Arithmetic of algebraic varieties
Project/Area Number  09640011 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
Algebra

Research Institution  University of Tokyo 
Principal Investigator 
SAITO Takeshi Univ.of Tokyo, Ass.Professor, 大学院・数理科学研究科, 助教授 (70201506)

CoInvestigator(Kenkyūbuntansha) 
KURIHARA Masato Tokyo Metro.Univ., Ass.Professor, 理学部, 助教授 (40211221)
SAITO Shuji Tokyo Inst.of Tech.Professor, 理学部, 教授 (50153804)
ODA Takayuki Univ.of Tokyo, Professor, 大学院・数理科学研究科, 教授 (10109415)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥2,500,000 (Direct Cost : ¥2,500,000)
Fiscal Year 1998 : ¥1,100,000 (Direct Cost : ¥1,100,000)
Fiscal Year 1997 : ¥1,400,000 (Direct Cost : ¥1,400,000)

Keywords  Hilbert modular form / Langlands correspondence / Galois representations / Shimura curves / padic Hodge theory / etale cohomology / StiefelWhitney class / conductor / Hilbert保型形式 / Langlands対応 / Galois表現 / 志村 曲線 / p進Hodge理論 / エタール・コホモロジー / StiefelWhitney類 / 導手 / Hilbert 保型形式 / Langlands 対応 / Galois 表現 / エチールコホモロジー / P進Hodge理論 
Research Abstract 
The main result is obtained in the research on the ladic representation associated a Hilbert modular form. It is a 2dimensional ladic representation of the absolute Galois group G_F of a totally real field F.For a finite place upsilon * l, it is shown by Carayol that the representation of the WeilDeligne group defined by its restriction to the decomposition group at upsilon corresponds to the upsiloncomponent of the automorphic representation of GL_2 (A_F) determined by f. Using recent results in padic Hodge theory, I formulated a similar statement and proved it. The second year of the project was spent to write a paper and paper is nearly completed. As a byproduct of the proof, I proved the monodromyweight conjecture for Galois representation associated to modular forms. It seems to have been overlooked even for the case F=Q.The paper is already to appear in a Proceedings. I also obtained some other results. For an algebraic variety X over a field K, the second StiefelWhitney class of its ladic etale cohomology is defined in the second Galois cohomology H^2 (K, Z/2Z). I formulated a conjecture between the class with the HasseWitt class of de Rham cohomology and proved it in several cases.

Report
(4results)
Research Output
(7results)