Explicit formulas of special values of automorphic L-functions and their applications

• Project/Area Number: 17K14166
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Kobe University
• Principal Investigator: Morimoto Kazuki 神戸大学, 理学研究科, 准教授 (20725254)
• Project Period (FY): 2017-04-01 – 2024-03-31
• Project Status: Completed (Fiscal Year 2023)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 保型L関数 / 保型形式の周期 / p進代数群の表現論 / 保型L関数の特殊値 / 保型L函数 / L函数 / 代数学

Outline of Final Research Achievements

I studied a relationship between special values of L-functions and periods automorphic forms. As a consequence, I proved the Gross-Prasad conjecture and the refined Gross-Prasad conjecture in the case of (SO(5), SO(2)). Also, I proved Lapid-Mao conjecture, which conjectures Ichino-Ikeda type formula of Whittaker periods, for any irreducible cuspidal automorphic representation when it has Whittaker periods.

Academic Significance and Societal Importance of the Research Achievements

現代数論の研究において、ゼータ関数一般化であるL関数は最も重要な研究対象の一つだと考えれている。BSD予想で期待されているように、L関数の特殊値は多くの数論的情報を含むと考えられている。本研究では、保型表現に付随するL関数に関して特殊値の明示公式を考察した。特に、ジーゲル保型形式に付随するL関数の中心値の明示公式に関するBoechrer予想を解決することが出来た。楕円保型形式が楕円曲線と対応(志村-谷山予想)したように、ジーゲル保型形式はアーベル曲面との対応が期待されており、この対応が確立されれば、本研究で得られた公式の様々な数論的が期待される。

