To IUM main page

Ilhan Ikeda

Artin $L$-functions (Spring 1997)

Brief outline of the course

This will be an introductory course on Artin $L$-functions (no prerequisites assumed). After reviewing basic theory of Galois representations, we will define Artin $L$-functions, which is the non-abelian generalization of Hecke $L$-functions, and study the elementary properties of them. Our third topic will be the statement of Stark's conjectures about the leading term of the Taylor series expansion of Artin $L$-functions at $s=0$. This conjecture can be viewed as the non-abelian generalization of the classical analytic class number formula. We will reproduce a proof of Stark's conjectures for rational characters (following T. Chinburg). Finally we will discuss generalizations of Stark's conjecture; in particular, we will study the unpublished work of B. Gross, on the values of Artin $L$-functions utilizing higher $K$-theory.

More precisely, we will discuss the following topics (In what follows, let $F$ denote a number field; $S$ a finite set consisting of primes of $F$ including the archimedean ones; $K/F$ a finite Galois extension, and $V$ a finite-dimensional $\Bbb C$-vector space.):

Rambler's Top100