nLab
regulator of a number field

Contents

Idea

Given a number field KK, with ring of integers 𝒪 K\mathcal{O}_K, then the regular Reg KReg_K is a number which measure the size of the group of units GL 1(𝒪 K)GL_1(\mathcal{O}_K).

Since the determinant constitutes a group homomorphism K 1GL 1K_1 \to GL_1 from first algebraic K-theory to the group of units, the reguöator may also be thought of as extracting information on the first algebraic K-theory group. This is the perspective taken in the generalization to higher regulators (Beilinson regulators) which are effectivlely Chern characters for algebraic K-theory.

Properties

Relation to zeta function and class number formula

The Dedekind zeta function ζ K\zeta_K of KK has a simple pole at s=1s = 1. The class number formula says that its residue there is proportional the the product of the regulator with the class number of KK

lims1(s1)ζ K(s)ClassNumber KRegulator K. \underset{s\to 1}{\lim} (s-1) \zeta_K(s) \propto ClassNumber_K \cdot Regulator_K \,.

References

Created on August 25, 2014 at 23:25:35. See the history of this page for a list of all contributions to it.