2018 Fiscal Year Research-status Report
Arithmetic cohomology over local fields
| Project/Area Number |
18K03258
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| Research Institution | Rikkyo University |
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Principal Investigator |
ガイサ トーマス 立教大学, 理学部, 教授 (30571963)
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Keywords | Weil etale cohomology / Arithmetic cohomology / BSD conjecture / Duality / Class field theory |
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| Outline of Annual Research Achievements |
In the first year of the project Imade progress in two aspects of Weil etale cohomology (also called arithmetic cohomology).
For regular flat and proper schemes over henselian discrete valuation rings, Morin and I gave a preliminary definition of arithmetic cohomology and made calculations in some specific weights, notably weights zero, one and d+1, where d is the relative dimension of the scheme. it turns out that the groups we calculated have the properties that we expect, if we assume that Weil-etale cohomology over finite fields is well behaved. For example in weight d+1 we obtain a cohomology group whose completion agrees with the fundamental group, and which
we expect to be finitely generated (up to the part coming from the base field).
Over gobal function fields of characteristic p, there is the problem of generalizing Weil-etale cohomology from motivic coefficients to other coefficients. In joint work with T.Suzuki we did this for coefficients in the Neron model of an abelian variety , and gave a Weil etale version of the Birch and Swinnerton-Dyer conjecture. This gives some hope to be able to give a Weil-etale version of the Birch and Swinnerton Dyer conjecture even over number fields.
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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 calcuations we made over henselian discrete valuation fields are in accordance with the expecte results. For example, we recover Lichtenbaum's duality for curves over a local field as a special case, and we can prove that arithmetic cohomology is an integral model of the fundamental group in weight d+1.
Over a global function field we succeeded in using a result of Kato-Trihan to show that the Weil etale version of the Birch and Swinnerton-Dyer conjecture holds if the Tate-Shafarevich group is finite.
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| Strategy for Future Research Activity |
It turns out that over local fields we need some information on properties of etale motivic cohomology of the model as input, and not much is known. This is a very interesting question, and I hope to obtain some result.
In the function field case the next step would be to find a Weil-etale conjecture of the special value for other objects, like lattices, tori, or even one-motives. In order to do this one first has to find a good theory of Neron-models, and then relate the cohomology with coefficients in the Neron model to the special value of the zeta-function.
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| Causes of Carryover |
３月の出張精算を次年度に行うため未使用額が発生した。
使用計画：３月の出張の精算，国内研究集会の出席，海外の大学（ドイツ）での共同研究，ボールドー大学（フランス）の共同研究者Morin氏の招待．
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