2022 Fiscal Year Final Research Report
Study of L-functions and regulators of motives via special functions
| Project/Area Number |
18K03234
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 11010:Algebra-related
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| Research Institution | Chiba University |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Keywords | L関数 / モチーフ / 超幾何関数 / 周期 / レギュレーター |
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| Outline of Final Research Achievements |
We described the periods and regulators of hypergeometric fibrations in terms of hypergeometric functions (joint work with M. Asakura). As an application, we gave formulas expressing hypergeometric functions and their special values in terms of the logarithm of algebraic functions and algebraic numbers (joint work with M. Asakura and T. Terasoma). We formulated l-adic and p-adic analogues of the Gross-Deligne conjecture on the periods of motives with complex multiplication, and proved them for motives of abelian type (joint work with B. Kahn). We discovered a method to prove in a uniform manner the various transformation formulas for classical hypergeometric functions. We gave a definition of generalized hypergeometric functions (including those with many variables) over finite fields, and established various fundamental properties.
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| Free Research Field |
数論
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| Academic Significance and Societal Importance of the Research Achievements |
超幾何関数は数学のさまざまな分野や物理学で現れる非常に重要な対象である．超幾何関数は数論においても重要であり，数論的なL関数（リーマン・ゼータ関数の一般化）との関係からも注目されている．L関数の特殊値は現代の数論における最も重要な研究対象の一つであり，周期やレギュレーターという幾何学的な不変量との関係が予想されている．本研究では，モチーフの周期やレギュレーターと超幾何関数との関係について多角的に調べた．特に，周期，レギュレーター，超幾何関数たちの数論的（l進的，p進的, モチーフ的）な類似物にについて基礎的な研究を行った．古典的な超幾何関数論に新たな視点を与えるということも期待できる．
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