Math 254a: Evaluating the L-series

Math 254a: Dirichlet L-series

1 n : (n, f) = 1 0 : otherwise , the absolute convergence for s > 1 follows from the convergence of ζQ(s) for s > 1. (ii) Given the convergence for s > 1, the proof of the product expansion is identical to the case ζQ(s). We also observe that L(s, χ1) = ζQ(s). As we will see, for non-trivial characters, the L-series behave rather differently. We will analyze these functions further, but first w...

Math 254a: Zeta Functions and L-series

Theorem 1.1. Let G be a group of Dirichlet characters modulo n, and KG the associated field. Fix a prime number p, and let H1 ⊂ H2 ⊂ G be the groups of χ such that χ(p) = 1 or χ(p) 6= 0 respectively. Then G/H2 ∼= IKG,p ⊂ Gal(KG/Q), and G/H1 ∼= DKG,p (note that these are non-canonical isomorphisms). We are now ready to prove the theorem on factoring Dedekind zeta functions of sub-cyclotomic fiel...

Math 254a: the Analytic Class Number Formula I

Math 254a: the Analytic Class Number Formula, and Dirichlet Characters

Math 254a: Class Field Theory: an Overview

Class field theory relates abelian extensions of a given number field K to certain generalized ideal class groups ofK. The fundamental tool for doing this is Frobenius elements, and the Artin map obtained from them. Recall from lecture 24 that if L/K is an abelian extension (i.e., a Galois extension with abelian Galois group), and p is a prime of OK unramified in OL, then we defined Fr(p) ∈ Gal...