Integral Ternary Quadratic Forms Research Articles

Near-miss identities and spinor genus classification of ternary quadratic forms with congruence conditions

In this paper, near-miss identities for the number of representations of some integral ternary quadratic forms with congruence conditions are found and proven. The genus and spinor genus of the corresponding lattice cosets are then classified. Finally, a complete genus and spinor genus classification for all conductor 2 lattice cosets of 2-adically unimodular lattices is given.

On the analytic theory of isotropic ternary quadratic forms

A new local-global result about the primitive representations of zero by integral ternary quadratic forms is proven. By an extension of a result of Kneser (given in the Appendix), it yields a quantitative supplement to the Hasse principle on the number of automorphic orbits of primitive zeros of a genus of forms. One ingredient in its proof is an asymptotic formula for a count of the zeros of a given form in such an orbit.

Explicit class number formulas for Siegel–Weil averages of ternary quadratic forms

In this paper, we investigate the interplay between positive-definite integral ternary quadratic forms and class numbers. We generalize a result of Jones relating the theta function for the genus of a quadratic form to the Hurwitz class numbers, obtaining an asymptotic formula (with a main term and error term away from finitely many bad square classes t j Z 2 t_j\mathbb {Z}^2 ) relating the number of lattice points in a quadratic space of a given norm with a sum of class numbers related to that norm and the squarefree part of the discriminant of the quadratic form on this lattice.

Counting integral points on indefinite ternary quadratic equations over number fields

Abstract We study an asymptotic formula for counting integral points over an equation defined by a non-degenerate indefinite integral ternary quadratic form f representing a non-zero integer a such that $-a\cdot det(\,f)$ is square over a number field. In particular, we prove that the leading coefficient of this asymptotic formula is given by the product of local densities normalized by $1-p^{-1}$ over all finite primes p.

Class numbers and representations by ternary quadratic forms with congruence conditions

In this paper, we are interested in the interplay between integral ternary quadratic forms and class numbers. This is partially motivated by a question of Petersson.

Local densities of diagonal integral ternary quadratic forms at odd primes

We give formulas for local densities of diagonal integral ternary quadratic forms at odd primes. Exponential sums and quadratic Gauss sums are used to obtain these formulas. These formulas (along with 2-adic densities and Siegel’s mass formula) can be used to compute the representation numbers of certain ternary quadratic forms.

A conjecture on the zeta functions of pairs of ternary quadratic forms

We consider the prehomogeneous vector space of pairs of ternary quadratic forms. For the lattice of pairs of integral ternary quadratic forms and its dual lattice, there are six zeta functions associated with the the prehomogeneous vector space. We present a conjecture which states that there are simple relations among the six zeta functions. We prove that the coefficients coincide on fundamental discriminants.

On the singular points of the orbifolds arising from integral, ternary quadratic forms

Mennicke (Proc R Soc Edinb Sect A 67:309–352, 1963; Proc R Soc Edinb Sect A 88:151–157, 1981) investigated the number of conic points with isotropies of orders 3, 4 and 6 that can appear in the orientable twofold orbifold covering of the orbifold $$Q_{f}$$ associated with an integral ternary quadratic form. Mennicke made essential use of a theorem of Jones (The Arithmetic Theory of Quadratic Forms. The Carus Mathematical Monographs, vol. 10. MAA, Baltimore, 1950, Theorem 86). In this paper we revisit Mennicke’s results and we extend them to $$Q_{f},$$ that is, even when $$Q_{f}$$ is non-orientable. Our method is new and independent of Jones’ Theorem. We also study the possible cusp points.

Spinor representations of positive definite ternary quadratic forms

For a positive definite integral ternary quadratic form [Formula: see text], let [Formula: see text] be the number of representations of an integer [Formula: see text] by [Formula: see text]. The famous Minkowski–Siegel formula implies that if the class number of [Formula: see text] is one, then [Formula: see text] can be written as a constant multiple of a product of local densities which are easily computable. In this paper, we consider the case when the spinor genus of [Formula: see text] contains only one class. In this case the above also holds if [Formula: see text] is not contained in a set of finite number of square classes which are easily computable. By using this fact, we prove some extension of the recent results on both the representations of generalized Bell ternary forms and the representations of ternary quadratic forms with some congruence conditions.

