Shimura varieties, local Shimura varieties and their etale cohomology

2019 Fiscal Year Final Research Report

• Project/Area Number: 15H03605
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: The University of Tokyo
• Principal Investigator: MIEDA Yoichi 東京大学, 大学院数理科学研究科, 准教授 (70526962)
• Project Period (FY): 2015-04-01 – 2020-03-31
• Keywords: 志村多様体 / 局所志村多様体 / ラングランズ対応 / エタールコホモロジー / リジッド幾何

Outline of Final Research Achievements

Shimura varieties are algebraic varieties obtained as arithmetic quotients of Hermitian symmetric spaces, and local Shimura varieties are their local counterparts. I studied them mainly by using p-adic geometric technique. First, for a fairly wide class of Shimura varieties, called the preabelian type, I constructed the potentially good reduction loci and proved that they have almost the same etale cohomology as that of the whole Shimura varieties. I also introduced a new method computing the etale cohomology of local Shimura varieties by means of their modulo p reductions, and apply it to study the explicit local Langlands correspondence for GL(n). I investigated the local Shimura variety for GSp(4) in detail, and obtained results on relation between its etale cohomology and the local Langlands correspondence.

Free Research Field

整数論

Academic Significance and Societal Importance of the Research Achievements

この研究のテーマである志村多様体や局所志村多様体は，保型表現とGalois表現が結び付くことを主張するラングランズ対応と関係しているがゆえに，整数論において特に興味を持たれている幾何学的対象である．本研究で得た成果によって，GL(n)の局所ラングランズ対応の具体的なふるまいを詳しく調べる手段が得られたことになる．また，GSp(4)の局所志村多様体についての成果は，局所ラングランズ対応の精密化である局所Arthur分類もまた局所志村多様体と関係しているだろうという示唆を与えており，今後の当該分野の研究の指針となることが期待される．