Local Langlands correspondence and Lubin-Tate perfectoid space

2014 Fiscal Year Final Research Report

• Project/Area Number: 25887009
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: The University of Tokyo
• Principal Investigator: TSUSHIMA TAKAHIRO 東京大学, 数理(科)学研究科(研究院), 助教 (70583912)
• Project Period (FY): 2013-08-30 – 2015-03-31
• Keywords: ガロワ表現 / Lubin-Tate perfectoid空間 / 局所ラングランズ対応 / 局所ジャッケ・ラングランズ対応 / タイプ理論

Outline of Final Research Achievements

We have studied Galois representations in a view point of Langlands program. More concretely, we have studied the local Langlands correspondence and the local Jacquet-Langlands correspondence geometrically through geometry of Lubin-Tate perfectoid space. An epipelagic representation means an easiest cuspidal representation of ramified type of a general linear group over a non-archimedean local field. For such representations, we have constructed a family of affinoids and its formal models, and studied the middle cohomology of the reductions of the models in a representation theoretic view point. As a result, we have constructed two correspondences between representations. We have proved that one of them corresponds to the local Jacquet-Langlands correspondence for epipelagic representations. As a by-product, we have shown the B-S-S conjecture for epipelagic representations. This conjecture asserts the compatibility of types under the local Jacquet-Langlands correspondence.

Free Research Field

数論幾何学