Math 254a: First Lecture

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...

Lecture Notes 1 for 254a

The aim of this course is to tour the highlights of arithmetic combinatorics the combinatorial estimates relating to the sums, differences, and products of finite sets, or to related objects such as arithmetic progressions. The material here is of course mostly combinatorial, but we will also exploit the Fourier transform at times. We will also discuss the recent applications of this theory to ...

Math 254a: Evaluating the L-series

We had fixed the notation ζ f = e 2πi f , and τ (χ) = f k=1 χ(k)ζ k f , if χ has conductor f. We want to show: Theorem 0.1. Let χ be a Dirichlet character of conductor f > 1. Then for χ even, we have L(1, χ) = −