Research Products

• [Int'l Joint Research] Weizmann Institute of Science(イスラエル)
• [Int'l Joint Research] Rutgers University(米国)
• [Int'l Joint Research] Tel-Aviv University(イスラエル)
• [Int'l Joint Research] Tel Aviv University(イスラエル)
• [Journal Article] ON A CERTAIN LOCAL IDENTITY FOR LAPID?MAO’S CONJECTURE AND FORMAL DEGREE CONJECTURE : EVEN UNITARY GROUP CASE (2022)
  • Author(s): Morimoto Kazuki
  • Journal Title: Journal of the Institute of Mathematics of Jussieu Volume: 4 Issue: 4 Pages: 1107-1161
  • DOI: 10.1017/s1474748020000523
  • Peer Reviewed
• [Journal Article] Refined global Gross-Prasad conjecture on special Bessel periods and Boecherer’s conjecture (2020)
  • Author(s): Furusawa Masaaki、Morimoto Kazuki
  • Journal Title: Journal of the European Mathematical Society Volume: 23 Issue: 4 Pages: 1295-1331
  • DOI: 10.4171/jems/1034
  • Peer Reviewed
• [Journal Article] On L-functions for U_{2n+1}×Res_{E/F}GL_m (m>n) (2020)
  • Author(s): Morimoto Kazuki、Soudry David
  • Journal Title: Journal of Number Theory Volume: 216 Pages: 83-141
  • DOI: 10.1016/j.jnt.2020.05.002
  • Peer Reviewed
• [Journal Article] Refined global Gross-Prasad conjecture on special Bessel periods and Boecherer's conjecture (2019)
  • Author(s): Masaaki Furusawa, Kazuki Morimoto
  • Journal Title: Journal of the European Mathematical Society Volume: 印刷中
  • Peer Reviewed
• [Journal Article] Model transition for representations of unitary type (2018)
  • Author(s): Kazuki Morimoto
  • Journal Title: International Mathematics Research Notices Volume: 印刷中 Pages: 1112-1203
  • DOI: 10.1093/imrn/rny048
  • Peer Reviewed
• [Journal Article] On tensor product L-functions for quaternion unitary groups and GL(2) over totally real number fields: mixed weight cases (2018)
  • Author(s): Kazuki Morimoto
  • Journal Title: Advances in Mathematics Volume: 337 Pages: 317-362
  • DOI: 10.1016/j.aim.2018.08.009
  • Peer Reviewed
• [Presentation] On Ichino-Ikeda type formula of Whittaker periods for unitary groups (2023)
  • Author(s): Kazuki Morimoto
  • Organizer: Automorphic Forms and L-functions of higher rank, Queen Mary University of Londnon
  • Int'l Joint Research / Invited
• [Presentation] On Ichino-Ikeda type formula of Bessel periods for (U(2n), U(1)) and (GL(2n), GL(1)) (2023)
  • Author(s): Kazuki Morimoto
  • Organizer: RIMS共同研究(公開型)「保型表現の解析的・数論的研究」
  • Int'l Joint Research / Invited
• [Presentation] On Ichino-Ikeda type formula of Whittaker periods for even unitary groups (2022)
  • Author(s): Kazuki Morimoto
  • Organizer: 第9回京都保型形式研究集会
  • Int'l Joint Research / Invited
• [Presentation] (SO(5), SO(2))のBessel周期の市野-池田型の公式と一般化されたBoecherer予想 (2021)
  • Author(s): 森本 和輝
  • Organizer: 早稲田大学整数論セミナー
  • Invited
• [Presentation] On Ichino-Ikeda type formula of Whittaker periods for even unitary groups (2021)
  • Author(s): 森本 和輝
  • Organizer: National University of Singapore, Representation Theory and Number Theory
  • Int'l Joint Research / Invited
• [Presentation] (SO(5), SO(2)) の Bessel 周期の市野-池田型公式について (2021)
  • Author(s): 森本 和輝
  • Organizer: RIMS研究集会「保型形式,保型表現, ガロア表現とその周辺」
  • Int'l Joint Research
• [Presentation] On Gan-Gross-Prasad conjecture for (U(2n), U(1)) and (SO(5), SO(2)) (2020)
  • Author(s): Kazuki Morimoto
  • Organizer: RIMS研究集会「保型形式とL関数の解析的, 幾何的, p進的研究」
  • Int'l Joint Research / Invited
• [Presentation] On Gan-Gross-Prasad conjecture for (U(2n), U(1)) (2019)
  • Author(s): Kazuki Morimoto
  • Organizer: Representation Theory Workshop 2019
  • Int'l Joint Research / Invited
• [Presentation] On Gross-Prasad conjecture for (SO(2n+1), SO(2)) (2019)
  • Author(s): Kazuki Morimoto
  • Organizer: KIAS seminar
  • Int'l Joint Research / Invited
• [Presentation] On Gan-Gross-Prasad conjecture for (U(2n), U(1)) and (SO(5), SO(2)) (2019)
  • Author(s): Kazuki Morimoto
  • Organizer: 南大阪保型表現セミナー
  • Int'l Joint Research / Invited
• [Presentation] 対称積L関数の特殊値とRamakrishnan-Shahidi liftの周期関係式について (2018)
  • Author(s): 森本和輝
  • Organizer: 京都大学数学教室談話会
  • Invited
• [Presentation] U(2n) の Whittaker 周期の明示公式について (2018)
  • Author(s): 森本和輝
  • Organizer: 九州代数的整数論2018
  • Invited
• [Presentation] On a certain local identity for Lapid-Mao's conjecture in the even unitary group case (2017)
  • Author(s): 森本和輝
  • Organizer: 神戸整数論集会2017
  • Int'l Joint Research / Invited
• [Presentation] 次数 2 の Siegel モジュラー形式の L 函数の中心値の明示公式について (2017)
  • Author(s): 森本和輝
  • Organizer: 東京電機大学数学講演会
  • Invited
• [Presentation] On a certain local identity for an explicit formula of Whittaker periods on the even unitary group (2017)
  • Author(s): 森本和輝
  • Organizer: Workshop ``Special values of automorphic L-functions, periods of automorphic forms and related topics''
  • Int'l Joint Research
• [Funded Workshop] RIMS conference ``Automorphic form, automorphic L-functions and related topics" (2022)
• [Funded Workshop] Workshop ``Special values of automorphic L-functions, periods of automorphic forms and related topics'' (2017)