2022 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 | Class number / Function field / Number theory |
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| Outline of Annual Research Achievements |
During the academic year 2022-23 we were able to complete one paper on the study of the variation of the mu-invariant of abelian variety over function field. The paper is now submitted and available at this address https://arxiv.org/abs/2301.09073 . Besides this, we tried to extend the method of Brinon-T to a Galois equivariant version. We are facing a difficulty : the construction there is not functorial enough. Therefore, we understood the importance of strengthening first the construction of Brinon-T. On an other hand, we have now the formulation for a conjectural p-adic L-function attached to automorphic p-adic coefficients. We have done a talk about this at the IISc Bangalore in March 2023.
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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
The present objective is to :-functorialize the construction of [Brinon-Trihan].-Generalize [Brinon-Trihan] to a Galois equivariant analogue using the method of Burns and Kakde.-Construct a p-adic L-function associated to an automorphic overconvergent F-isocrystal-Consider related problems like a chi-BSD formula where chi is an Artin character of a Galois group.-Pursue further our study of mu-invariant of abelian variety.
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| Strategy for Future Research Activity |
In order to achieve our objective we plan the following for the future research:1) -functorialize the construction of [Brinon-Trihan] :my co-author will visit me this April 2023 and we shall consider this question. 2) -Generalize [Brinon-Trihan] to a Galois equivariant analogue using the method of Burns and Kakde: i plan to visit again Prof Kakde to help me to achieve this goal (probably end of January 2024). 3) Construct a p-adic L-function associated to an automorphic overconvergent F-isocrystal: I might need the help of an expert in the field of automorphic form over function field. My plan at the moment is to find the right person.4)Consider related problems like a chi-BSD formula where chi is an Artin character of a Galois group. This project will be studied during the visit of my co-author Prof Vauclair in April. 5) Pursue further our study of mu-invariant of abelian variety. This project will be done with my two co-authors in Taiwan Prof Tsoi and Tan during this summer 2023.
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| Causes of Carryover |
The remaining amount does not take into account my last business trip in India that ended March 31. This last trip expanse was settled in April 2023. This explains why I have still money left.
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