Related papers: Imaginary quadratic number fields with class group…

We improve a result of H. L. Montgomery and J. P. Weinberger by establishing the existence of infinitely many fundamental discriminants $d>0$ for which the class number of the real quadratic field $\mathbb{Q}(\sqrt{d})$ exeeds…

We present an improved algorithm for tabulating class groups of imaginary quadratic fields of bounded discriminant. Our method uses classical class number formulas involving theta-series to compute the group orders unconditionally for all…

The theory of continued fractions of functions $ \sqrt D $ is used to give lower bound for class numbers $h(D)$ of general real quadratic function fields $K=k(\sqrt D)$ over $k={\bf F}_q(T)$. For five series of real quadratic function…

We prove that there are >>X^{1/30}/(log X) imaginary quadratic number fields with an ideal class group of 3-rank at least 5 and discriminant bounded in absolute value by X. This improves on an earlier result of Craig, who proved the…

In this paper, we obtain an asymptotic formula for the number of imaginary quadratic fields with prime discriminant and class number up to $H$, as $H\to \infty$. Previously, such an asymptotic was only known under the assumption of the…

Let $K$ be a number field, $\mathfrak{q}$ be an integral ideal, and $\mathrm{Cl}(\mathfrak{q})$ be the associated ray class group. Suppose $\mathrm{Cl}(\mathfrak{q})$ possesses a real exceptional character $\psi$, possibly principal, with a…

We prove that, for any $\varepsilon>0$, the number of real quadratic fields $\mathbb{Q}(\sqrt{d})$ of discriminant $d<x$ whose class number is $\ll \sqrt{d}(\log{d})^{-2}(\log\log{d})^{-1}$ is at least $x^{1/2-\varepsilon}$ for $x$ large…

For any integer $l\geq 1$, let $p_1, p_2, \ldots, p_{l+2}$ be distinct prime numbers $\geq 5.$ For all real numbers $X>1,$ we let $N_{3,l}(X)$ denote the number of real quadratic fields $K$ whose absolute discriminant $d_K\leq X$ and $d_K$…

In this paper, we construct certain infinite families of imaginary quadratic fields whose class number is divisible by a given positive integer.

For a square-free integer $t$, Byeon \cite{byeon} proved the existence of infinitely many pairs of quadratic fields $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{tD})$ with $D > 0$ such that the class numbers of all of them are indivisible…

Let $E$ be an elliptic curve over $\mathbb{Q}$, $p$ an odd prime number and $n$ a positive integer. In this article, we investigate the ideal class group $\mathrm{Cl}(\mathbb{Q}(E[p^n]))$ of the $p^n$-division field $\mathbb{Q}(E[p^n])$ of…

Let $D<0$ be a fundamental discriminant and $h(D)$ be the class number of $\mathbb{Q}(\sqrt{D})$. Let $R(X,D)$ be the number of classes of the binary quadratic forms of discriminant $D$ which represent a prime number in the interval…

Let n be an odd number and F an imaginary quadratic field with odd discriminant. We show that there exists infinitely many cubic fields K such that the class number of K is divisible by n and the Galois closure of K contains F.

Here we study algebraic function fields K, give necessary and sufficient condition for the ideal class group $H(K)$ of any real quadratic function field $K$ to have a cyclic subgroup of order $n$, and obtain eight series of such fields $K$,…

Let $d$ be a square free integer and $L_d:=\mathbb{Q}(\zeta_{8},\sqrt{d})$. In the present work we determine all the fields $L_d$ such that the $2$-class group, $\mathrm{Cl}_2(L_d)$, of $L_d$ is of type $(2,4)$ or $(2,2,2)$.

In this paper we obtain a complete list of imaginary $n$-quadratic fields with class groups of exponent $3$ and $5$ under ERH for every positive integer $n$ where an $n$-quadratic field is a number field of degree $2^n$ represented as the…

We quantify a recent theorem of Wiles on class numbers of imaginary quadratic fields by proving an estimate for the number of negative fundamental discriminants down to -X whose class numbers are indivisible by a given prime and whose…

The goal is to obtain an asymptotic formula for the number of quadratic extensions with bounded discriminant of a some quadratic number field with odd class number. This extends an already known result for Q.

Let $k \geq 1$ be a cube-free integer with $k \equiv 1 \pmod {9}$ and $\gcd(k, 7\cdot 571)=1$. In this paper, we prove the existence of infinitely many triples of imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1})$…

Assume $x,\ y,\ n$ are positive integers and $n$ is odd. In this note, we show that the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{x^{2}-y^{n}})$ is divisible by $n$ for fixed $x, n$ if $\gcd(2x,y)=1$ and $y>C$ where…