|
| | | | |
| | | |
| Stat |
Members: 3645
Articles: 2'506'133
Articles rated: 2609
27 April 2024 |
|
| | | | |
|
Article overview
| |
|
|
Deficiency of p-Class Tower Groups and Minkowski Units | | Farshid Hajir
; Christian Maire
; Ravi Ramakrishna
; | | Date: |
17 Mar 2021 | | 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
$$0leq mathrm{Def}({
m G}_emptyset) leq dim (O_K^ imes/(O_K^{ imes
})^p),$$ relating the deficiency of $G_emptyset$ to the $p$-rank of the unit
group $O_K^ imes$ 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 $mathrm{Def}({
m
G}_emptyset)$ 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. | | Source: | arXiv, 2103.09508 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)
|
| |
|
|
|
|
News, job offers and information for researchers and scientists:
| |
REGISTER