Uwe Jannsen

Extensions of motives and a new p-adic cohomology theory in characteristic p.

Abstract: This is report on joint work in progress with Jean-Marc Fontaine. We define a refinement of crystalline cohomology by using the theory of so-called gauges as introduced earlier by Mazur and Kato and certain syntomic sheaves studied earlier by Fontaine and Messing. This theory can be related to F-zips as introduced by Moonen and Wedhorn, to displays as introduced by Langer and Zink, and to the Fontaine-Laffaille theory. A first application is given by generalizing results of Milne and Ramachandran on special values and extensions of motives over finite fields from effective to arbitrary motives.