Regulators, Deligne’s conjecture and Beilinson’s first conjecture

Abstract

In this chapter we describe two examples of maps from algebraic K-theory to Deligne-Beilinson cohomology that can be considered as a first motivation for Beilinson’s conjectures. These conjectures are then formulated in such a way that they generalize, at the same time, a conjecture of Deligne on the values of L-functions of motives at so-called critical points. We will state the conjectures only for smooth projective varieties defined over the rational numbers, but it should be observed that almost everything can be formulated for motives over arbitrary number fields, the statements becoming just more complicated. Also, it is shown how a conjecture of Zagier on polylogarithms fits the framework of Beilinson’s conjectures and the (conjectural) theory of mixed (Tate) motives. The last section contains some numerical results due to Mestre & Schappacher.