regulator

Let $K$ be a number field with $[K:\mathbb{Q}]=n=r_{1}+2r_{2}$ . Here $r_{1}$ denotes the number of real embeddings:

$\sigma_{i}\colon K\hookrightarrow\mathbb{R},\quad 1\leq i\leq r_{1}$

while $r_{2}$ is half of the number of complex embeddings:

$\tau_{j}\colon K\hookrightarrow\mathbb{C},\quad 1\leq j\leq r_{2}$

Note that $\{\tau_{j},\bar{\tau}_{j}\mid 1\leq j\leq r_{2}\}$ are all the complex embeddings of $K$ . Let $r=r_{1}+r_{2}$ and for $1\leq i\leq r$ define the “norm” in $K$ corresponding to each embedding:

$\parallel\cdot\parallel_{i}\colon K^{\times}\to\mathbb{R}^{+}$

$\parallel\alpha\parallel_{i}=\mid\sigma_{i}(\alpha)\mid,\quad 1\leq i\leq r_{1}$

$\parallel\alpha\parallel_{r_{1}+j}=\mid\tau_{j}(\alpha)\mid^{2},\quad 1\leq j% \leq r_{2}$

Let $\mathcal{O}_{K}$ be the ring of integers of $K$ . By Dirichlet’s unit theorem, we know that the rank of the unit group $\mathcal{O}_{K}^{\times}$ is exactly $r-1=r_{1}+r_{2}-1$ . Let

$\{\epsilon_{1},\epsilon_{2},\ldots,\epsilon_{r-1}\}$

be a fundamental system of generators of $\mathcal{O}_{K}^{\times}$ modulo roots of unity (this is, modulo the torsion subgroup). Let $A$ be the $r\times(r-1)$ matrix

$A=\left(\begin{array}[]{cccc}\log\parallel\epsilon_{1}\parallel_{1}&\log% \parallel\epsilon_{2}\parallel_{1}&\ldots&\log\parallel\epsilon_{r-1}\parallel% _{1}\\ \log\parallel\epsilon_{1}\parallel_{2}&\log\parallel\epsilon_{2}\parallel_{2}&% \ldots&\log\parallel\epsilon_{r-1}\parallel_{2}\\ \vdots&\vdots&\ddots&\vdots\\ \log\parallel\epsilon_{1}\parallel_{r}&\log\parallel\epsilon_{2}\parallel_{r}&% \ldots&\log\parallel\epsilon_{r-1}\parallel_{r}\\ \end{array}\right)$

and let $A_{i}$ be the $(r-1)\times(r-1)$ matrix obtained by deleting the $i$ -th row from $A$ , $1\leq i\leq r$ . It can be checked that the determinant of $A_{i}$ , $\det{A_{i}}$ , is independent up to sign of the choice of fundamental system of generators of $\mathcal{O}_{K}^{\times}$ and is also independent of the choice of $i$ .

Definition.

The regulator of $K$ is defined to be

$\operatorname{Reg}_{K}=\mid\det{A_{1}}\mid$

The regulator is one of the main ingredients in the analytic class number formula for number fields.

References

1 Daniel A. Marcus, Number Fields, Springer, New York.
2 Serge Lang, Algebraic Number Theory. Springer-Verlag, New York.

Title	regulator
Canonical name	Regulator
Date of creation	2013-03-22 13:54:34
Last modified on	2013-03-22 13:54:34
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	8
Author	alozano (2414)
Entry type	Definition
Classification	msc 11R27
Related topic	NumberField
Related topic	DirichletsUnitTheorem
Related topic	ClassNumberFormula
Related topic	RegulatorOfAnEllipticCurve
Defines	regulator of a number field