On the existence of integral ternary quadratic forms without automorphs of finite order

An example of an integral ternary quadratic form [Formula: see text] such that its associated orbifold [Formula: see text] is a manifold is given. Hence, the title is proved.

On the orbifold coverings associated to integral, ternary quadratic forms

The group of (integral) automorphs of a ternary integral quadratic form f acts properly discontinuously as a group of isometries of the Riemann’s sphere (resp. the hyperbolic plane) if f is definite (resp. indefinite) and the quotient has a natural structure of spherical (resp. hyperbolic) orbifold, denoted by \(Q_{f}\). Then f is a B-covering of the form g if \(Q_{f}\) is an orbifold covering of \(Q_{g}\), induced by $$\begin{aligned} T^{\prime }fT=\rho g \end{aligned}$$ where T is an integral matrix and \(\rho >0\) is an integer. Given an integral ternary quadratic form f a number \(\Pi _{f}\), called the B-invariant of the form f, is defined. It is conjectured that if f is a B-covering of the form g then both forms have the same B-invariant. The purpose of this paper is to reduce this conjecture to the case in which g is a form with square-free determinant. This reduction is based in the following main Theorem. Any definite (resp. indefinite) form f is a B-covering of a, unique up to genus (resp. integral equivalence), form g with square-free determinant such that f and g have the same B-invariant. To prove this, a normal form of any definite (resp. indefinite) integral, ternary quadratic form f is introduced. Some examples and open questions are given.

Multiplicative property of representation numbers of ternary quadratic forms

Let f be a positive definite integral ternary quadratic form and $$\theta (z;f)=\sum _{n=0}^{\infty }a(n;f)q^n$$ its theta function. For any fixed square-free positive integer t with $$a(t;f)\ne 0$$ , we define $$\rho (n;t,f):=a(tn^2;f)/a(t;f)$$ . For the case when $$f=x_1^2+x_2^2+x_3^2$$ and $$t=1$$ , Hurwitz proved that $$\rho (n;t,f)$$ is multiplicative and he gave its expression. Cooper and Lam proved four similar formulas and proposed a conjecture for some other cases. Using the results given in this paper, we can check the multiplicative property of $$\rho (n;t,f)$$ for many cases. All cases in Cooper and Lam’s conjecture are included in ours.

The number of representations of squares by integral ternary quadratic forms

Let f be a positive definite integral ternary quadratic form and let r(k,f) be the number of representations of an integer k by f. In this article we study the number of representations of squares by f. We say the genus of f, denoted by gen(f), is indistinguishable by squares if for every integer n, r(n2,f)=r(n2,f′) for every quadratic form f′∈gen(f). We find some non-trivial genera of ternary quadratic forms which are indistinguishable by squares. We also give some relation between indistinguishable genera by squares and the conjecture given by Cooper and Lam, and we resolve their conjecture completely.

Genus-correspondences respecting spinor genus

For two positive definite integral ternary quadratic forms f and g and a positive integer n, if n⋅g is represented by f and n⋅dg=df, then the pair (f,g) is called a representable pair by scaling n. The set of all representable pairs in gen(f)×gen(g) is called a genus-correspondence. In [6], Jagy conjectured that if n is square free and the number of spinor genera in the genus of f equals to the number of spinor genera in the genus of g, then such a genus-correspondence respects spinor genus in the sense that for any representable pairs (f,g),(f′,g′) by scaling n, f′∈spn(f) if and only if g′∈spn(g). In this article, we show that by giving a counter example, Jagy's conjecture does not hold. Furthermore, we provide a necessary and sufficient condition for a genus-correspondence to respect spinor genus.

