Arithmetic of automorphic forms, periods and L-values

In number theory, a period is a complex number, whose real and imaginary part can be written as the integral of rational functions with rational coefficients, over a domain inR n , given by polynomial inequalities with rational coefficients. Periods arise in various contexts in mathematics and play a decisive role in the study of special values of automorphic L-functions, as documented by famous conjectures of Deligne and Beilinson, but also the global Gross-Prasad conjecture. In turn, these L-values encode significant information about the arithmetic of automorphic forms and Galois representations. It is the aim of this project is to investigate the arithmetic of automorphic forms and the connection of periods to special values of automorphic L-functions. In the strategy, which I propose, the analysis of various cohomology theories of automorphic representations p will play a decisive role: The periods that I will consider will come from a comparison of a rational structure on a model (such as a Whittaker model, a Shalika model or a Bessel model) of pf and a rational structure on a realization of p f in a cohomology space (such as (g,K)-cohomology or -cohomology). Analyzing the connection of such cohomologically defined periods and special values of L-functions is projected joint work with Michael Harris and with A. Raghuram. As another aim of this project, I plan to investigate the arithmetic of cohomological automorphic forms by studying the effect of Galois automorphisms s on automorphic representations p and various instances of Langlands- transfers. Here, it seems that only now, after J. Arthur presented a proof of his conjectures on the discrete automorphic spectrum, such questions may be reasonably treated for classical groups and the Langlands lift to the general linear group. Another interesting instance of a Langlands transfer, for which these questions shall be treated (at least in certain low-dimensional cases), is the Asgari-Shahidi lift from general spin groups to the general linear group. A last, rather Lie-theoretical aspect of this project is to study the -cohomological unitary dual, i.e., the unitary representations, which may contribute to the coherent cohomology of a Shimura variety. This could have interesting applications to a structural description of coherent cohomology in terms of the automorphic spectrum of the underlying Shimura variety, like in the case of J. Frankes proof of the Borel-conjecture for (g,K)-cohomology.

Put in a flamboyant way, L-functions are something like the holy grail of number theory. L-functions inherit their outstanding role in mathematics form the fact that they encode very deep, if not to say all knowledge about the prime numbers. For example, the most difficult and surprisingly at the same time most basic L-function is the famous Riemann zeta-function, maybe the best-known and most important function in mathematics. Its zeros, i.e., the complex numbers where the zeta-function simply yields the value 0 as its outcome, are known to describe the still all mysterious pattern of how the prime numbers are distributed among all natural numbers. The very location of these zeros, however, is yet to be discovered.It is hence not surprising that trying to understand the values of an L-function (at least at some special arguments) is a very important goal in modern number theory. This research-project was devoted to this intriguing endeavour and its associated tasks. We succeeded in basically all objectives, which we had formulated in the projects application, exploring the nature of L-functions and the arithmetic of what one calls automorphic forms. The numerous results of this research-project hence shade some new light on the still so much challenging puzzle, which number theory provides us.