Let D be a central simple division algebra over a number field k of index d. The group G`=GL(n)/D is a connected
reductive algebraic group over k and an inner form of the split general linear group G=GL(nd)/k. To a cuspidal
automorphic representation p of G` one can associate an L-function L(s, p). This function encodes almost the entire
arithmetic information of p, e.g., even in the easiest case n=d=1 and the associated L-function is L(s,1)= p^(-
s/2)G(s/2)(s), where (s) is the Riemannian -function. The question, whether the special values of an automorphic
L-function are non-zero or even rational numbers is therefore of highest arithmetic interest.
In my project I plan to establish new rationality results of automorphic L-functions attached to cuspidal
automorphic representation of the group G`. These results can be viewed as generalizations of rationality results of
automorphic L-functions of the split group G due to Harder, Mahnkopf and Raghuram-Shahidi. As a first step, I
plan to generalize (together with Prof. Raghuram) a well-known theorem of Clozel and Waldspurger on regular
algebraic cuspidal automorphic representations from the special cases d=1 (Clozel) and d=2, n=1 (Waldspurger) to
general n and d: If p is a regular, algebraic cuspidal automorphic representation of G` then I expect that its finite
part p f is defined over a number field and that for every automorphism s of the field of complex numbers the s-
twist of p is again a regular, algebraic cuspidal automorphic representation. The key-tools for that should be the
theory of degenerate Whittaker models (which should replace the lack of multiplicity one for new-vectors) and the
recently established global Jacquet-Langlands correspondence of Badulescu.
The first major part of my project will then be to define periods p(p) for regular, algebraic cuspidal automorphic
representations p of G`, analogously to the split case, which was treated in the works of Harder, Mahnkopf and
Raghuram-Shahidi. These periods relate rational structures on p f , cuspidal cohomology and the Whittaker model of
p f . As an immediate consequence of this one can deduce a rationality result for algebraic Hecke characters. Prof.
Raghuram and I already planed to work on these questions together.
Further, I then want to use these periods, in order to obtain rationality results for L-functions attached to cuspidal
automorphic representations of G`. As in Mahnkopf`s work, one of the key-ingredients for this will be to obtain an
appropriate zeta-integral-description of these special values, which itself has a cohomological interpretation as
Poincaré-duality. This is a difficult taks and requires a precise understanding of the cohomological CAP-
representations of G`. Therefore, I also plan to alternatively use a method of Harder, in order to obtain rationality-
results on special values of automorphic L-functions. It consists in analyzing the constant term of holomorphic
Eisenstein-classes with respect to maximal-parabolic subgroups. In the special case n=d=2 and k being the field of
rational numbers, I expect to be able to reprove results of Hida in a much shorter way and give new insights in
them.
Among the main motivations to study rationality results of automorphic L-functions attached to cuspidal
automorphic representation of G` one has to name the conjectures of Delinge and the conjectures of Gross-Prasad.
Recently, spectacular progress was made on the local version of the Gross-Prasad conjecture by Waldspurger and
Moeglin-Waldspurger in Paris. Among the most urgent open problems in rationality results and the Gross-Prasad
conjectures, one has to face the question if certain archimedean Rankin-Selbgerg integrals vanish at 1/2. I hope to
make some progress on this problem during my stay in Paris, where I plan to spend the second year of my project.
Back in Vienna I expect fruitful scientific interactions with Mahnkopf, who significantly contributed to the field of
rationality results and Schwermer, who is a leading expert in the theory of cuspidal and Eisenstein cohomology.