2023 Fiscal Year Research-status Report
Class number formula over global field of characteristic p and with coefficients.
| Project/Area Number |
21K03186
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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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| Keywords | Iwasawa theory / Elliptic curve / Function fields |
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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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| Causes of Carryover |
A trip to see my co-authors in Taiwan was cancelled due to some duty regarding Entrance examination. This trip is reported to August 2024.
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