The number of representations of squares by integral ternary quadratic forms (II)

Let f be a positive definite ternary quadratic form. We assume that f is non-classic integral, that is, the norm ideal of f is Z. We say f is strongly s-regular if the number of representations of squares of integers by f satisfies the condition in Cooper and Lam's conjecture in [3]. In this article, we prove that there are only finitely many strongly s-regular ternary forms up to isometry if the minimum of the nonzero squares that are represented by the form is fixed. In particular, we show that there are exactly 207 non-classic integral strongly s-regular ternary forms that represent one (see Tables 1 and 2). This result might be considered as a complete answer to a natural extension of Cooper and Lam's conjecture.

Parametrizing quartic algebras over an arbitrary base

We parametrize quartic commutative algebras over any base ring or scheme (equivalently finite, flat degree four $S$-schemes), with their cubic resolvents, by pairs of ternary quadratic forms over the base. This generalizes Bhargava's parametrization of quartic rings with their cubic resolvent rings over $\mathbb{Z}$ by pairs of integral ternary quadratic forms, as well as Casnati and Ekedahl's construction of Gorenstein quartic covers by certain rank 2 families of ternary quadratic forms. We give a geometric construction of a quartic algebra from any pair of ternary quadratic forms, and prove this construction commutes with base change and also agrees with Bhargava's explicit construction over $\mathbb{Z}$.

Spinor Regular Positive Ternary Quadratic Forms

Refining the notion of regularity introduced by Dickson, an integral quadratic form is said to be spinor regular if it represents all integers represented by its spinor genus. Examples of positive definite primitive integral ternary quadratic forms which have this property are presented, and it is proved that there exist only finitely many equivalence classes containing such forms.

Ternary quadratic forms and Brandt matrices

In a recent paper [9] the author showed (among other results) estimates on the asymptotic behaviour of the representation numbers of positive definite integral ternary quadratic forms, in particular, that for n in a fixed square class tZ2 and lattices L, K in the same spinor genus one has . The main tool utilized for the proof was the theory of modular forms of weight 3/2, especially Shimura’s lifting from the space of cusp forms of weight 3/2 to the space of modular forms of weight 2.

Representation of spinor exceptional integers by ternary quadratic forms

The existence and basic properties of what are now referred to as spinor exceptional integers for a genus of integral ternary quadratic forms were first observed in the 1950’s by Jones and Watson [7] and Kneser [8] in the context of indefinite forms. The study of these integers and their generalizations has been undertaken by a number of authors in recent years, and has contributed significantly to the understanding of representation properties unique to ternary forms. In this direction, the present author proved in a previous paper [4] that if c is a primitive spinor exceptional integer for a genus of integral ternary quadratic forms and f is some form in this genus, then a form in the spinor genus of f primitively represents c if and only if f primitively represents an integer of the type ct2, for some odd positive integer t, relatively prime to the discriminant d, which satisfies the condition that the Jacobi symbol (–cd/t) equals 1.

Worst-case complexity bounds for algorithms in the theory of integral quadratic forms

The problem of factoring an integer and many other number-theoretic problems can be formulated in terms of binary quadratic Diophantine equations. This class of equations is also significant in complexity theory, subclasses of it having provided most of the natural examples of problems apparently intermediate in difficulty between P and NP-complete problems, as well as NP-complete problems [2, 3, 22, 26]. The theory of integral quadratic forms developed by Gauss gives some of the deepest known insights into the structure of classes of binary quadratic Diophantine equations. This paper establishes explicit polynomial worst-case running time bounds for algorithms to solve certain of the problems in this theory. These include algorithms to do the following: (1) reduce a given integral binary quadratic form; (2) quasi-reduce a given integral ternary quadratic form; (3) produce a form composed of two given integral binary quadratic forms; (4) calculate genus characters of a given integral binary quadratic form, when a complete prime factorization of its determinant D is given as input; (5) produce a form that is the square root under composition of a given form (when it exists), when a complete factorization of D and a quadratic nonresidue for each prime dividing D is given as input.