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

Beilinsons Conjectures on targeted Values of L-Functions bargains with Alexander Beilinsons conjectures on distinct values of L-functions. subject matters coated diversity 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 off with an advent to the Beilinson conjectures and the idea of Chern sessions from greater 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 booklet concludes via reviewing the definition and homes of Deligne homology, in addition to Hodge-D-conjecture.

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

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FiP"1 — > 0 (where TL (p) is in degree zero) and the Deligne cohomology as U V an(X'Z(P)) :=mq(X 'Z(p)P,an) ' For simplicity, in this paragraph, we drop the sub-script "an" and write Z(p)p and H 2 . 2. y x U y = jxAdy [0 if deg x = 0 if deg x > 0 otherwise . and deg y = p' U is a morphism of complexes. In fact, if we denote the differential in Z(p)p by d (where, of course, d : Z(p) — > 0 is the inclusion) and \i = deg x and ^i1 = deg y, we have: d(x U y) = x-dy \L = 0, ji' < p' x • dy -■ dxAdy |i = 0 , \L ' = p ■ \ = dx liy + (-D^xJdy dxAdy ji > 0 , |i' = p ' 0 otherwise It is quite easy to show that U is associative.

In §6 we recall the definition and some properties of the cycle class in the De Rham cohomology (following [2], [9] and [1]). Especially we explain the behaviour of those classes with respect to the Hodge filtration. These constructions are needed in §7. There we first ex­ plain the relations between the Deligne cohomology of a projective manifold and the intermediate Jacobian of Griffiths. We reproduce Deligne's definition of the cycle class in the D - B - cohomology ([10]) and we compare it to the Abel-Jacobi map.

We have res d log = ord , kernel (J [ord ) = 0* xes and x denote the order of a zero denote the Cauchy-Poincare kernel (J |_r e s ) = fiv . 9) ^ , l o g s)) / (ord,res) The components of t h e i n d u c e d map 3H 1 (Y,0*(*S) —> Y ftldog * S)) - ^ 3 H 1 ( Y , U _

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