Automorphe Modelle und L-Werte für globale Algebren

The study of special values of L-functions is a very active and central topic in modern number theory. While there is an extensive literature for automorphic L-functions of GL(N) over number fields F, the much more general case of GL(N) over a central division F-algebra D is yet far from being explored in this generality.
This research project aims to start a new wave of research on special values of L-functions, exploring indepth the fundamental case of GL(N)/ D over a central division algebra D over a number field F.
Our final results will be broad and systematic extensions of various important, recent results in the field, hence also (a) laying the foundation for a further, more conceptual analysis of special values and (b) providing a deeper insight in the special, split case of GL(N)/ F itself.
Our key-method will be a thorough analysis of various model-spaces of automorphic representations. With this guideline the main focus of our research-project will be on three particular aspects:
(1) WHITTAKER MODELS: We will establish a theory of Whittaker-periods for isobaric automorphic representations over totally real fields F and launch a series of new results for special values of Rankin-Selberg L-functions of GL(N)xGL(M) over F.
(2) SHALIKA MODELS: We will lay the foundations of a theory of Shalika-models for GL(N)/ D, generalizing the well-known concept from the split case (and the few known non-split cases). This will have a variety of applications to residues of standard L-functions of GL(N) in the spirit of Dirichlet’s Class Number Formula.
(3) COHOMOLOGICAL MODELS: The PI and his team will explore cohomological models of residual automorphic representations of GL(N)/ D, and hence establish an indispensable tool for the structural analysis of the arithmetic of residual representations. Our results will complement Borel’s famous result on
the injectivity of cuspidal cohomology in the space of automorphic cohomology.
The PI has actively contributed to this area of research over the last couple of years, proving rationalityresults for special L-values of automorphic representations in many cases. As an example for the PI’s contribution, our projected research sketched in (1) above is a profound extension of the PI’s recent work with Harris and Lin in the CM-case and hence should be considered as the first step in showing Deligne’s conjecture on critical values of motivic L-functions for tensor-products of motives of arbitrary rank in the
totally real case.
The funding of this project shall be used to employ two of the PI’s new PhD-students, who will carry out large parts of the above project. The PI is convinced that their results will be of interest to most researches on this topic and an excellent preparation for a career in research.