Motives : Motivic L - functions

The exposition here follows the lecture delivered at the summer school, and hence, contains neither precision, breadth of comprehension, nor depth of insight. The goal rather is the curious one of providing a loose introduction to the excellent introductions that already exist, together with scattered parenthetical commentary. The inadequate nature of the exposition is certainly worst in the third section. As a remedy, the article of Schneider [37] is recommended as a good starting point for the complete novice, and that of Nekovar [34] might be consulted for more streamlined formalism. For the Bloch-Kato conjectures, the paper of Fontaine and Perrin-Riou [19] contains a very systematic treatment, while Kato [25] is certainly hard to surpass for inspiration. Kings [28], on the other hand, gives a nice summary of results (up to 2003). 1 Motivation Given a variety X over Q, it is hoped that a suitable analytic function ζ(X, s), a ζ-function of X, encodes important arithmetic invariants of X. The terminology of course stems from the fundamental function ζ(Q, s) = Σ ∞ 0 n −s named by Riemann, which is interpreted in this general context as the zeta function of Spec(Q). A general zeta function should generalize Riemann's function in a manner similar to Dedekind's extension to number fields. Recall that the latter can be defined by replacing the sum over positive integers by a sum over ideals: