Strategic research to construct motivic units using new symmetry

• Project/Area Number: 18H05233
• Research Category: Grant-in-Aid for Scientific Research (S)
• Allocation Type: Single-year Grants
• Review Section: Broad Section B
• Research Institution: Keio University
• Principal Investigator: 坂内 健一 慶應義塾大学, 理工学部(矢上), 教授 (90343201)
• Co-Investigator(Kenkyū-buntansha): 志甫 淳 東京大学, 大学院数理科学研究科, 教授 (30292204) ; 寺杣 友秀 法政大学, 理工学部, 教授 (50192654) ; 勝良 健史 慶應義塾大学, 理工学部(矢上), 教授 (50513298) ; 小林 真一 九州大学, 数理学研究院, 教授 (80362226) ; 安田 正大 北海道大学, 理学研究院, 教授 (90346065) ; 山本 修司 慶應義塾大学, 理工学研究科(矢上), 准教授 (20635370)
• Project Period (FY): 2018-06-11 – 2023-03-31
• Project Status: Completed (Fiscal Year 2022)
• Budget Amount: ¥119,470,000 (Direct Cost: ¥91,900,000、Indirect Cost: ¥27,570,000); Fiscal Year 2022: ¥24,050,000 (Direct Cost: ¥18,500,000、Indirect Cost: ¥5,550,000); Fiscal Year 2021: ¥24,050,000 (Direct Cost: ¥18,500,000、Indirect Cost: ¥5,550,000); Fiscal Year 2020: ¥24,050,000 (Direct Cost: ¥18,500,000、Indirect Cost: ¥5,550,000); Fiscal Year 2019: ¥24,050,000 (Direct Cost: ¥18,500,000、Indirect Cost: ¥5,550,000); Fiscal Year 2018: ¥23,270,000 (Direct Cost: ¥17,900,000、Indirect Cost: ¥5,370,000)
• Keywords: 数論幾何 / L関数 / 総実代数体 / ポリログ / プレクティック構造 / 同変性 / 整数論 / 代数トーラス / L関数の特殊値 / 数論幾何学

Outline of Annual Research Achievements

本研究の目的は、代表者が新しく注目した総実代数多様体に付随する「代数トーラス」と呼ばれる高次元の代数多様体に対して、総実代数体の単数群による同変性を考えて、この場合に新たに「ポリログ」を構成し、このポリログのHodge実現やp進実現などを組織的に研究することである。総実代数体に付随する代数トーラス上の新谷生成類を等分点に制限すると、Lerchゼータ関数と呼ばれる関数の臨界値が与えられる。この事実を用いて、同変ポリログと新谷生成類との関係から、プレクティック混合Hodge構造を用いて同変ポリログを等分点に制限すると、Lerchゼータ関数の非臨界値を与えると予想できる。今年度はこの予想を具体的に定式化することに従事した。有理数体の場合にBeilinson-Deligneが証明したことの自然な総実代数体版となっており、他の状況証拠的な各種予想との整合性が極めて高い。この予想を仮定すると、総実代数体のHecke L関数のBeilinson予想が導かれることも証明できた。技術的に困難だったのは、単数群の作用による同変Deligne-Beilinsonコホモロジーを定義するところであった。Hodge加群の理論などはある程度整備されているために、対応する同変コホモロジーの理論を作り上げることは簡単であると誤認していたが、実際、関手性が十分保障されておらず、工夫が必要であった。単数群の作用を表す図式を用いて定義する方法を編み出し、無事に論文としてまとめて投稿することに成功した。以上の成果より、総実代数体に付随する代数トーラスの同変ポリログから総実代数体のBeilinson予想を導くための道筋を与えることに成功した。

Research Progress Status

令和4年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

令和4年度が最終年度であるため、記入しない。

Assessment Rating

Interim Assessment Comments (Rating)

A+: In light of the aim of introducing the research area into the research categories, more progress has been made in research than expected.

