Related papers: Imaginary multiquadratic number fields of exponent…

In this article, we study necessary conditions for certain square-free integers to be congruent numbers. Our method uses divisibility properties of class numbers of related imaginary quadratic fields. We first consider positive square-free…

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…

I discuss a variety of results involving s(n), the number of representations of n as a sum of three squares. One of my objectives is to reveal numerous interesting connections between the properties of this function and certain modular…

The modern theory of class field towers has its origins in the study of the p-class field tower over a quadratic imaginary number field, so it is fitting that this problem be the first in the discipline to be nearing a solution. We survey…

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…

Suppose $E$ is an elliptic curve defined over $\Q$. At the 1983 ICM the first author formulated some conjectures that propose a close relationship between the explicit class field theory construction of certain abelian extensions of…

Let $\ell$ be a prime number and $q$ be a power of $\ell$. Given an odd prime number $p$ and an imaginary quadratic extension $F$ of the rational function field $\mathbb{F}_q(T)$, let $\lambda_p(F)$ denote the Iwasawa $\lambda$-invariant of…

In this paper, we study additively indecomposable quadratic forms over real biquadratic and simplest cubic fields. In particular, we show that over these fields, we can always find such a classical form in 2 variables, which differs from…

Let $(\alpha,\mathcal{N}_{\alpha})$ and $(\beta,\mathcal{N}_{\beta})$ be two canonical number systems for an imaginary quadratic number field $K$ such that $\alpha$ and $\beta$ are multiplicatively independent. We provide an effective lower…

The aim of this paper is to study the group of elliptic units of a cyclic extension $L$ of an imaginary quadratic field $K$ such that the degree $[L:K]$ is a power of an odd prime $p$. We construct an explicit root of the usual top…

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…

In this paper we consider certain quaternary quadratic forms and octonary quadratic forms and by using the theory of modular forms, we find formulae for the number of representations of a positive integer by these quadratic forms.

The number F(h) of imaginary quadratic fields with a given class number h is of classical interest: Gauss' class number problem asks for a determination of those fields counted by F(h). The unconditional computation of F(h) for h up to 100…

A positive-definite integral quadratic form is called regular if it represents every positive integer which is locally represented. In this article, we classify all regular diagonal quadratic forms of rank greater than 3.

A curious identity of Bunyakovsky (1882), made more widely known by P\'olya and Szeg{\H o} in their ``Problems and Theorems in Analysis", gives an evaluation of a sum of the floor function of square roots involving primes $p\equiv…

Cohn asks if for every real quadratic field Q(m) with discriminant d there exists a non-maximal order corresponding to f > 1 such that the relative class number Hd(f) = h(f2d)/h(d) is one. We prove that when m = 46 (and in seven other…

Algebraic integers in totally imaginary quartic number fields are not discrete in the complex plane under a fixed embedding, which makes it impossible to visualize all integers in the plane, unlike the quadratic imaginary algebraic…

Let $A$ be a real quadratic order of discriminant $p$ or $4p$ with a prime $p$. In this paper we classify all proper totally imaginary quadratic $A$-orders $B$ with index $w(B)=[B^\times: A^\times]>1$. We also calculate numerical invariants…

Recently, the authors showed that for every irrational number $\alpha$, there exist infinitely many positive integers $n$ represented by any given positive definite binary quadratic form $Q$, satisfying $||\alpha n||<n^{-(1/2-\varepsilon)}$…

We show that there are only finitely many finite fields whose members are the sum of an $n$-potent element and a $5$-potent element. Combining this with the algorithmic results provided by S.D. Cohen {\it et al.}, we confirm in the…