Galois representations arising from etale cohomology

1999年和2000年。这一时期的主要成果是导体公式的证明。这是一个联合工作的调查员之一，K。加藤局部域上的一个变量的不好的归约会引起惯性作用的分歧。导体公式告诉我们如何在规则模型上用微分形式来测量它。我们证明了它的假设下，约化的封闭纤维是一个除数与正常的交叉。我还研究了局部域与不完全剩余域的分歧，并定义了绝对伽罗瓦群上的分歧群的过滤。1.对于局部域上具有半稳定约化的簇。Rapoport和Zink定义了权重谱序列。给出了谱序列的一个初等构造，并证明了它的函性。从这一点上，我推导出的独立性1的交替和的痕迹组成的行动的一个元素的韦伊群和代数对应1进上同调。作为应用，证明了代数曲面的Hasse-Weil L函数的坏Euler因子与1无关的经典猜想被简化为Kunneth投影由代数圈和权单值映射定义的Tate猜想的一部分.证明了定义在对数正则格式内部的光滑曲线族，如果在边界的每一个类属点上延拓，则该曲线族延拓到边界。

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.