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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 Finally we will discuss generalizations of Stark's conjecture; in particular, we will study the unpublished work of B. Gross, on the values of Artin $L$-functions utilizing higher $K$-theory.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.):

• Review of elementary algebraic number theory (Dedekind domains, primes in $F$, ramification theory, properties of $\text{Gal}(K/F)$, inertia groups, decomposition groups, ramification groups etc.), adele ring $F_{\Bbb A}$ and idele group $F_{\Bbb A}^\times$ of $F$.
• Characters of the ideal group $I(F)$ of $F$. Definition of Hecke (abelian) $L$-function of $F$, basic analytic properties, idelic interpretation: Hecke characters $F_{\Bbb A}^\times\rightarrow\Bbb T=\{z\in\Bbb C\mid~\mid z\mid=1\}$ of $F$.
• Important example: $F=\Bbb Q$, Dirichlet characters and Dirichlet $L$-functions. Class field theory of $\Bbb Q$.
• A transcendental invariant of $F$: regulator $r_F$ of $F$.
• Finiteness of the ideal class group $\text{Cl}(F)$, class number $h_F$ of $F$. Analytic class number formula of Dirichlet.
• $\bullet$ Review of linear representations of finite groups (character of representations, virtual characters, irreducible representations, orthogonality relations, Frobenius reciprocity, induced representations, contragradiant representations, Brauer's theorem), linear representations of $\text{Gal}(K/F)$, Artin conductor $\frak f(\chi)$ of the character $\chi$ of a representation $\text{Gal}(K/F)\rightarrow GL(V)$.
• Definition of Artin (non-abelian) $L$-functions. Basic formal properties of Artin $L$-functions (additivity, induction, inflation). Analytic properties of Artin $L$-functions (application of Brauer's theorem), ``archimedean factors'' ($\Gamma$-factors), the functional equation satisfied by Artin $L$-functions. Some conjectures (results of Arthur, Clozel and Langlands-without proof!).
• $S$-units in $F$, Herbrand isomorphism, Stark's regulator and the statement of Stark's conjecture, connections with Dirichlet's class number formula. Equivalent forms of Stark's conjecture.
• Stark's conjecture for rational characters i.e $\text{Im}(\chi) \subset\Bbb Q$ (the work of T. Chinburg): cohomology of finite groups: Nakayama-Swan theorems, cohomological methods in class field theory.
• Preliminary notions on $K$-theory, regulators of number fields and the conjecture of B. Gross.