Research Products

• [Journal Article] Canonical equivariant cohomology classes generating zeta values of totally real fields (2023)
  • Author(s): Kenichi Bannai, Kei Hagihara, Kazuki Yamada, Shuji Yamamoto
  • Journal Title: Transactions of the American Mathematical Society, Series B Volume: 10 Issue: 19 Pages: 613-635
  • DOI: 10.1090/btran/144
  • Peer Reviewed / Open Access
• [Journal Article] On the refined Kaneko?Zagier conjecture for general integer indices (2023)
  • Author(s): Ono Masataka、Yamamoto Shuji
  • Journal Title: Mathematische Zeitschrift Volume: 305 Issue: 1 Pages: 1-13
  • DOI: 10.1007/s00209-023-03318-2
  • Peer Reviewed / Open Access
• [Journal Article] Sum formulas for Schur multiple zeta values (2023)
  • Author(s): Henrik Bachmann, Shin-ya Kadota, Yuta Suzuki, Shuji Yamamoto, Yoshinori Yamasaki
  • Journal Title: Journal of Combinatorial Theory, Series A Volume: 200 Pages: 105781-105781
  • DOI: 10.1016/j.jcta.2023.105781
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Duality of One-variable Multiple Polylogarithms and Their q-analogues (2023)
  • Author(s): YAMAMOTO Shuji
  • Journal Title: Tokyo Journal of Mathematics Volume: 46 Issue: 2 Pages: 301-311
  • DOI: 10.3836/tjm/1502179378
  • Peer Reviewed
• [Journal Article] Quadrature formulas for Bessel polynomials (2023)
  • Author(s): Matsumura Hideki
  • Journal Title: Indagationes Mathematicae Volume: 34 Issue: 3 Pages: 622-636
  • DOI: 10.1016/j.indag.2023.01.001
  • Peer Reviewed
• [Journal Article] Asymptotic behavior of class groups and cyclotomic Iwasawa theory of elliptic curves (2023)
  • Author(s): Toshiro Hiranouchi, Tatsuya Ohshita
  • Journal Title: Journal de theorie des nombres de Bordeaux Volume: 35 Issue: 2 Pages: 591-657
  • DOI: 10.5802/jtnb.1258
  • Peer Reviewed / Open Access
• [Journal Article] p-adic polylogarithms and p-adic Hecke L-functions for totally real fields (2022)
  • Author(s): Kenichi Bannai, Kei Hagihara, Kazuki Yamada, Shuji Yamamoto
  • Journal Title: Journal fur die reine und angewandte Mathematik (Crelles Journal) Volume: 2022 Issue: 791 Pages: 53-87
  • DOI: 10.1515/crelle-2022-0040
  • Peer Reviewed
• [Journal Article] MULTIPLE ZETA FUNCTIONS OF KANEKO-TSUMURA TYPE AND THEIR VALUES AT POSITIVE INTEGERS (2022)
  • Author(s): YAMAMOTO Shuji
  • Journal Title: Kyushu Journal of Mathematics Volume: 76 Issue: 2 Pages: 497-509
  • DOI: 10.2206/kyushujm.76.497
  • ISSN: 1340-6116, 1883-2032
  • Peer Reviewed
• [Journal Article] Counterexamples to the local-global principle for non-singular plane curves and a cubic analogue of Ankeny-Artin-Chowla-Mordell conjecture (2022)
  • Author(s): Hirakawa Yoshinosuke、Shimizu Yosuke
  • Journal Title: Proceedings of the American Mathematical Society Volume: 150 Pages: 1821-1835
  • DOI: 10.1090/proc/15306
  • Peer Reviewed
• [Journal Article] Infinitely many hyperelliptic curves with exactly two rational points (2021)
  • Author(s): Hirakawa Yoshinosuke、Matsumura Hideki
  • Journal Title: Rocky Mountain Journal of Mathematics Volume: 51 Issue: 3 Pages: 883-889
  • DOI: 10.1216/rmj.2021.51.883
  • Peer Reviewed
