Motives and Motivic L-functions

This report aims to be an exposition of the theory of L-functions from the motivic point of view. The classical theory of pure motives provides a category consisting of ‘universal cohomology theories’ for smooth projective varieties defined over – for instance – number fields. Attached to every motive we can define a function which is holomorphic on a subdomain of C which at least conjecturally satisfies similar properties to the Riemann zeta function: for instance meromorphic continuation and functional equation. The properties of this function are expected to encode deep arithmetic properties of the underlying variety. In particular, we will discuss a conjecture of Deligne and its more general forms due to Beilinson and others, that relates certain “special” values of the L-function – values at integer points – to a subtle arithmetic invariant called the regulator, vastly generalising the analytic class number formula. Finally I aim to describe potentially fruitful ways in which these exciting conjectures might be rephrased or reinterpreted, at least to make things clearer to me.