DescriptionThis project is a natural continuation of the Number Theory III module. There one learns that the class group of a number field K, defined as the quotient group of the fractional ideals modulo the principal ideals of K, is a finite abelian group, and its size is usually called the class number of K. This number can be thought as a measure of how far is the ring of integers of K from being a Unique Factoraization Domain.In this project we will relate in a rather explicit way the class number of K to a complex function attached to K, namely the Dedekind zeta function of K. Given a field K, the Dedekind zeta function of K is defined as $$\zeta_K(s) = \prod_{\mathfrak{p}} (1-N(\mathfrak{p})^{-s})^{-1},$$ where the product is taken over all prime ideals of K, N() denotes the norm of an ideal, and s is a complex variable. This infinite product converges absolutely when Re(s) > 1 and hence defines an complex analytic function there. When one takes K=Q, the field of rational numbers, then the Dedekind zeta function is nothing else than the familiar Riemann zeta function. The first aim of this project will be to establish the meromorphic continuation of the Dedekind zeta function to the entire complex plane, when K above is taken to be a quadratic field. It turns out that the function has a pole of order one at s=1, and our aim will be to compute its residue. We will obtain an explicit formula for this, the so called Class Number Formula. For example when K is taken to be an imaginary quadratic field we have that, $$ Res_{s=1} \zeta_K(s) = \frac{2 \, \pi\, h}{w \, \sqrt{|D|}},$$ where h is the class number of K, w is the number of roots of unity in K and D is the discriminant of K over the rationals. The formula is slightly more compicated when one takes K to be a real quadratic field due to the existence of units of infinite order. Actaully we will see that one can obtain an even more precise formula by involving the value at s=1 of a particular Dirichlet series attached to a quadratic character. Depending on interest the project may take various directions. The following list is by all means not complete. 1) Understand the values of the Dedekind zeta function at negative integers when K is a real quadratic field. 2) Given two number fields K and F with K a finite extension of F, one can establish a relation between the Dedekind zeta functions of the two fields involving the so-called Artin L functions. 3) Study the properties of Dirichlet L series and their application as for example Dirichlet's theorem on arithmetic progressions.
ResourcesThere are many reference for zeta functions of number fields covering the Class Number Formula. Here are some of them
[2] J. Neukirch, Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, 1999, [3] D Zagier, Zetafunktionen und quadratische Koerper, Hochschultext, Springer-Verlag 1981 (this is an excellent resource but unfortunately available only in German).
Pre-requisites
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