Zeta functions associated with automorphic distributions and an analytic number theory of quartic forms

2014 Fiscal Year Final Research Report

• Project/Area Number: 24540029
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Rikkyo University
• Principal Investigator: SATO Fumihiro 立教大学, 理学部, 教授 (20120884)
• Research Collaborator: SUGIYAMA Kazunari 千葉工業大学, 情報科学部, 准教授 (90375395) ; MIYAZAKI Tadashi 北里大学, 一般教育部, 講師 ; KOGISO Takeyoshi 城西大学, 理学部, 教授 (20282296)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: 概均質ベクトル空間 / ゼータ関数 / 保型超関数 / アイゼンシュタイン級数 / 二次写像

Outline of Final Research Achievements

A prehomogeneous vector space of commutative parabolic type is determined by a certain pair of a simple algebraic group G and its maximal parabolic subgroup P, We studied a relation between automorphic pairs of distributions on a prehomogeneous vector space of this type and (quasi-) automorphic distribution vectors belonging to degenerate principal series representations of G using associated zeta functions. The case G = SL(2) was analyzed in detail and as an application we obtained a Weil-tye converse theorem for Maass wave forms. We applied the theory of automorphic pairs to quadratic mappings and constructed zeta functions associated to Clifford quartic forms. It is a conjecture that the zeta functions coincide with the Koecher-Maass zeta functions of Eisenstein series of orthogonal groups. We obtained partial affirmative results on the conjecture.

Free Research Field

代数学、とくに代数群の整数論