• [Journal Article] How to calculate the proportion of everywhere locally soluble diagonal hypersurfaces (2021)
  • Author(s): Hirakawa Yoshinosuke、Kanamura Yoshinori
  • Journal Title: International Journal of Number Theory Volume: 17 Issue: 10 Pages: 2361-2377
  • DOI: 10.1142/s1793042121500925
  • Peer Reviewed
• [Journal Article] A generalized regularization theorem and Kawashima's relation for multiple zeta values (2021)
  • Author(s): Masanobu Kaneko, Ce Xu, Shuji Yamamoto
  • Journal Title: Journal of Algebra Volume: 580 Pages: 247-263
  • DOI: 10.1016/j.jalgebra.2021.04.005
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Truncated t-adic symmetric multiple zeta values and double shuffle relations (2021)
  • Author(s): Masataka Ono, Shin-ichiro Seki, Shuji Yamamoto
  • Journal Title: Research in Number Theory Volume: 7 Issue: 1
  • DOI: 10.1007/s40993-021-00241-5
  • Peer Reviewed
• [Journal Article] Multivariable Hoffman-Ihara operators and the operad of formal power series (2020)
  • Author(s): Shuji Yamamoto
  • Journal Title: Journal of Algebra Volume: 556 Pages: 634-648
  • DOI: 10.1016/j.jalgebra.2020.04.006
  • Peer Reviewed
• [Journal Article] Counterexamples to the local-global principle associated with Swinnerton-Dyer's cubic form (2020)
  • Author(s): Hirakawa Yoshinosuke
  • Journal Title: Rocky Mountain Journal of Mathematics Volume: 50 Issue: 6
  • DOI: 10.1216/rmj.2020.50.2097
  • Peer Reviewed
• [Journal Article] The Hodge realization of the polylogarithm on the product of multiplicative groups (2020)
  • Author(s): Bannai Kenichi、Hagihara Kei、Yamada Kazuki、Yamamoto Shuji
  • Journal Title: Mathematische Zeitschrift Volume: 296 Issue: 3-4 Pages: 1787-1817
  • DOI: 10.1007/s00209-020-02483-y
  • Peer Reviewed
• [Journal Article] Category of mixed plectic Hodge structures (2020)
  • Author(s): Bannai Kenichi、Hagihara Kei、Kobayashi Shinichi、Yamada Kazuki、Yamamoto Shuji、Yasuda Seidai
  • Journal Title: Asian Journal of Mathematics Volume: 24 Issue: 1 Pages: 31-76
  • DOI: 10.4310/ajm.2020.v24.n1.a2
  • Peer Reviewed
• [Journal Article] On infiniteness of integral overconvergent de Rham-Witt cohomology modulo torsion (2020)
  • Author(s): Veronika Ertl and Atsushi Shiho
  • Journal Title: Tohoku Mathematical Journal Volume: 72 Issue: 3 Pages: 395-410
  • DOI: 10.2748/tmj/1601085622
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] A unique pair of triangles (2019)
  • Author(s): Hirakawa Yoshinosuke、Matsumura Hideki
  • Journal Title: Journal of Number Theory Volume: 194 Pages: 297-302
  • DOI: 10.1016/j.jnt.2018.07.007
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Presentation] On the Equivariant Polylogarithm Class for the Algebraic Torus associated with a Totally Real Field, (2022)
  • Author(s): Kenichi Bannai
  • Organizer: RIMS Workshop: Algebraic Number Theory and Related Topics
  • Int'l Joint Research / Invited
• [Presentation] 総実代数体のゼータ関数の値と新谷生成類 (2022)
  • Author(s): 山本修司
  • Organizer: 第16回多重ゼータ研究集会＆第58回関西多重ゼータ研究会
  • Invited
• [Presentation] ゼータ値の生成関数とコホモロジー解釈 (2022)
  • Author(s): 山本修司
  • Organizer: 第3回 数学と諸分野の連携に向けた若手数学者交流会
  • Invited
