Related papers: On certain quotients of $p$-class tower groups of …
The modern theory of class field towers has its origins in the study of the p-class field tower over a quadratic imaginary number field, so it is fitting that this problem be the first in the discipline to be nearing a solution. We survey…
We determine some properties of the narrow 2-class field tower of those real quadratic number fields whose discriminants are not a sum of two squares and for which their 2-class groups are elementary of order $4$. Here in Part I, we…
Cyclic number fields of odd prime degree are constructed as ray class fields over the rational number field. They are collected in multiplets sharing a common conductor and discriminant. The algorithms are implemented in Magma and applied…
The $p$-group generation algorithm from computational group theory is used to obtain information about large quotients of the pro-2 group $G = \text{Gal} (k^{nr,2}/k)$ for $k = \mathbb{Q}(\sqrt{d})$ with $d = -445, -1015, -1595, -2379$. In…
Let $p$ be an odd prime. For a number field $K$, we let $K_\infty$ be the maximal unramified pro-$p$ extension of $K$; we call the group $\mathrm{Gal}(K_\infty/K)$ the $p$-class tower group of $K$. In a previous work, as a non-abelian…
Given a random real quadratic field from $\{ \mathbb{Q}(\sqrt{p}\,) ~|~ p \text{ primes} \}$, the conjectural probability $\mathbb{P}(h=q)$ that it has class number $q$ is given for all positive odd integers $q$. Some related conjectures of…
This article is the first in a series devoted to computing the class groups of real quadratic fields. We present a new relation between the class number and the index of unit groups. This relation generalizes Hilbert class field theory for…
Recent examples of periodic bifurcations in descendant trees of finite p-groups with p in {2,3} are used to show that the possible p-class tower groups G of certain multiquadratic fields K with p-class group of type (2,2,2), resp. (3,3),…
In this paper, a new criterion is given to determine the $p-$rationality of some complex cubic number fields in terms of $ p-$divisibility of certain terms of a third-order recurrence sequence, several illustrated examples are…
Cohen and Lenstra have given a heuristic which, for a fixed odd prime $p$, leads to many interesting predictions about the distribution of $p$-class groups of imaginary quadratic fields. We extend the Cohen-Lenstra heuristic to a…
We prove a conjecture of Boston that if $p\geq 5$, all $p$-central quotients of the free group on two generators and of the free product of two cyclic groups of order $p$ are Beauville groups. In the case of the free product, we also…
In our work we investigate quotient structures and quotient spaces of a space of orderings arising from subgroups of index two. We provide necessary and sufficient conditions for a quotient structure to be a quotient space that, among other…
The purpose of this paper is to present the notion of quotient of supergroups in different categories using the unified treatment of the functor of points and to examine some physically interesting examples.
All current techniques for showing that a number field has an infinite p-class field tower depend on one of various forms of the Golod-Shafarevich inequality. Such techniques can also be used to restrict the types of p-groups which can…
We compute the Euler-Poincar\'e characteristic of quotients of the Bruhat-Tits building of PGL(n) under the action of arithmetic groups arising from central division algebras over rational function fields of positive characteristic. We use…
We give simple proofs of the Davenport--Heilbronn theorems, which provide the main terms in the asymptotics for the number of cubic fields having bounded discriminant and for the number of 3-torsion elements in the class groups of quadratic…
For unitary groups associated to a ramified quadratic extension of a $p$-adic field, we define various regular formal moduli spaces of $p$-divisible groups with parahoric levels, characterize exceptional special divisors on them, and…
In this article we continue the investigation of the length of the narrow $2$-class field tower of real quadratic number fields $\mathrm{k}$ whose discriminants are not a sum of two squares and for which their $2$-class groups are…
We define the notion of complex stratification by quasifolds and show that such spaces occur as complex quotients by certain nonclosed subgroups of tori associated to convex polytopes. The spaces thus obtained provide a natural…
The geometric conjecture developed by the authors in [1,2,3,4] applies to the smooth dual Irr(G) of any reductive p-adic group G. It predicts a definite geometric structure - the structure of an extended quotient - for each component in the…