domain.
