Related papers: Composition of Binary Quadratic Forms over Number …

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.

Let $ n $ be an integer and $ n\ge 2 $. A classic integral quadratic form over local fields is called classic $ n $-universal if it represents all $n$-ary classic integral quadratic forms. We determine the equivalent conditions and minimal…

Positive and negative quadratic forms are well known and widely used. They are multivariate homogeneous polynomials of degree two taking positive or negative values respectively for any values of their arguments not all zero. In the present…

For any integer $k\ge 1$, we show that there are infinitely many complex quadratic fields whose 2-class groups are cyclic of order $2^k$. The proof combines the circle method with an algebraic criterion for a complex quadratic ideal class…

In this paper we obtain an asymptotic formula for the number of $\operatorname{SL}_2(\mathbb{Z})$-equivalence classes of positive definite binary quadratic forms over $\bZ$ having bounded discriminant $\Delta = 1-4p$, with $p$ a prime. We…

A (positive definite integral) quadratic form is called almost 2-universal if it represents all (positive definite integral) binary quadratic forms except those in only finitely many equivalence classes. Oh [7] determined all almost…

We consider the question of determining whether two binary cubic forms over an arbitrary field $K$ whose characteristic is not $2$ or $3$ are equivalent under the actions of either GL$(2,K)$ or SL$(2,K)$, deriving two necessary and…

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 an explicit upper bound on the number of real quadratic fields that admit a universal quadratic form of a given rank, thus establishing a density zero statement. More generally, we obtain such a result for totally positive definite…

We call a positive definite Hermitian lattice regular if it represents all integers which can be represented locally by the lattice. We investigate binary regular Hermitian lattices over imaginary quadratic fields $\mathbb{Q}(\sqrt{-m})$…

A Lie algebra is said to be quadratic if it admits a symmetric invariant and non-degenerated bilinear form. Semisimple algebras with the Killing form are examples of these algebras, while orthogonal subspaces provide abelian quadatric…

The association of algebraic objects to forms has had many important applications in number theory. Gauss, over two centuries ago, studied quadratic rings and ideals associated to binary quadratic forms, and found that ideal classes of…

In analogy with the 290-Theorem of Bhargava-Hanke, a criterion set is a finite subset $C$ of the totally positive integers in a given totally real number field such that if a quadratic form represents all elements of $C$, then it…

We describe first-degree prime ideals of biquadratic extensions in terms of first-degree prime ideals of two underlying quadratic fields. The identification of the prime divisors is given by numerical conditions involving their ideal norms.…

Let $n>1$ be an odd integer. We prove that there are infinitely many imaginary quadratic fields of the form $\mathbb{Q}(\sqrt{x^2-2y^n})$ whose ideal class group has an element of order $n$. This family gives a counter example to a…

For a field extension $L/K$ we consider maps that are quadratic over $L$ but whose polarisation is only bilinear over $K$. Our main result is that all such are automatically quadratic forms over $L$ in the usual sense if and only if $L/K$…

We classify all quadratic imaginary number fields that have a Euclidean ideal class. There are seven of them, they are of class number at most two, and in each case the unique class that generates the class-group is moreover norm-Euclidean.

We provide, explicitly, equivalences and dual equivalences between categories of abstract quadratic forms theories and subcategories of multifields and multirings, that will bring new perspectives and methods to the abstract theories of…

We investigate the average distribution of primes represented by positive definite integral binary quadratic forms, the average being taken over negative fundamental discriminants in long ranges. In particular, we prove corresponding…

We provide an algorithm that, given any order $O$ in a quaternion algebra over a global field, computes representatives of all right equivalence classes of right $O$-ideals, including the non-invertible ones. The theory is developed for a…