The Project
The Project “New Aspects of Automorphic and Smooth-Automorphic Forms” is based at the Faculty of Mathematics of the University of Vienna. The project will run from February 2024 until January 2028.
Research Context
Central to our research are the qualitative and quantitative questions surrounding the existence of (smooth-)automorphic forms. The existence of cuspidal automorphic representations with prescribed local behavior remains largely elusive, even for well-studied groups like GL(n) and its outer forms. Moreover, the extent to which the theory of Eisenstein series, as developed by Franke, can be extended to accommodate smooth-automorphic forms—that is, whether every smooth-automorphic form is a finite sum of derivatives of residues of smooth-automorphic Eisenstein series—remains an open question.
Objectives
This research project seeks to synergistically advance the field of (smooth-)automorphic representation theory by combining recent developments with more classical aspects. Our primary goals include:
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Growth conditions for the dimension of spaces of cusp forms: Developing new approaches to understand the growth behavior of these spaces and applications to rationality of special values of L-functions.
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Constructions of new p-adic automorphic L-functions and the “trivial zero conjecture”.
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Analytic properties of smooth-automorphic Eisenstein series: Investigating the analytic properties of these series, which are essential for understanding the structure of automorphic forms.
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A Paley-Wiener theorem for the adelic Schwartz space: Establishing a Paley-Wiener theorem, a fundamental tool in harmonic analysis, for this important space.
Approach
We will employ a combination of existing techniques, such as those presented in the PI’s recent book on smooth-automorphic forms, and develop novel approaches to address the aforementioned objectives. The PI and his international collaborators will focus on the latter three themes, while the first theme will be the subject of a PhD thesis.
Innovation
The results of this research program will be entirely original and contribute significantly to the forefront of automorphic forms research.
Primary Researchers Involved
H. Grobner (PI), R. Beuzart-Plessis, N. Grbac, M. Harris, J. Lin, A. Raghuram, S. Žunar; A. Betina; M. Ez-Zarraq.
Find out more about the team here.
