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

• 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
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000); Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2013: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2012: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000)
• 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 / 関数等式

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.

Research Products

• [Journal Article] (GL(1)×SO(q),V(q))上の超関数の保型対 （佐藤文広氏との共同研究） (2015)
  • Author(s): 宮崎直
  • Journal Title: 数理解析研究所講究録 Volume: 1934 Pages: 90-103
  • Open Access
• [Journal Article] Automorphic pairs of distributions and its application to explicit constructions of Maass forms (2015)
  • Author(s): 杉山和成
  • Journal Title: 数理解析研究所講究録 Volume: 1934 Pages: 83-89
  • Open Access
• [Journal Article] Local functional equations associated with decomposable graphs (2013)
  • Author(s): Fumihiro Sato
  • Journal Title: Josai Mathematical Monographs Volume: 6 Pages: 59-69
  • NAID: 120005520756
  • Peer Reviewed
• [Journal Article] Representations of Clifford algebras and local functional equation (2012)
  • Author(s): Fumihiro Sato and Takeyoshi Kogiso
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: 36 Pages: 53-66
  • Peer Reviewed
• [Presentation] Integrals of Wishart-Siegel type and local functional equations (2014)
  • Author(s): 佐藤文広
  • Organizer: Analysis, Geometry and Representations on Lie groups and Homogeneous Spaces
  • Place of Presentation: Ryad Mogador Opera (Marrakech, Morocco)
  • Year and Date: 2014-12-09
  • Invited
• [Presentation] Automorphic pairs of distributions on prehomogeneous vector spaces and zeta functions (2014)
  • Author(s): 佐藤文広
  • Organizer: Prehomogeneous Vector Spaces and Related Topics (JSPS-CNRS joint seminar）
  • Place of Presentation: 立教大学(東京都豊島区)
  • Year and Date: 2014-09-04
  • Invited
• [Presentation] 超関数の保型形式とMaass形式 (2014)
  • Author(s): 杉山和成
  • Organizer: 大阪大学整数論＆保型形式セミナー
  • Place of Presentation: 大阪大学理学部(大阪府豊中市）
  • Year and Date: 2014-06-20
  • Invited
• [Presentation] Spherical homogeneous spaces -Definition, Classification, Weyl group-
  • Author(s): 佐藤文広
  • Organizer: 16-th Autumn Workshop in Number Theory
  • Place of Presentation: 白馬ハイマウントホテル
  • Invited
• [Presentation] Zeta functions of quadratic mappings
  • Author(s): 佐藤文広
  • Organizer: Modular functions and quadratic forms -Number theoretic delights-
  • Place of Presentation: 大阪大学中之島センター
  • Invited
• [Presentation] 局所関数等式、超関数の保型対、ゼータ関数
  • Author(s): 佐藤文広
  • Organizer: 2013年度表現論シンポジウム
  • Place of Presentation: マホロバ・マインズ三浦
  • Invited
• [Presentation] On non-prehomogeneous local functional equation
  • Author(s): 佐藤 文広
  • Organizer: Kyoto Conference on automorphic forms
  • Place of Presentation: 京都大学理学研究科
  • Invited
• [Presentation] 分解可能グラフに付随する局所関数等式
  • Author(s): 佐藤 文広
  • Organizer: 2012年度城西大学理学研究科ワークショップ、Representation theory of algebraic groups and related topics
  • Place of Presentation: 城西大学紀尾井町キャンパス
  • Invited