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Let $D<0$ be a fundamental discriminant and denote by $E(D)$ the exponent of the ideal class group $\text{Cl}(D)$ of $K={\mathbb Q}(\sqrt{D})$. Under the assumption that no Siegel zeros exist we compute all such $D$ with $E(D)$ is a divisor…

We prove results towards classifying the possible torsion subgroups of elliptic curves over quadratic fields $\mathbb{Q}(\sqrt{d})$, where $0<d<100$ is a square-free integer, and obtain a complete classification for 49 out of 60 such…

Let $D$ be a square-free integer other than 1. Let $K$ be the quadratic field ${\mathbb Q}(\sqrt D)$. Let $\delta \in \{1,2\}$ with $\delta=2$ if $D\equiv 1 \pmod 4$. To each prime ideal $\mathcal P$ in $K$ that splits in $K/\mathbb Q$ we…

For a binary quadratic form $Q$, we consider the action of $\mathrm{SO}_Q$ on a two-dimensional vector space. This representation yields perhaps the simplest nontrivial example of a prehomogeneous vector space that is not irreducible, and…

In this work we consider constructions of genus three curves $X$ such that $\mathrm{End}(\mathrm{Jac} (X))\otimes Q$ contains the totally real cubic number field $Q(\zeta _7 +\bar{\zeta}_7 )$. We construct explicit three-dimensional…

Let $k$ and $n$ be fixed positive integers. For each prime power $q\geqslant k\geqslant 3$, we show that any subset $A\subseteq \mathbb{Z}_q^n$ free of $k$-term arithmetic progressions has size $|A|\leqslant c_k(q)^n$ with a constant…

We present a survey of recent results, scattered in a series of papers that appeared during past five years, whose common denominator is the use of cubic relations in various algebraic structures. Cubic (or ternary) relations can represent…

Let $5 \leq k \leq 11$ and $0\leq i \leq k-1$ be integers. We determine all solutions to the equation \begin{align*} n(n+d)(n+2d)\cdots(n+(i-1)d)(n+(i+1)d) \cdots (n+(k-1)d) = y^3 \end{align*} in integers $n,d,y$ with $ny \neq 0$, $d\geq…

We propose a description of colour triplets of quarks by entangled Z3-graded Lee-Wick type fields, one with real mass and the two remaining ones with mutually conjugate complex masses. This is obtained by attributing colour degrees of…

Let $K/\mathbb{Q}$ be a quadratic extension. In this paper we study the $4$-rank of the class group $\text{Cl}(K(\sqrt{n}))$, where $n$ varies over squarefree rational integers. We show that for $100\%$ of squarefree $n$, the $4$-rank is…

In this work we consider constructions of genus three curves $X$ such that $\mathrm{End}(\mathrm{Jac}(X)) \otimes Q$ contains the totally real cubic number field $Q(\zeta _ 7 + \overline{\zeta}_7)$. We construct explicit two-dimensional…

Let $d$ be a positive square-free integer. In this paper we shall investigate the structure of the $2$-class group of the cyclotomic $\mathbb{Z}_2$-extension of the imaginary biquadratic number field $\mathbb{Q}(\sqrt{d},\sqrt{-1})$.…

Let $K=\mathbb{Q}(\sqrt[n]{a})$ be an extension of degree $n$ of the field $\Q$ of rational numbers, where the integer $a$ is such that for each prime $p$ dividing $n$ either $p\nmid a$ or the highest power of $p$ dividing $a$ is coprime to…

Let ${\mathbb F}_q$ be the finite field with $q=p^k$ elements with $p$ being a prime and $k$ be a positive integer. For any $y, z\in\mathbb{F}_q$, let $N_s(z)$ and $T_s(y)$ denote the numbers of zeros of $x_1^{3}+\cdots+x_s^3=z$ and…

Let $f(x)=x^n+ax^2+bx+c \in \Z[x]$ be an irreducible polynomial with $b^2=4ac$ and let $K=\Q(\theta)$ be an algebraic number field defined by a complex root $\theta$ of $f(x)$. Let $\Z_K$ deonote the ring of algebraic integers of $K$. The…

Let $H(\lambda_4)$ be the Hecke group $\langle x,y\,:\, x^2=y^4=1 \rangle$ and, for a square-free positive integer $n$, consider the subset $\mathbb{Q}^*(\sqrt{-n})=\left\{(a+\sqrt{-n})/c \, | \, a,b=(a^2+n)/c \in \mathbb{Z},\, c\in…

The $p$-group generation algorithm is used to verify that the Hilbert $3$-class field tower has length $3$ for certain imaginary quadratic fields $K$ with $3$-class group $\mathrm{Cl}_3(K) \cong [3,3]$. Our results provide the first…

Let $K$ be a number field with ring of integers $\mathcal{O}_K$. Let $\mathcal{N}_K$ be the set of positive integers $n$ such that there exist units $\varepsilon, \delta \in \mathcal{O}_K^\times$ satisfying $\varepsilon + \delta = n$. We…

Let $p\equiv -q \equiv 5\pmod 8$ be two prime integers. In this paper, we investigate the unit groups of the fields $ L_1 =\mathbb{Q}(\sqrt 2, \sqrt{p}, \sqrt{q}, \sqrt{-1} )$ and $ L_1^+=\mathbb{Q}(\sqrt 2, \sqrt{p}, \sqrt{q} )$.…

Let $n$ be a squarefree positive odd integer. We will show that there exist infinitely many imaginary quadratic number fields with discriminant divisible by $n$ and-at the same time-having an element of order $n$ in the class group. We then…