transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
Given a number field $K$, with ring of integers $\mathcal{O}_K$, then the regulator $Reg_K$ is a number which measures the size of the group of units $GL_1(\mathcal{O}_K)$.
Since the determinant constitutes a group homomorphism $K_1 \to GL_1$ 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.
The Dedekind zeta function $\zeta_K$ of $K$ has a simple pole at $s = 1$. The class number formula says that its residue there is proportional to the product of the regulator and the class number of $K$
Last revised on October 20, 2022 at 09:16:47. See the history of this page for a list of all contributions to it.