Projektdetails
Abstract
Wider research context:
The theory of automorphic forms is one of the pillars of the Langlands program. Fundamental results of Franke imply that every automorphic form of a reductive group is the finite sum of derivatives of residues of Eisenstein series. Whereas very recent exciting developments on the refined GGP-conjecture underline the necessity of following Bernstein-Casselman-Wallach’s approach, which suggests to extend the theory of automorphic forms to include also those smooth functions, to be called smooth-automorphic forms, which are of uniform moderate growth, but not necessarily K-finite.
In both of these aspects we encounter qualitative as well as quantitative problems on the existence of (smooth-)automorphic forms. To remark on two of the most fundamental related questions, it is still largely unclear, if cuspidal automorphic reps with a prescribed local behavior exits (even for well-studied groups, like GL(n) and its outer forms). While it is also unknown, if the theory of Eisenstein series, as pushed by Franke to a certain limit, allows a “smooth-automorphic” analogue, i.e. whether every smooth-automorphic form is a finite sum of derivatives of residues of smooth-automorphic Eisenstein series – i.e. Eisenstein series, which are attached to cusp forms that are not necessarily K-finite.
Objectives:
This research project aims to bring several exciting recent developments in (smooth)-automorphic representation theory and some of its more classical aspects together – and so to push both of them in a synergetic way to another level. We will suggest several clear strategies for a profound, detailed analysis of qualitative as well as quantitative problems in automorphic and smooth-automorphic forms: Among them, we will provide new results and approaches to
(i) Growth conditions for the dimension of spaces of cusp forms,
(ii) Analytic properties of smooth-automorphic Eisenstein series and
(iii) A Paley-Wiener theorem for the adelic Schwartz space.
Approach:
The PI and his collaborators will combine the latest available techniques (e.g. such as developed in the PI’s book on smooth-automorphic forms, which has just appeared and which is the first book on this topic) and present entirely new, original approaches to the themes of (i) – (iii): The latter two shall be worked on by the PI and his international collaborators (see below), potentially joined by one of the PI’s new postdocs, which shall be hired by the fundings of this project. The first theme shall be the challenging topic of the PhD-thesis of one of the PI’s new students.
Innovation:
All results of this research-program will be entirely novel and original. They will add as new and strong contributions to the research-front of the field of automorphic forms.
Primary researchers involved:
H. Grobner (PI), his int. collaboration partners R. Beuzart-Plessis, N. Grbac, M. Harris, and S. Zunar, and one of the PI's new postdocs and one new PhD-student.
The theory of automorphic forms is one of the pillars of the Langlands program. Fundamental results of Franke imply that every automorphic form of a reductive group is the finite sum of derivatives of residues of Eisenstein series. Whereas very recent exciting developments on the refined GGP-conjecture underline the necessity of following Bernstein-Casselman-Wallach’s approach, which suggests to extend the theory of automorphic forms to include also those smooth functions, to be called smooth-automorphic forms, which are of uniform moderate growth, but not necessarily K-finite.
In both of these aspects we encounter qualitative as well as quantitative problems on the existence of (smooth-)automorphic forms. To remark on two of the most fundamental related questions, it is still largely unclear, if cuspidal automorphic reps with a prescribed local behavior exits (even for well-studied groups, like GL(n) and its outer forms). While it is also unknown, if the theory of Eisenstein series, as pushed by Franke to a certain limit, allows a “smooth-automorphic” analogue, i.e. whether every smooth-automorphic form is a finite sum of derivatives of residues of smooth-automorphic Eisenstein series – i.e. Eisenstein series, which are attached to cusp forms that are not necessarily K-finite.
Objectives:
This research project aims to bring several exciting recent developments in (smooth)-automorphic representation theory and some of its more classical aspects together – and so to push both of them in a synergetic way to another level. We will suggest several clear strategies for a profound, detailed analysis of qualitative as well as quantitative problems in automorphic and smooth-automorphic forms: Among them, we will provide new results and approaches to
(i) Growth conditions for the dimension of spaces of cusp forms,
(ii) Analytic properties of smooth-automorphic Eisenstein series and
(iii) A Paley-Wiener theorem for the adelic Schwartz space.
Approach:
The PI and his collaborators will combine the latest available techniques (e.g. such as developed in the PI’s book on smooth-automorphic forms, which has just appeared and which is the first book on this topic) and present entirely new, original approaches to the themes of (i) – (iii): The latter two shall be worked on by the PI and his international collaborators (see below), potentially joined by one of the PI’s new postdocs, which shall be hired by the fundings of this project. The first theme shall be the challenging topic of the PhD-thesis of one of the PI’s new students.
Innovation:
All results of this research-program will be entirely novel and original. They will add as new and strong contributions to the research-front of the field of automorphic forms.
Primary researchers involved:
H. Grobner (PI), his int. collaboration partners R. Beuzart-Plessis, N. Grbac, M. Harris, and S. Zunar, and one of the PI's new postdocs and one new PhD-student.
| Kurztitel | Aspekte autom. und glatt-autom. Formen |
|---|---|
| Status | Laufend |
| Tatsächlicher Beginn/ -es Ende | 1/02/24 → 31/01/28 |