| Project/Area Number |
21K03186
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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 | Sophia University |
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Principal Investigator |
TRIHAN FABIEN 上智大学, 理工学部, 准教授 (60738300)
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| Project Period (FY) |
2021-04-01 – 2026-03-31
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| Project Status |
Granted (Fiscal Year 2023)
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| Budget Amount *help |
¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
Fiscal Year 2025: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2024: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2023: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2022: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
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| Keywords | Iwasawa theory / Elliptic curve / Function fields / Class number / Function field / Number theory |
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| Outline of Research at the Start |
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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| Outline of Annual Research Achievements |
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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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
Both projects are proceeding smoothly.
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| Strategy for Future Research Activity |
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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