Ternary Quadratic Forms Research Articles

Some exact formulae for the numbers of representations of integers by ternary quadratic forms

In this paper we shall give some exact formulae for the numbers of representations of integers by certain ternary quadratic forms whose genus have class numbers large 1.

Positive ternary quadratic forms with finitely many exceptions

An integral quadratic form f is said to be almost regular if f globally represents all but finitely many integers that are represented by the genus of f. In this paper, we study and characterize all almost regular positive definite ternary quadratic forms.

A Bailey lattice

We exhibit a technique for generating new Bailey pairs which leads to deformations of classical q-series identities, multiple series identities of the Rogers-Ramanujan type, identities involving partial theta functions, and a variety of representations for q-series by number-theoretic objects such as weight 3/2 modular forms, ternary quadratic forms, and weighted binary quadratic forms.

On the Dimension of Some Spaces of Generalized Ternary Theta-Series

Abstract The upper bound of dimension of vector spaces of generalized theta-series corresponding to some ternary quadratic forms is established. In a number of cases, the dimension of vector spaces of generalized theta-series is established and bases of these spaces are constructed.

SOME VARIANTS OF THE CONGRUENT NUMBER PROBLEM II

for square-free integers m, and we have obtained that the special value L(Em, 1) of L-function for Em is described in terms of positive definite ternary quadratic forms. It is an analogue result of Tunnell’s [Tun] for the classical congruent number problem. In this paper, we use a result of Gauss on ternary quadratic forms to give more useful criterion for non-2π/3and non-2π/3-congruentness. Some of our results are the following.

The values of ternary quadratic forms at prime arguments

MathematikaVolume 48, Issue 1-2 p. 137-149 Research Article The values of ternary quadratic forms at prime arguments R. J. Cook, R. J. Cook Department of Pure Mathematics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield, S3 7RH Stoneygate, Bessie Lane, Bradwell, Hope Valley, S33 9HZSearch for more papers by this authorA. Fox, A. Fox 2 Clarence Road Monk Bretton, Barnsley, South Yorkshire, S71 2NLSearch for more papers by this author R. J. Cook, R. J. Cook Department of Pure Mathematics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield, S3 7RH Stoneygate, Bessie Lane, Bradwell, Hope Valley, S33 9HZSearch for more papers by this authorA. Fox, A. Fox 2 Clarence Road Monk Bretton, Barnsley, South Yorkshire, S71 2NLSearch for more papers by this author First published: 01 December 2001 https://doi.org/10.1112/S002557930001439XCitations: 10AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume48, Issue1-2December 2001Pages 137-149 RelatedInformation

Elliptic curves and positive definite ternary forms

For two given ternary quadratic forms f(x,y,z) and g(x,y,z) , let r(f,n) and r(g,n) be the numbers of representations of n represented by f(x,y,z) and g(x,y,z) respectively. In this paper we study the following problem: when will we have r(f,n)=r(g,n) or r(f,n)≠r(g,n) . Our method is to use elliptic curves and the corresponding new forms.

Representation masses of spinor genera

In this paper we establish C. Siegel's mass formulas with quadratic characters which give a mass formula for spinor genera. For ternary quadratic forms, the splitting numbers are completely determined by local Jordan decompositions.

Representation masses of spinor genera

In this paper we establish C. Siegel's mass formulas with quadratic characters which give a mass formula for spinor genera. For ternary quadratic forms, the splitting numbers are completely determined by local Jordan decompositions.

Eisenstein series of 3/2 weight and one conjecture of Kaplansky

We shall prove thatf1 =x2 +y4 + 7z2 represents all eligible numbers congruent to 2 mod 3 except 14 × 72k which was conjectured by Kanplansky. Our method is to use modular forms of weight 3/2. Our method can also be applied to other ternary quadratic forms.

Representation of Integers by Ternary Quadric Forms

In this paper we give a formula for the number of representations of some square-free integers by certain ternary quadratic forms and estimate the lower bound of the 2-power appearing in this number.

