Related papers: A note on Euclidean cyclic cubic fields

Let $K$ be a composite field of some real quadratic fields. We give a sufficient condition on $K$ such that all elliptic curves over $K$ is modular.

Motivated by classical results of Aubry, Davenport and Cassels, we define the notion of a Euclidean quadratic form over a normed integral domain and an ADC form over an integral domain. The aforementioned classical results generalize to:…

We give a complete proof of Thurston's Orbifold Theorem for very good 3-orbifolds of cyclic type. An orbifold is said to be very good when it has a finite cover which is a manifold. A 3-orbifold is of cyclic type if the singular set is a…

In this paper we give two different proofs of Bobenko and Springborn's theorem of circle pattern: there exists a hyperbolic (or Euclidean) circle pattern with proscribed intersection angles and cone angles on a cellular decomposed surface…

Let p denote an odd prime. For all p-admissible conductors c over a quadratic number field \(K=\mathbb{Q}(\sqrt{d})\), p-ring spaces \(V_p(c)\) modulo c are introduced by defining a morphism \(\psi:\,f\mapsto V_p(f)\) from the divisor…

In this article we explain how to construct cyclic octic unramfied extensions of the real quadratic number field $k = {\mathbb Q}(\sqrt{2p}\,)$, where $p \equiv 1 \bmod 8$ is a prime number such that $h_2(k) \equiv 0 \bmod 8$. The…

We show that the vector of period ratios of a cubic surface is rational over $Q(\omega)$, where $\omega = \exp(2\pi i/3)$ if and only if the associate abelian variety is isogeneous to a product of Fermat elliptic curves. We also show how to…

This paper is devoted to the units of integral group rings of cyclic $2$-groups of small orders, namely, the orders of $2^n$ for $n<8$. Immediately we should note the issues our consideration describe in the introduction in more detail.…

We present computational results on the divisor class number and the regulator of a cubic function field over a large base field. The underlying method is based on approximations of the Euler product representation of the zeta function of…

Let $m\in\mathbb{Z}$ be an integer and $L_m=\mathbb{Q}(\alpha)$ be the simplest cubic field with class number $h_m$ and conductor $\mathfrak{f}_m$ where $\alpha$ is a root of $f_m(X)=X^3-mX^2-(m+3)X-1$. Let $\mathcal{O}_{L_m}$ be the ring…

We prove that the integral Hodge conjecture holds for 1-cycles on irreducible holomorphic symplectic varieties of K3 type and of Generalized Kummer type. As an application, we give a new proof of the integral Hodge conjecture for cubic…

Let $K/\mathbb{Q}$ be an algebraic number field of class number one and $\mathcal{O}_K$ be its ring of integers. We show that there are infinitely many non-Wieferich primes with respect to certain units in $\mathcal{O}_K$ under the…

We introduce a new class of surfaces in Euclidean $3$-space, called surfaces of osculating circles, using the concept of osculating circle of a regular curve. These surfaces contain a uniparametric family of planar lines of curvature. In…

We prove that homoclinic classes for a residual set of C^1 vector fields X on closed n-manifolds are maximal transitive and depend continuously on periodic orbit data. In addition, X does not exhibit cycles formed by homoclinic classes. We…

In 2024, M. K. Ram proved that the class number of an imaginary cyclic quartic number field is never equal to a prime $p\equiv 3\pmod 4$. Here we greatly generalize this result to the case of the non-quadratic imaginary cyclic number fields…

We show that if two division $p$-algebras of prime degree share an inseparable field extension of the center then they also share a cyclic separable one. We show that the converse is in general not true. We also point out that sharing all…

We discuss continued fractions on real quadratic number fields of class number 1. If the field has the property of being 2-stage euclidean, a generalization of the euclidean algorithm can be used to compute these continued fractions.…

Let F be the cubic field of discriminant -23 and let O be its ring of integers. By explicitly computing cohomology of congruence subgroups of GL(2,O), we computationally investigate modularity of elliptic curves over F.

A number field K is a finite extension of rational number field Q. A circulant digraph integral over K means that all its eigenvalues are algebraic integers of K. In this paper we give the sufficient and necessary condition for circulant…

We focus on two important classes of lattices, the well-rounded and the cyclic. We show that every well-rounded lattice in the plane is similar to a cyclic lattice, and use this cyclic parameterization to count planar well-rounded…