By M. Rapoport, N. Schappacher, P. Schneider

Beilinsons Conjectures on distinct Values of L-Functions bargains with Alexander Beilinsons conjectures on distinctive values of L-functions. themes coated variety from Pierre Delignes conjecture on serious values of L-functions to the Deligne-Beilinson cohomology, besides the Beilinson conjecture for algebraic quantity fields and Riemann-Roch theorem. Beilinsons regulators also are in comparison with these of Émile Borel.

Comprised of 10 chapters, this quantity starts with an advent to the Beilinson conjectures and the speculation of Chern sessions from better k-theory. The "simplest" instance of an L-function is gifted, the Riemann zeta functionality. The dialogue then turns to Delignes conjecture on serious values of L-functions and its connection to Beilinsons model. next chapters concentrate on the Deligne-Beilinson cohomology; ?-rings and Adams operations in algebraic k-theory; Beilinson conjectures for elliptic curves with advanced multiplication; and Beilinsons theorem on modular curves. The publication concludes by means of reviewing the definition and houses of Deligne homology, in addition to Hodge-D-conjecture.

This monograph will be of substantial curiosity to researchers and graduate scholars who are looking to achieve a greater realizing of Beilinsons conjectures on distinct values of L-functions.

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42 DELIGNE-BEILINSON COHOMOLQGY Hele*ne Esnault* , Eckart Viehweg In these notes we describe the Deligne cohomology of a complex manifold as well as Beilinson's algebraic cohomology theory of a quasiprojective complex manifold and some of its properties. In fact, most of the content of our manuscript can be found (in a more compressed form) in the first paragraph of Beilinson's article [3], We tried to include all details needed, and we hope that our presentation is sufficiently "down to earth" to serve as an introduction to this theory.

We need to know in the following is that the action of the Adams operations i/>k, for k > 1, on it can be determined explicitly. ) and k > 1 we have r/>kx = (rMVxj)j>i) . Proof: We freely use the notations of SGA 6 exp. V. cit. cit. ) = ( 0 ) l , . . , f c ^ O , . . ) f o r a l U > l . cit. 3). ) = ( 0 , 1 , ( 4 ^ ) , - > ! ))t+1 holds true for all £ > 1 which proves the assertion. All the important properties of Chern classes now can be expressed by the following statement. ) of augmented # ° ( y .

3). ,P) :x0 = l} j>0 which obviously forms an abelian group with respect to the cup-product as addition (it is suggestive to think of elements in the second factor as being power series in one variable with constant coefficient 1). ) in a natural way can be made into an augmented # ° ( y . , Z ) — A-algebra, too. The interested reader should consult SGA 6 exp. 0 App. I §3 or exp. V §6 for the details. ) we need to know in the following is that the action of the Adams operations i/>k, for k > 1, on it can be determined explicitly.