Distribution of units of an algebraic number field

Abstract

Let F be an algebraic number field with maximal order $0F$ and let $0\mathop x\limits_F$ be the group of all units of F We assume that $0\mathop x\limits_F$ is an infinite group and we are interested in their distribution. For an integral ideal n of F, we set $E(n): = \{ modn|u \in 0\mathop x\limits_F \} ( \subset {(0F/n)^x})$ and $I\left( n \right): = \left[ {{{\left( {{O_F}/n} \right)}^X}:E\left( n \right)} \right]$ which is equal to the extension degree of the ramified part of the ray class field corresponding to the ideal n over F Let K be a Galois extension of the rational number field $\mathbb{Q}$, which contains the field F, and fix an element $\eta$ of Gal $\left( {K/\mathbb{Q}} \right)$. We consider a primitive integral polynomial g(x) such that $\left\{ {{\varepsilon ^{g\left( \eta \right)}}\left| {\varepsilon \in O_F^X} \right.} \right\}$ is a finite group with order ${\delta _1}$ . Here we choose such a polynomial g(x) of minimal degree. Then $h\left( x \right): = \left( {{x^d} - 1} \right)/g\left( x \right)$ is in $\mathbb{Z}\left[ x \right]$ for $d: = \left[ {\left\langle \eta \right\rangle :\left\langle \eta \right\rangle \cap Gal\left( {K/F} \right)} \right]$. Next we take a maximal natural number ${\delta _0}$ such that ${\sqrt[{{\delta _0}}]{\varepsilon }^{{\delta _1}g\left( \rho \right)}} = 1$ holds for every $\varepsilon \in O_F^X$ and any extension $\rho$ of $\eta$ . Let p be a prime number and $\mathfrak{p}$ a prime ideal of F lying above p such that there is a prime ideal $\mathfrak{P}$ of K with $\mathfrak{p} = \mathfrak{P} \cap F$ whose Frobenius automorphism is $\eta$ Then $\ell \left( p \right): = {\delta _0}h\left( p \right)/{\delta _1}$ is an integer dividing $I\left( \mathfrak{p} \right)$ and we conjecture that the set of such prime ideals with $I\left( \mathfrak{p} \right) = \ell \left( p \right)$ has a positive density $k\left( \eta \right)$ We will give the conjectural density $k\left( \eta \right)$ by the Euler product and study its positivity. Degrees of prime ideals in question are greater than or equal to 1 and so our density is a modified form of a natural density.

Partially supported by Grant-in-Aid for Scientific Research (C), The Ministry of Education, Science, Sports and Culture.