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2016 Fiscal Year Final Research Report

Arithmetic D-modules and Langlands correspondence

Research Project

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Project/Area Number 25800004
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Algebra
Research InstitutionThe University of Tokyo

Principal Investigator

Abe Tomoyuki  東京大学, カブリ数物連携宇宙研究機構, 准教授 (70609289)

Project Period (FY) 2013-04-01 – 2017-03-31
Keywordsp進コホモロジー / 関数体のラングランズ対応 / 数論的D加群
Outline of Final Research Achievements

We constructed a Langlands type correspondence for p-adic coefficient theory for function fields, and as a result, I resolved Deligne's crystalline companion conjecture for curves. This was nothing but the goal of this research, and we may say that I attained it. This correspondence enables us an p-adic interpretation of cohomology theories over finite fields, and opened a new door of the theory.
The main result states that there exists a correspondence between overconvergent F-isocrystals and cuspidal automorphic representations. However, in the proof, the category of overconvergent isocrystals is too small, and we need to deal with a much wider class called the arithmetic D-modules. In the work, by completing the program of Berthelot on the construction of such theory, we were able to obtain the desired result.

Free Research Field

数論幾何学

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Published: 2018-03-22  

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