Related papers: Lower bound for class number of certain real quadr…

In this paper, the second Kronecker ``limit" formula for a real quadratic field is established for the first time. More precisely, we obtain the second Kronecker limit formula of Zagier's zeta function. Using the reduction theory of Zagier,…

Let $d \in \{-4, -8, 8\}$. We study the $8$-part of the narrow class group in the thin families of quadratic number fields of the form $\mathbb{Q}(\sqrt{dpq})$, where $p\equiv q \equiv 1\bmod 4$ are prime numbers, and we prove new lower…

We establish necessary and sufficient conditions for a quadratic polynomial to be irreducible in the ring $Z[[x]]$ of formal power series with integer coefficients. For $n,m\ge 1$ and $p$ prime, we show that $p^n+p^m\beta x+\alpha x^2$ is…

We determine precisely when the length of the narrow 2-class field tower is $2$ for most of those real quadratic number fields whose discriminant is not a sum of two squares and for which their 2-class groups are elementary of order $4$.

We give an upper bound for the norm of the determinant of additively indecomposable, totally positive definite quadratic forms defined over the ring of integers of totally real number fields. We apply these results to find lower and upper…

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 $\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1})$…

We study the number of limit cycles bifurcating from a piecewise quadratic system. All the differential systems considered are piecewise in two zones separated by a straight line. We prove the existence of 16 crossing limit cycles in this…

In this paper, we revisit the theory of perfect unary forms over real quadratic fields. Specifically, we deduce an infinite family of real quadratic fields $\mathbb{Q}(\sqrt{d})$ when $d=2$ or $3$ mod $4$, such that there are three classes…

A standard formula (1) leads to a proof of HT90, but requires proving the existence of $\theta$ such that $\alpha\ne 0$, so that $\beta=\alpha/\sigma(\alpha)$. We instead impose the condition (M), that taking $\theta=1$ makes $\alpha=0$.…

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…

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…

We prove a version of Hilbert's Irreducibility Theorem in the quadratic case, giving a quantitative improvement to a result of Bilu-Gillibert in this restricted setting. As an application, we give improvements to several quantitative…

Let $\zeta_K(s)$ denote the Dedekind zeta-function associated to a number field $K$. In this paper, we give an effective upper bound for the height of first non-trivial zero other than $1/2$ of $\zeta_K(s)$ under the generalized Riemann…

Let $k\geq 3$ and $n\geq 3$ be odd integers, and let $m\geq 0$ be any integer. For a prime number $\ell$, we prove that the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{\ell^{2m}-2k^n})$ is either divisible by $n$ or by a…

For a prime number $p \geq 5$, we explicitly construct a family of imaginary quadratic fields $K$ with ideal class groups $Cl_{K}$ having $p$-rank ${{\rm{rk}}_{p}(Cl_{K})}$ at least $2$. We also quantitatively prove, under the assumption of…

Unit-generated orders of a quadratic field are orders of the form $\mathcal{O} = \mathbb{Z}[\varepsilon]$, where $\varepsilon$ is a unit in the quadratic field. If the order $\mathcal{O}$ is a maximal order of a real quadratic field, then…

The class numbers $h^{+}$ of the real cyclotomic fields are very hard to compute. Methods based on discriminant bounds become useless as the conductor of the field grows and methods employing Leopoldt's decomposition of the class number…

It is shown that the class number for negative discriminant $D$ can be expressed in terms of the base $B$ expansions of reduced fractions $\frac{x}{|D|}$, where $B$ is an integer prime to $D$. This result is then formulated to obtain…

Let $n\geq 3$ be an integer and $d$ an odd square-free integer. We shall compute the rank of the $2$-class group of $L_{n,d}:=\mathbb{Q}(\zeta_{2^n},\sqrt{d})$, when all the prime divisors of $d$ are congruent to $\pm 3\pmod 8$ or…

This note provides an effective lower bound for the number of primes in the quadratic progression $p=n^2+1 \leq x$ as $x \to \infty$.