| 研究課題/領域番号 |
21K03186
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| 研究種目 |
基盤研究(C)
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| 配分区分 | 基金 |
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| 応募区分 | 一般 |
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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 | 上智大学 |
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研究代表者 |
TRIHAN FABIEN 上智大学, 理工学部, 准教授 (60738300)
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| 研究期間 (年度) |
2021-04-01 – 2026-03-31
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| 研究課題ステータス |
交付 (2023年度)
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| 配分額 *注記 |
4,030千円 (直接経費: 3,100千円、間接経費: 930千円)
2025年度: 780千円 (直接経費: 600千円、間接経費: 180千円)
2024年度: 780千円 (直接経費: 600千円、間接経費: 180千円)
2023年度: 780千円 (直接経費: 600千円、間接経費: 180千円)
2022年度: 780千円 (直接経費: 600千円、間接経費: 180千円)
2021年度: 910千円 (直接経費: 700千円、間接経費: 210千円)
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| キーワード | Iwasawa theory / Elliptic curve / Function fields / Class number / Function field / Number theory |
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| 研究開始時の研究の概要 |
Our goal is to make significant progress in the l=p-part of the (Equivariant) Tamagawa Number Conjecture for general p-adic coefficients as well as to extend the results of Trihan-Vauclair and Lai-Longhi-Tan-Trihan to new cases.
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| 研究実績の概要 |
Our research is currently twofold. First, Burns-Castillo have established that the equivariant Birch and Swinnerton-Dyer conjecture implies a refined version of BSD for each character. We demonstrate that their approach extends to the function field of characteristic p, where the equivariant BSD conjecture is already known for tamely ramified extensions and semistable abelian varieties. In a second project, we delve into the study of mu-invariants, specifically examining their behavior with respect to a finite Galois p-extension of an ordinary abelian variety A over a Zp-extension of global fields L/K (whose characteristic is not necessarily positive). This extension may ramify at a finite number of places where A has ordinary reductions.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
Both projects are proceeding smoothly.
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| 今後の研究の推進方策 |
In direction of the geometric class number formula and under semisimplicity of the Frobenius acting on the rigid cohomology of the Dieudonne crystal associated to the Neron model of an abelian variety over an open of good reduction we have proved in an earlier work the Main conjecture of Iwasawa assuming that the Pontryagin dual of the Selmer group computed over the unramified Zp-extension was a finitely generated Zp-module. Next, we hope to generalize this to general log F-crystal over varieties over finite fields and unramified everywhere finite extensions of such varieties. The new ingredient is the use of the syntomic complex associated to F-crystals of Kato.
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