2007 Fiscal Year Final Research Report Summary
Relation between automorphic forms and zeta functions associated with prehomogeneous vector spaces
| Project/Area Number |
16340012
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Rikkyo University |
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Principal Investigator |
SATO Fumihiro Rikkyo University, College of Science, Professor (20120884)
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| Co-Investigator(Kenkyū-buntansha) |
HIGA Tatsuo Rikkyo Universtiy, College of Science, Professor (00150748)
KAKEI Saburo Rikkyo Universtiy, College of Science, Associate Professor (60318798)
IBUKIYAMA Tomoyoshi Osaka University, Graduate School of Science, Professor (60011722)
HIRONAKA Yumiko Waseda University, School of Education, Professor (10153652)
KIMURA Tatsuo Tsukuba University, Graduate School of pure and applied Sciences, Professor (30022726)
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| Project Period (FY) |
2004 – 2007
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| Keywords | prehomogeneous vector space / zeta function / automorphic form / Eisenstein series / Clifford algebra / spherical function |
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| Research Abstract |
The main problems we investigated in this research project are
(1) To identify the zeta functions associated with prehomogeneous vector spaces with some kind of zeta functions attached to automorphic forms
(2) To construct a theory of(local) functional equations which is not covered by the theory of prenomogeneous vector spaces. The results we obtained are as follows :
(1) According to the classification theory due to Sato and Kimura, irreducible regular prehomogeneous vector spaces are classified into 5 series of classical type and 24 spaces of sporadic type. We identified the zeta functions associated with 4 series of prehomogeneous vector spaces of classical type with the standard L-functions or Koecher-Maass zeta functions of certain real analytic Eisenstein series. One of the results which are necessary for the proof of these results is a new integral representation of the Siegel series (= p-part of the Fourier coefficients of Eisenstein series). As another application of the new integral representation, we proved a formula which connects the Siegel series to spherical functions on a p-adic semisimple symmetric space of the orthogonal groups.
(2) We proved that, given a pair of homogeneous polynomials on Ra satisfying a local functional equation and a pair of nondegenerate dual quadratic mappings of R^m to R^n, then, the pull backs of the polynomials by the quadratic mappings also satisfy a local functional equation. This generalizes a result due to Faraut-Koranyi-Clerc and we can construct many examples of functional equations which are not covered by the theory of prehomogeneous vector spaces. We also classified nondegenerate dual quadratic mappings over quadratic spaces and proved that such quadratic mappings are in one to one correspondence to representations of a tensor product of 2 Clifford algebras.
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Research Products
(86 results)
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[Presentation] Dimensions of Siegel Modular Forms2008
Author(s)
Tomoyoshi Ibukiyama
Organizer
22-th Annual Workshop on Automorphic Forms and Related Topics
Place of Presentation
Blocker Hall, Texas A & M University, USA
Year and Date
2008-03-10
Description
「研究成果報告書概要(欧文)」より
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