2022 Fiscal Year Final Research Report
Special values of automorphic L-functions and periods
| Project/Area Number |
19K03407
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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 | Osaka Metropolitan University (2022)
Osaka City University (2019-2021) |
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Principal Investigator |
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| Project Period (FY) |
2019-04-01 – 2023-03-31
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| Keywords | 保型L函数 / L函数の特殊値 / テータ対応 |
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| Outline of Final Research Achievements |
Boecherer proclaimed a conjecture concerning degree two Siegel cusp forms whch are Hecke eighenforms in the mid 1980's, which was about a relationship between a finite sum over an ideal class group of an imaginary quadratic field of Fourier coefficients and the central value of the spinor L-function twisted by a quadratic character corresponding to the imaginary quadratic field. Many specialists were interested in it but it remained unsolved for a long time. In a joint work with Kazuki Morimoto at Kobe University, we proved the original conjecture. Moreover we proved its generalization to the case corresponding to a finite sum of Fourier coefficients weighted by a character of the ideal class group.
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| Free Research Field |
保型L函数
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| Academic Significance and Societal Importance of the Research Achievements |
数論的に定義されたL函数の特殊値は、対応する数論的対象の重要な情報を含んでいると予想されている。特に函数等式の中心における特殊値は、Birch & Swinnerton-Dyer予想及びその一般化にみられるように、興味深い。本研究の成果である一般化されたBoecherer予想は、GL(2)に関するWaldspurgerの定理の自然な一般化であり、Waldspurgerの定理がこれまでに、楕円曲線及びGL(2)の保型形式の数論に、重要な応用をもたらしたように、今後、アーベル曲面及びGSp(2)の保型形式の数論への様々な応用が期待できる。本研究の研究成果の学術的意義は大きい。
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