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

Beilinsons Conjectures on exact Values of L-Functions offers with Alexander Beilinsons conjectures on precise values of L-functions. themes coated variety from Pierre Delignes conjecture on serious values of L-functions to the Deligne-Beilinson cohomology, in addition to 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 off with an creation 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 severe 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 e-book concludes by way 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 figuring out of Beilinsons conjectures on designated values of L-functions.

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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 . , Z ) — X-algebras. Furthermore, the family of these homomorphisms (for all (simplicial) schemes in V) is uniquely characterized by the fact that c([E]) = ( l , l , c ( £ ) , 0 , . . ) for line bundles E . Proof: See [Gro 1] §3. The reader will realize that the purpose of the weak Gysin property is to ensure the validity of the corollary on p.

Is the theory of the N^ron-Tate height, and part c. is part of the conjecture of Birch and Swinnerton-Dyer. In [Tat 2] the reader may find a discussion of this case in which the conjectural picture is even more precise insofar as £*(M,ra) itself (not only mod(Q x ) is predicted in terms of arithmetic invariants of X. The general conjecture certainly is modeled on this case. Beilinson ([Bei 1,3]), Bloch ([Bio 2]), and Gillet/Soute ([GS]) construct - all three by different techniques - a natural height pairing for any X which has certain geometric properties (conjecturally it always should have those).

The first conjecture says that the regulator map reg:2Ti(X,Q(*))—> 2T£(X / R , R(«)) constructed in §4 leads to a ) mod Q x . In case m = | the regulator map alone is not sufficient to induce a Q-structure.