Arithmetic D-modules and Langlands correspondence

2016 Fiscal Year Final Research Report

• Project/Area Number: 25800004
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: The 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
• Keywords: p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