Related papers: Computing the Hilbert Class Fields of Quartic CM F…

In this article we make an explicit approach to the higher degree case of the problem: " For a given $CM$ field $M$, construct its maximal abelian extension $C(M)$ (i.e. the Hilbert class field) by the adjunction of special values of…

In previous work, the first author developed an algorithm for the computation of Hilbert modular forms. In this paper, we extend this to all totally real number fields of even degree and nontrivial class group. Using the algorithm over…

One can define class invariants for a quartic primitive CM field K as special values of certain Siegel (or Hilbert) modular functions at CM points corresponding to K. We provide explicit bounds on the primes appearing in the denominators of…

We introduce an algorithm that computes explicit class fields of an imaginary quadratic field $K$ for a given modulus $\mathfrak{f}\subset\mathcal{O}_K$ more efficiently than the use of their classical counterparts. Therein, we prove the…

We present and analyze two algorithms for computing the Hilbert class polynomial $H_D$ . The first is a p-adic lifting algorithm for inert primes p in the order of discriminant D < 0. The second is an improved Chinese remainder algorithm…

We consider the problem of characterizing all number fields $K$ such that all algebraic integers $\alpha\in K$ can be written as the sum of distinct units of $K$. We extend a method due to Thuswaldner and Ziegler that previously did not…

Let $F$ be a totally real number field of class number one, and let $K$ be a CM-field with $F$ as its maximal real subfield. For each positive integer $N$, we construct a class group of certain binary quadratic forms over $F$ which is…

In this paper, we examine the general algorithm for class group computations, when we do not have a small defining polynomial for the number field. Based on a result of Biasse and Fieker, we simplify their algorithm, improve the complexity…

We show that we can develop from scratch and using only classical language a theory of relative quadratic extensions of a given number field $K$ which is as explicit and easy as for the well-known case that $K$ is the field of rational…

This is the first of a series of two papers in which we present a solution to Manin's Real Multiplication program -- an approach to Hilbert's 12th problem for real quadratic extensions of $\mathbb{Q}$ -- in positive characteristic, using…

Let $\ell$ be an odd prime, $q$ an odd prime power such that $q \not\equiv 0 \pmod \ell$, and $m$ the order of $q$ in $\F_\ell^\times$. We propose an explicit $L$-polynomial of hyperelliptic function field $K:=\F_q(T,…

Given a number field, it is an important question in algorithmic number theory to determine all its subfields. If the search is restricted to abelian subfields, one can try to determine them by using class field theory. For this, it is…

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…

This paper develops explicit class field theory for orders: of rank 1 in any global function field -- Hayes theory -- and of rank 2 in real quadratic function fields -- Real Multiplication. The essential ingredient in the development of the…

For a cyclic Kummer extension $K$ of a rational function field $k$ is considered, via class field theory, the extended Hilbert class field $K_H^+$ of $K$ and the corresponding extended genus field $K_g^+$ of $K$ over $k$, along the lines of…

Class field theory furnishes an intrinsic description of the abelian extensions of a number field that is in many cases not of an immediate algorithmic nature. We outline the algorithms available for the explicit computation of such…

Bruinier and Yang conjectured a formula for an intersection number on the arithmetic Hilbert modular surface, CM(K).T_m, where CM(K) is the zero-cycle of points corresponding to abelian surfaces with CM by a primitive quartic CM field K,…

We construct some families of quadratic fields whose class numbers are divisible by $3.$ The main tools used are a trinomial introduced by Kishi and a parametrization of Kishi and Miyake of a family of quadratic fields whose class numbers…

It is proved that the Hilbert class field of a real quadratic field ${\Bbb Q}(\sqrt{D})$ modulo a power $m$ of the conductor $f$ is generated by the Fourier coefficients of the Hecke eigenform for a congruence subgroup of level $fD$.

In this paper, we use the theory of genus fields to study the Euclidean ideals of certain real biquadratic fields $K.$ Comparing with the previous works, our methods yield a new larger family of real biquadratic fields $K$ having Euclidean…