| 研究課題/領域番号 |
22KF0214
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| 補助金の研究課題番号 |
22F22316 (2022)
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| 研究種目 |
特別研究員奨励費
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| 配分区分 | 基金 (2023)
補助金 (2022) |
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| 応募区分 | 外国 |
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| 審査区分 |
小区分11010:代数学関連
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| 研究機関 | 京都大学 |
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研究代表者 |
市野 篤史 京都大学, 理学研究科, 教授 (40347480)
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| 研究分担者 |
CHEN SHIH-YU 京都大学, 理学研究科, 外国人特別研究員
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| 研究期間 (年度) |
2023-03-08 – 2024-03-31
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| 研究課題ステータス |
中途終了 (2023年度)
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| 配分額 *注記 |
2,200千円 (直接経費: 2,200千円)
2024年度: 500千円 (直接経費: 500千円)
2023年度: 1,100千円 (直接経費: 1,100千円)
2022年度: 600千円 (直接経費: 600千円)
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| キーワード | Special L-values / Deligne's conjecture / Betti-Whittaker periods |
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| 研究開始時の研究の概要 |
We proposed to prove new cases of Deligne's conjecture for the following L-functions:
(1) Symmetric power L-functions for GL(2).
(2) Tensor product L-functions for GL(2).
(3) Rankin-Selberg L-functions for GSp(4) x GSp(4) and GSp(4) x GL(2) x GL(2).
For the first two classes of L-functions, the algebraicity is expressed in terms of the motivic periods of elliptic newforms. For (3), the algebraicity is expressed in terms of special value of adjoint L-functions and Petersson norm of arithmetic holomorphic Siegel cusp forms of degree 2.
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| 研究実績の概要 |
In the literature, most known results on Deligne's conjecture or its automorphic analogue were obtained by cohomological interpretation of integral representation of the L-functions. The purpose of this research project is to investigate another approach to Deligne's conjecture. We consider ratios of Rankin-Selberg L-functions of algebraic automorphic representations of general linear groups. Under some regularity conditions, we prove that these ratios are algebraic at critical points. As applications, we prove new cases of Deligne's conjecture for symmetric power L-functions and tensor product L-functions of elliptic modular forms, which are previously known only for small degree. One technical difficulty in the proof of our main result is the non-vanishing of certain archimedean pairing. We extend the non-vanishing result of B. Sun to non-unitary cohomologically induced representations. The results of this research have been compiled into a paper and published as a preprint.
We also prove the trivialness of the relative period associated to a regular algebraic cuspidal automorphic representation of GL(2n) of orthogonal type. Together with the result of G. Harder and A. Raghuram, this implies the algebraicity of the ratios of successive critical L-values for GSpin(2n) x GL(n’). The results of this research were compiled into a paper and will be published by the International Mathematics Research Notices.
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