Eisenstein Series of 3/2 Weight and Eligible Numbers of Positive Definite Ternary Forms

A general algorithm is given for the number of representations for a positive integer n by the genus of a positive definite ternary quadratic form with form ax2 + by2 + cz2. Using this algorithm, we study several nontrivial genera of positive ternary forms with small discriminants in the paper. As a conclusion we prove that f1 = x2 + y2 + 7z2 represents all eligible numbers congruent to 2 mod 3 except 14 * 72k which was conjectured by Kaplansky in [K]. Our method is to use Eisenstein series of weight 3/2.

Kaplansky′s ternary quadratic form

This paper proves that if N is a nonnegative eligible integer, coprime to 7, which is not of the form x2 + y2 + 7z2, then N is square‐free. The proof is modelled on that of a similar theorem by Ono and Soundararajan, in which relations between the number of representations of an integer np2 by two quadratic forms in the same genus, the pth coefficient of an L‐function of a suitable elliptic curve, and the class number formula prove the theorem for large primes, leaving 3 cases which are easily numerically verified.

Representations of indefinite ternary quadratic forms over number fields

Representations of indefinite ternary quadratic forms over number fields

On integral representations by totally positive ternary quadratic forms

Soit K un corps de nombres algebrique totalement reel dont l'anneau d'entiers R est un anneau principal. Soit f(x 1 , x 2 , x 3 ) une forme quadratique ternaire totalement definie a coefficients dans R. Nous etudions les representations par f d'entiers totalement positifs N E R. Nous demontrons une formule qui lie le nombre de representations de N par des classes differentes dans le genre de f, au nombre de classes de R[√-c f N], ou cf E R est une constante qui depend seulement de f. Nous donnons une demonstration algebrique du resultat classique de H. Maass sur les representations comme sommes de trois carres d'entiers de Q(√5) et une dependance explicite entre le nombre de representations et le nombre de classes du corps biquadratique correspondant. Nous donnons egalement des formules analogues pour certaines formes quadratiques provenant d'ordres quaternioniques maximaux de nombre de classe 1, sur les entiers de corps de nombres quadratiques reels.

Almost Periodicity of the Normalized Representation Numbers Associated to Positive Definite Ternary Quadratic Forms

It is shown that the function which associates to each natural numbernthe appropriately normalized number of representations ofnby a positive definite ternary quadratic form is almost periodic. Furthermore the mean value of this function on the squarefree numbers is calculated and it is shown that the function has a smooth limiting distribution.

Multiplicities of non-arithmetic ternary quadratic forms and elliptic curves of positive rank

In this note we will characterize all non-arithmetic ternary quadratic forms with unbounded multiplicities. It turns out that their existence is closely related with certain classes of elliptic curves whose investigation is interesting on its own.

Joint Invariants of a Triplet of Coplanar Conics: Stability and Discriminating Power for Object Recognition

Joint invariants of a pair of coplanar conics have played a central role in the initial work on the application of invariants to object recognition. In this paper, we are concerned with the invariants of a triplet of coplanar conics. It will be shown that, in addition to the invariants associated to all pairs of three conics, there exists one new invariant which is only associated to the triplet of conics. This new invariant of the three conics can therefore be used to discriminate triplets of conics which might be pair-wise similar. All joint invariants of a triplet of coplanar conics will first be derived based on the invariant algebra of a net of ternary quadratic forms. Then, experimentations for discriminating power, accuracy, and stability of joint invariants are conducted both for simulated and real images. Finally, it is also shown that the method developed for a triplet of coplanar conics can be extended to any number of coplanar conics.

A constructive theory of triple and quintuple product identities of the second degree

The groundwork for a theory of quadratic identities involving the classical triple and quintuple products is layed. The approach is through the study and use of affine maps that act on indexing lattices associated with the terms (double sums) in the given identity. The terms of the identity are found to be connected by the invariant of a ternary quadratic form.

Ramanujan's ternary quadratic form

do not seem to obey any simple law.” Following I. Kaplansky, we call a non-negative integer N eligible for a ternary form f(x, y, z) if there are no congruence conditions prohibiting f from representing N. By the classical theory of quadratic forms, it is well known that any given genus of positive definite ternary quadratic forms represents every eligible integer. Consequently if a genus consists of a single class with representative f(x, y, z), then f represents every eligible integer. In the case of Ramanujan’s form, this only implies that an eligible integer, one not of the form 4(16μ+ 6), is represented by φ1 or φ2.