Galois representations arising from etale cohomology

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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.