The general formalism of L-functions, Deligne cohomology and Poincaré duality theories

Abstract

In this chapter we consider smooth projective varieties defined over Q and we define their L-functions. The whole formalism depends on several conjectures, suggested by the zero- and one-dimensional cases. The main ingredient of this chapter, Deligne-Beilinson cohomology, is introduced and shown to be a Poincaré duality theory. Such a (co)homology theory has the right properties to admit a formalism of characteristic classes which will generalize the classical regulator. This will be further explained in the next chapter.