Periods of automorphic forms and special values of L-functions

• Project/Area Number: 26287003
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Kyoto University
• Principal Investigator: Ichino Atsushi 京都大学, 理学研究科, 准教授 (40347480)
• Co-Investigator(Kenkyū-buntansha): 池田 保 京都大学, 理学研究科, 教授 (20211716)
• Project Period (FY): 2014-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥15,990,000 (Direct Cost: ¥12,300,000、Indirect Cost: ¥3,690,000); Fiscal Year 2018: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000); Fiscal Year 2017: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000); Fiscal Year 2016: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000); Fiscal Year 2015: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000); Fiscal Year 2014: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
• Keywords: 周期 / 保型表現 / テータ対応 / 保形表現 / 形式次数

Outline of Final Research Achievements

In this research, we have studied periods of automorphic forms on quaternion algebras. In particular, we aimed at an explicit realization of the functoriality called the Jacquet-Langlands correspondence in terms of absolute Hodge cycles.
Also, we have studied the automorphic discrete spectrum of metaplectic groups. In particular, we aimed at a generalization of the Arthur classification for classical groups and the theory of half-integral weight modular forms by Shimura and Waldspurger.

Academic Significance and Societal Importance of the Research Achievements

Jacquet-Langlands対応の絶対Hodgeサイクルによる具体的な実現に向けて、大きな成果を上げることができた。この成果は30年以上の間停滞していた分野に著しい進展をもたらす、学術的意義の高いものである。またメタプレクティック群の保型離散スペクトルについて、緩増加部分を精密に記述することに成功した。この成果は、重さ半整数保型形式の理論を一新するものである。

Research Products

• [Int'l Joint Research] ミシガン大学(米国)
• [Int'l Joint Research] シンガポール国立大学(シンガポール)
• [Int'l Joint Research] ウィーン大学(オーストリア)
• [Int'l Joint Research] 国立台湾大学(台湾)
• [Int'l Joint Research] ミシガン大学(米国)
• [Int'l Joint Research] シンガポール国立大学(シンガポール)
• [Int'l Joint Research] 国立台湾大学(台湾)
• [Int'l Joint Research] ジュシュー数学研究所(フランス)
• [Journal Article] On the irreducibility of some induced representations of real reductive Lie groups (2019)
  • Author(s): Wee Teck Gan、Atsushi Ichino
  • Journal Title: Tunisian Journal of Mathematics Volume: 1 Issue: 1 Pages: 73-107
  • DOI: 10.2140/tunis.2019.1.73
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] The Shimura?Waldspurger correspondence for Mp_2<em>n</em> (2018)
  • Author(s): Wee Teck Gan、Atsushi Ichino
  • Journal Title: Annals of Mathematics Volume: 188 Issue: 3 Pages: 965-965
  • DOI: 10.4007/annals.2018.188.3.5
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] On the formal degrees of square-integrable representations of odd special orthogonal and metaplectic groups (2017)
  • Author(s): Atsushi Ichino, Erez Lapid, Zhengyu Mao
  • Journal Title: Duke Mathematical Journal Volume: 166 Issue: 7 Pages: 1301-1348
  • DOI: 10.1215/00127094-0000001x
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] The Gross-Prasad conjecture and local theta correspondence (2016)
  • Author(s): Wee Teck Gan, Atsushi Ichino
  • Journal Title: Inventiones mathematicae Volume: 206 Issue: 3 Pages: 705-799
  • DOI: 10.1007/s00222-016-0662-8
  • NAID: 130008004975
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Periods of automorphic forms: the case of (GLn+1 × GLn, GLn) (2015)
  • Author(s): Atsushi Ichino, Shunsuke Yamana
  • Journal Title: Compositio Mathematica Volume: 151 Issue: 4 Pages: 665-712
  • DOI: 10.1112/s0010437x14007362
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Formal degrees and local theta correspondence (2014)
  • Author(s): Wee Teck Gan, Atsushi Ichino
  • Journal Title: Invent. Math. Volume: 195 Issue: 3 Pages: 509-672
  • DOI: 10.1007/s00222-013-0460-5
  • Peer Reviewed
• [Presentation] Hodge classes and the Jacquet-Langlands correspondence (2018)
  • Author(s): Atsushi Ichino
  • Organizer: Simons Symposium, Relative Trace Formulas
  • Int'l Joint Research / Invited
• [Presentation] Hodge classes and the Jacquet-Langlands correspondence (2018)
  • Author(s): Atsushi Ichino
  • Organizer: Langlands functoriality and Trace formula, including relative aspects
  • Int'l Joint Research / Invited
• [Presentation] The Shimura-Waldspurger correspondence for Mp(2n) (2017)
  • Author(s): Atsushi Ichino
  • Organizer: New directions in automorphic forms and L-functions
  • Int'l Joint Research / Invited
• [Presentation] Hodge classes and the Jacquet-Langlands correspondence (2017)
  • Author(s): Atsushi Ichino
  • Organizer: Special values of automorphic L-functions, periods of automorphic forms and related topics
  • Int'l Joint Research / Invited
• [Presentation] 保型形式と分岐則 (2017)
  • Author(s): 市野篤史
  • Organizer: 表現論シンポジウム
  • Invited
• [Presentation] The automorphic discrete spectrum of Mp(2n) (2016)
  • Author(s): Atsushi Ichino
  • Organizer: Relative Trace Formula, Periods, L-Functions and Harmonic Analysis
  • Place of Presentation: CIRM
  • Year and Date: 2016-05-26
  • Int'l Joint Research / Invited
• [Presentation] The automorphic discrete spectrum of Mp(2n) (2016)
  • Author(s): Atsushi Ichino
  • Organizer: New Developments in Representation Theory
  • Place of Presentation: National University of Singapore
  • Year and Date: 2016-03-17
  • Int'l Joint Research / Invited
• [Presentation] The automorphic discrete spectrum for Mp(2n) (2016)
  • Author(s): Atsushi Ichino
  • Organizer: Conference on Algebraic Number Theory
  • Place of Presentation: Tsinghua Sanya International Mathematics Forum
  • Year and Date: 2016-01-22
  • Int'l Joint Research / Invited
• [Presentation] The Gross-Prasad conjecture and local theta correspondence (2015)
  • Author(s): Atsushi Ichino
  • Organizer: Arithmetic and Algebraic Geometry 2015
  • Place of Presentation: 東京大学
  • Year and Date: 2015-01-30
  • Invited
• [Presentation] The Gross-Prasad conjecture and local theta correspondence (2014)
  • Author(s): Atsushi Ichino
  • Organizer: Gan-Gross-Prasad
  • Place of Presentation: Paris
  • Year and Date: 2014-06-23 – 2014-06-24
  • Invited