Mordell-Weil Groups of elliptically-fibered Calabi-Yau manifolds

• Project/Area Number: 19K03427
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Chuo University
• Principal Investigator: Kuwata Masato 中央大学, 経済学部, 教授 (00343640)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Project Status: Completed (Fiscal Year 2023)
• Budget Amount: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2023: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000); Fiscal Year 2022: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000); Fiscal Year 2021: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2020: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 楕円ファイブレーション / Mordell-Weil格子 / 楕円曲面 / Calabi-Yau多様体 / K3曲面 / Mordell-Weil群

Outline of Research at the Start

楕円曲面の切断のなす群，Mordell-Weil群は「高さ」による内積によって格子の構造を持ち，楕円曲面の幾何学的性質を解き明かす上で大きな役割を担う．近年，超弦理論において楕円ファイブレーションをもつ3次元以上のCalabi-Yau多様体が注目を集めており，とくにF-理論ではMordell-Weil群に物理的な意義が見出されている．本研究では，楕円ファイブレーションをもつ3次元以上のCalabi-Yau多様体上のMordell-Weil群について，楕円曲面での理論を拡張してMordell-Weil格子の理論に類似する理論を構築し，超弦理論などへの応用を目指す．

Outline of Final Research Achievements

We attempted to extend Mordell-Weil lattice theory for elliptic surfaces to higher-dimensional elliptically fibered varieties. We defined a lattice structure for the Mordell-Weil group of certain elliptic threefolds by introducing an inner product using the notion of height, and determined the structure in some cases. In particular, we have introduced a lattice structure to the rank 6 Mordell-Weil group of certain rational elliptic threefolds, and have shown that it is isomorphic to the root lattice of type E6. We also studied some Calabi-Yau threefolds with Mordell-Weil groups of rank 9 or 10.

Academic Significance and Societal Importance of the Research Achievements

楕円曲面の理論は代数幾何，解析学，数論など数学の様々な分野だけでなく，理論物理学まで密接に関連して発展してきた大変興味深い研究対象である．楕円曲面の切断のなす群は，交点形式に由来する内積によりMordell-Weil格子と呼ばれる格子の構造を持ち，その構造は楕円曲面の幾何学的性質を調べるなかで重要な役割を果たし，様々な応用を持つ．理論物理学では，高次元のCalabi-Yau楕円多様体のMordell-Weil群には物理的な意味あり，高次元の楕円多様体のMordell-Weil群に格子の構造を導入する試みは，広い分野に波及する可能性をもつ意義ある研究である．

Research Products

• [Journal Article] Ranks of elliptic curves in cyclic sextic extensions of Q (2024)
  • Author(s): Kisilevsky Hershy、Kuwata Masato
  • Journal Title: Indagationes Mathematicae Volume: - Issue: 4 Pages: 728-743
  • DOI: 10.1016/j.indag.2024.01.004
  • Peer Reviewed
• [Presentation] Toward the theory of Mordell-Weil lattices of elliptic threefolds (2023)
  • Author(s): Masato Kuwata
  • Organizer: Calabi-Yau Varieties and Related Topics 2023
  • Int'l Joint Research / Invited
• [Presentation] 楕円曲線の巡回拡大上定義された有理点について (2023)
  • Author(s): 鍬田 政人
  • Organizer: 2023 大分宮崎整数論研究集会
  • Invited
• [Presentation] Double cover of extremal rational elliptic surfaces (2023)
  • Author(s): 鍬田 政人
  • Organizer: 第 30 回 代数曲面ワークショップ
  • Invited
• [Presentation] Finding points defined over cyclic sextic extensions of an elliptic curve using a K3 surface (2022)
  • Author(s): Masato Kuwata
  • Organizer: Curves over finite fields and arithmetic of K3 surfaces (Bernoulli Institute, Rijksuniversiteit Groningen)
  • Int'l Joint Research / Invited
• [Presentation] Finding points defined over cyclic extensions of an elliptic curve: A geometric approach (2022)
  • Author(s): Masato Kuwata
  • Organizer: Brown Universiy Algebra seminar
  • Invited
• [Presentation] Jacobians of genus 2 curves with full level 3 structure and the related elliptic fibrations (2021)
  • Author(s): Masato Kuwata
  • Organizer: Boston University Number theory seminar
  • Invited
• [Presentation] Rational points on generalized Kummer varieties (2019)
  • Author(s): 鍬田 政人
  • Organizer: 高次元代数多様体の有理点
  • Int'l Joint Research / Invited