nLab regulator of a number field




Given a number field KK, with ring of integers 𝒪 K\mathcal{O}_K, then the regulator Reg KReg_K is a number which measures 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 regulator 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 effectively Chern characters for algebraic K-theory.


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 to the product of the regulator and 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 \,.


Last revised on October 20, 2022 at 09:16:47. See the history of this page for a list of all contributions to it.