Deficiency of p-Class Tower Groups and Minkowski Units

TL;DR

This paper investigates the deficiency of Galois groups of maximal unramified p-extensions of number fields, linking it to units and Minkowski units, and provides formulas and methods to analyze their structure and infiniteness.

Contribution

It derives an exact formula for the deficiency of Galois groups in terms of Minkowski units and introduces techniques to analyze their relations and infiniteness.

Findings

Exact formula for deficiency in terms of Minkowski units

Methods to determine the depth of relations in the Galois group

Evidence that the Shafarevich-Koch bound is often sharp

Abstract

Let $p$ be a prime. We define the deficiency of a finitely-generated pro-$p$ group $G$ to be $r(G)-d(G)$ where $d(G)$ is the minimal number of generators of $G$ and $r(G)$ is its minimal number of relations. For a number field $K$, let $K_\emptyset$ be the maximal unramified $p$-extension of $K$, with Galois group $G_\emptyset = Gal(K_\emptyset/K)$. In the 1960s, Shafarevich (and independently Koch) showed that the deficiency of $G_\emptyset$ satisfies $$0\leq \mathrm{Def}({\rm G}_\emptyset) \leq dim (O_K^\times/(O_K^{\times })^p),$$ relating the deficiency of $G_\emptyset$ to the $p$-rank of the unit group $O_K^\times$ of the ring of integers $O_K$ of $K$. In this work, we further explore connections between relations of the group $G_\emptyset$ and the units in the tower $K_\emptyset/K$, especially their Galois module structure. In particular, under the assumption that $K$ does not contain a primitive $p$th root of unity, we give an exact formula for $\mathrm{Def}({\rm G}_\emptyset)$ in terms of the number of independent Minkowski units in the tower. The method also allows us to infer more information about the relations of G$_\emptyset$, such as their depth in the Zassenhaus filtration, which in certain circumstances makes it easier to show that G$_\emptyset$ is infinite. We illustrate how the techniques can be used to provide evidence for the expectation that the Shafarevich-Koch upper bound is "almost always" sharp.