Study of the moduli of Galois representations of number fields and function fields

2015 Fiscal Year Final Research Report

• Project/Area Number: 25400016
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Kyushu University
• Principal Investigator: Taguchi Yuichiro 九州大学, 数理(科)学研究科(研究院), 准教授 (90231399)
• Co-Investigator(Kenkyū-buntansha): HATTORI Shin 九州大学, 大学院数理学研究院, 助教 (10451436)
• Co-Investigator(Renkei-kenkyūsha): KURIHARA Masato 慶應大学, 理工学部, 教授 (40211221) ; SAITO Takeshi 東京大学, 大学院数理科学研究科, 教授 (70201506) ; TAMAGAWA Akio 京都大学, 数理解析研究所, 教授 (00243105) ; YASUDA Seidai 大阪大学, 大学院理学研究科, 准教授 (90346065) ; HIRANOUCHI Toshiro 広島大学, 大学院理学研究科, 助教 (30532551)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Keywords: ガロア表現 / モジュライ / 有限性 / 代数体 / 函数体

Outline of Final Research Achievements

We have constructed a moduli scheme of Galois representations and studied its properties, and obtained some basic results. We have also obtained several related results, such as: (1) a vanishing theorem of the Galois-fixed subspace of a Galois representation of a rather general type of complete discrete valuation field (a generalization of a theorem of Imai) and its application to Iwasawa theory, (2) a result on the congruence of Galois representations and its application to non-existence theorems a la Rasmussen-Tamagawa, (3) proof of the fact that the Hecke field of a geometric Galois represntation is often (say, with density 1 primes, in certain cases) generated by the trace of the Frobenius for a single finite prime, (4) an upper bound of the number of the connected components of the Zariski closure of the image of a Galois representation.

Free Research Field

数論