2001 Fiscal Year Final Research Report Summary
Galois representations arising from etale cohomology
| Project/Area Number |
11640013
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | The University of Tokyo |
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Principal Investigator |
SAITO Takeshi School of Mathematical Sciences, The University of Tokyo, Professor, 大学院・数理科学研究科, 教授 (70201506)
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| Co-Investigator(Kenkyū-buntansha) |
KURIHARA Masato Tokyo Metropolitan University, Faculty of Sciences, Assistant Professor, 理学部, 助教授 (40211221)
TERASOMA Tomohide School of Mathematical Sciences, The University of Tokyo, Assistant Professor, 大学院・数理科学研究科, 助教授 (50192654)
KATO Kazuya Kyoto University, School of Sciences, Professor, 大学院・理学系研究科, 教授 (90111450)
FUJIWARA Kazuhiro Nagoya University, School of Polymathematics, Professor, 多元数理研究科, 教授 (00229064)
SAITO Shuji Nagoya University, School of Polymathematics, Professor, 多元数理研究科, 教授 (50153804)
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| Project Period (FY) |
1999 – 2001
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| Keywords | etale cohomology / Galois representation / weight spectral sequence / semi-stable reduction / log scheme / Hasse-Weil L function / resolution of singularities / conductor formula |
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| Research Abstract |
1999 and 2000. The main result in this period is a proof of the conductor formula. This is a joint work with one of the investigators, K. Kato. Bad reduction of a variety over a local field causes ramification of the inertia action. The conductor formula tells us how to measure it in terms of differential forms on a regular model. We proved it under the assumption that the reduced closed fiber is a divisor with normal crossings. I also studied ramification of local fields with imperfect residue field and defined a filtration by ramification groups on the absolute Galois group.
2001. 1. For a variety over a local field with semi-stable reduction. Rapoport and Zink defined the weight spectral sequence. I gave an elementary construction of the spectral sequence and proved its functoriality. From this, I deduced the independence of 1 of the alternating sum of the traces of the composition of the actions of an element of the Weil group and an algebraic correspondence on 1-adic cohomology. As an application, I proved that the classical conjecture that the bad Euler factor of the Hasse-Weil L function of an algebraic surface is independent of 1 is reduced to a part of Tate conjecture that the Kunneth projectors are defined by algebraic cycles and the weight-monodromy conjecture.
2. I proved that a smooth family of curves defined on the interior of log regular scheme is extended to the boundary if it is extended at each generic point of the boundary.
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