Summary:
The aim of this project is to investigate p-adic symmetric spaces and to find arithmetic +applications. Moreover we shall analyze certain related questions in the theory of p-adic L-functions.<br /> <br /> p-adic symmetric spaces have been introduced by M. Rapoport and T. Zink. Associated to every data (G, {μ}, b), consisting of an algebraic group G over Qp, a conjugacy class {μ} of cocharacters of G and an element b ∈ G(Qp), is a rigid analytic space cFwab which parameterizes certain filtered isocrystals. An example is Drinfeld\'s p-adic symmetric space Ω(d+1)<br /> <br /> <center>Ω(d+1) = PPd{/bQp} - ∪ Qp-rational hyperplanes.</center> <br /> Which parameterizes certain d-dimensional formal groups cal G. The space Ω(d+1) admits a family of ètale coverings Σ(d+1)n,n ≥ 1 which arise by trivializing the pn-division points {cal G}[pn] = mathrm{Ker}(pn: {cal G}to {cal G}). These coverings are of great importance. For example it is known that the dth L-adic cohomology group (ell ≠ p) of the Σ(d+1)n, n ≥ 1 realize the local Langlands correspondence for GLd(Qp).<br /> <br /> Many geometric objects which are of interest for number theorists occur as quotients of classical symmetric spaces by discrete subgroups of Lie groups. This allows us in certain cases to relate the L-functions of a motive to an automorphic L-functions (the easiest examples are elliptic curves over Q). Meanwhile it is known that many Shimura varieties admit uniformizations by p-adic symmetric spaces as well. The first known example of this phenomena is the case of a Shimura curve with completely split reduction at p which can be represented as a quotient of the p-adic upper half plane. A general theory of p-adic uniformization of Shimura varieties is due to Rapoport and Zink.<br /> <br /> P-adic L-functions can be associated to many objects of arithmetic interest (e.g. elliptic curves or modular forms). Their values at integral points should be related to arithmetic invariants of the objects. Recently interesting applications of the theory of p-adic uniformization to the theory of p-adic L-functions have emerged. For example M. Bertolini, H. Darmon and P. Schneider have introduced and investigated a new type of rigid analytic L-functions which are associated to modular forms on the p-adic upper half plane.<br /> <br /> In this project we plan to work concretely on the following subjects: <ol> <li> Construction of formal flat models for p-adic symmetric spaces (without PEL-structure).</li> <li> Construction of a tower of ètale coverings for general p-adic symmetric spaces (similar to the tower ... → Σn+1 → Σn → ...→ Ω(d+1)).</li> <li> Relations between ètale coverings of different p-adic symmetric spaces.</li> <li> Rigid analytic p-adic L-functions for automorphic forms.</li> <li> p-adic periods of modular forms of multiplicative type (at p).</li> <li> Investigation of values of p-adic L-functions of modular forms at integral points (especially at non-critical values). We want to investigate in particular the case of anticyclotomic p-adic L-functions.</li> </ol> Another topic which will be investigated mainly by one of the research assistants (Vytautas Paskunas) is the p-adic representation theory of p-adic linear algebraic groups over p-adic fields.
