Galois representations arising from etale cohomology

1999年和2000年。这一阶段的主要结果是导体公式的证明。这是与其中一名调查员K. Kato合作完成的。变量在局部场上的不良约化会引起惯性作用的分支。导体公式告诉我们如何在常规模型上用微分形式来测量它。我们在假设约化闭合纤维是法交除数的前提下证明了这一点。本文还研究了具有不完全残域的局部域的分枝，并在绝对伽罗瓦群上定义了分枝群的过滤。2001。1. 对于局部区域上具有半稳定还原的品种。Rapoport和Zink定义了权值谱序列。给出了谱序列的初等构造，并证明了它的函数性。由此，推导了Weil群中一个元的作用复合的迹的交替和与1进上同调上的代数对应的1的独立性。作为应用，证明了关于代数曲面的Hasse-Weil L函数的bad Euler因子与1无关的经典猜想被简化为Kunneth投影由代数循环和权一猜想2所定义的Tate猜想的一部分。证明了在对数正则格式内部定义的光滑曲线族，如果在边界的每一个一般点上进行扩展，就可以扩展到边界。

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.