• [Presentation] On the equivariant polylogarithm for the algebraic torus associated to totally real fields (joint with Hohto Bekki, Kei Hagihara, Tatsuya Oshita, Kazuki Yamada, and Shuji Yamamoto) (2021)
  • Author(s): Kenichi Bannai
  • Organizer: Eisenstein Series and Equivariant Cohomology
  • Int'l Joint Research / Invited
• [Presentation] Shintani Generating Class and the Equivariant Polylogarithm for Totally Real Fields (joint with Hohto Bekki, Kei Hagihara, Tatsuya Oshita, Kazuki Yamada, and Shuji Yamamoto) (2021)
  • Author(s): Kenichi Bannai
  • Organizer: Algebra Symposium
  • Int'l Joint Research / Invited
• [Presentation] On the Equivariant Polylogarithm Class for the Algebraic Torus associated with a Totally Real Field (2021)
  • Author(s): Kenichi Bannai
  • Organizer: RIMS Workshop: Algebraic Number Theory and Related Topics, RIMS
  • Int'l Joint Research / Invited
• [Presentation] 整p進コホモロジー理論について， (2021)
  • Author(s): 志甫淳
  • Organizer: 京都大学理学部数学教室談話会
  • Invited
• [Presentation] Shintani generating class and the p -adic polylogarithm for totally real fields (joint work with Kei Hagihara, Kazuki Yamada, and Shuji Yamamoto), (2020)
  • Author(s): Kenichi Bannai
  • Organizer: Seminaire de Geometrie Arithmetique Paris-Pekin-Tokyo
  • Invited
• [Presentation] Shintani Revisited (2019)
  • Author(s): Kenichi Bannai
  • Organizer: Boston University/Keio University Workshop 2019 in Number Theory
  • Int'l Joint Research / Invited
• [Presentation] On the p-adic polylogarithm function for totally real fields (2019)
  • Author(s): Kenichi Bannai
  • Organizer: p-adic Cohomology and Arithmetic Geometry 2019
  • Int'l Joint Research / Invited
• [Presentation] On the Shintani Generating Class of Algebraic Tori Associated to Totally Real Fields (2019)
  • Author(s): Kenichi Bannai
  • Organizer: Algebraic Number Theory and Related Topics, RIMS
• [Presentation] The polylogarithm associated to algebraic tori associated to totally real fields (2019)
  • Author(s): Kenichi Bannai
  • Organizer: Colloquium, Osaka University
  • Invited
• [Presentation] The p -adic polylogarithm and p-adic L-function for totally real fields (2019)
  • Author(s): Kenichi Bannai
  • Organizer: Seminar on Number Theory and Automorphic Forms, Osaka University
  • Invited
• [Presentation] Comparison between rigid syntomic and crystalline syntomic cohomology for strictly semistable log schemes (2018)
  • Author(s): Kazuki Yamada
  • Organizer: 第17回仙台広島整数論集会
• [Presentation] 混合プレクティックホッジ構造について (2018)
  • Author(s): 坂内健一, 萩原啓, 小林真一, 山田一紀, 山本修司, 安田正大
  • Organizer: ワークショップ「ホッジ理論と代数幾何学」
  • Invited
• [Presentation] 代数トーラスのポリログのde Rham実現 (2018)
  • Author(s): 坂内健一, 萩原啓, 山田一紀, 山本修司
  • Organizer: 第12回福岡数論研究集会
• [Presentation] On the de Rham realization of the polylogarithm for certain algebraic Tori (2018)
  • Author(s): Kenichi Bannai, Kei Hagihara, Kazuki Yamada and Shuji Yamamoto
  • Organizer: Keio-Yonsei Number Theory Workshop
  • Int'l Joint Research / Invited
• [Funded Workshop] BOSTON UNIVERSITY/KEIO UNIVERSITY WORKSHOP 2019 Number Theory (2019)
• [Funded Workshop] Conference on distribution of values of zeta functions and L-functions (2019)