Positive Definite Ternary Quadratic Forms Research Articles

English

In this note, we give an elementary proof of the following classical fact. Any positive definite ternary quadratic form over the rational numbers fails to represent infinitely many positive integers. For any ternary quadratic form (positive definite or indefinite), our method constructs certain congruence classes whose elements, up to a square factor, are the only elements not represented over the rational numbers by that form. In the case of a positive definite ternary form, we show that these classes are non-empty. This shows that the minimum number of variables in a positive definite quadratic form representing all positive integers is four. Our proof is very elementary and only uses quadratic reciprocity of Gauss.

Ternary quadratic forms representing the same integers

In 1997, Kaplansky conjectured that if two positive definite ternary quadratic forms with integer coefficients have perfectly identical integral representations, then they are isometric, both regular, or included either of two families of ternary quadratic forms. In this paper, we prove the existence of pairs of ternary quadratic forms representing the same integers which are not contained in Kaplansky’s list.

Positive-definite ternary quadratic forms with the same representations over ℤ

Kaplansky conjectured that if two positive-definite ternary quadratic forms have perfectly identical representations over [Formula: see text], they are equivalent over [Formula: see text] or constant multiples of regular forms, or is included in either of two families parameterized by [Formula: see text]. Our results aim to clarify the limitations imposed to such a pair by computational and theoretical approaches. First, the result of an exhaustive search for such pairs of integral quadratic forms is presented in order to provide a concrete version of the Kaplansky conjecture. The obtained list contains a small number of non-regular forms that were confirmed to have the identical representations up to 3,000,000 by computation. However, a strong limitation on the existence of such pairs is still observed, regardless of whether the coefficient field is [Formula: see text] or [Formula: see text]. Second, we prove that if two pairs of ternary quadratic forms have the identical simultaneous representations over [Formula: see text], their constant multiples are equivalent over [Formula: see text]. This was motivated by the question why the other families were not detected in the search. In the proof, the parametrization of quartic rings and their resolvent rings by Bhargava is used to discuss pairs of ternary quadratic forms.

Ternary quadratic forms and the class numbers of imaginary quadratic fields

We establish a series of relations between the numbers of solutions of some positive definite ternary quadratic forms and the class numbers of the corresponding imaginary quadratic fields. In particular, we obtain an explicit formula for the ternary quadratic form , where p and q are odd primes. Let h(d) be the class number of , and for a ternary quadratic form Q, we use for the number of integral solutions (x, y, z) of the equation . We give two examples to illustrate our results: (i) For , if m is a square-free positive integer with , then(ii) for , if m is a square-free positive integer with , then

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.

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.

Ternary quadratic forms over number fields with small class number

We enumerate all positive definite ternary quadratic forms over number fields with class number at most 2. This is done by constructing all definite quaternion orders of type number at most 2 over number fields. Finally, we list all definite quaternion orders of ideal class number 1 or 2.

On ternary quadratic forms

A result from 1988 on the square-free integers represented by a positive definite ternary quadratic form with integral coefficients is made uniform in the determinant of the form.

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.

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.

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.

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.

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.

On Gauss's proof of Seeber's Theorem

Gauss proved Seeber's Theorem, that the determinant of a reduced positive definite ternary quadratic form is at least half the product of its diagonal coefficients, by means of two determinantal identities whose origin has remained unclear. We examine Gauss's method from a general standpoint, as a method whereby, in certain circumstances, a polynomial in several variables may be shown to be non-negative on a convex polytope by representing it as a positive multilinear combination of the linear forms which determine the polytope. We show that Gauss's identities may be obtained in this manner and that the two identities can in fact be replaced by a simpler single identity which also gives Oppenheim's precise minimum for the determinant.

Levels of positive definite ternary quadratic forms

The level N and squarefree character q of a positive definite ternary quadratic form are defined so that its associated modular form has level N and character χ q {\chi _q} . We define à collection of correspondences between classes of quadratic forms having the same level and different discriminants. This makes practical a method for finding representatives of all classes of ternary forms having a given level. We also give a formula for the number of genera of ternary forms with a given level and character.

Levels of Positive Definite Ternary Quadratic Forms

The level N and squarefree character q of a positive definite ternary quadratic form are defined so that its associated modular form has level N and character ${\chi _q}$. We define à collection of correspondences between classes of quadratic forms having the same level and different discriminants. This makes practical a method for finding representatives of all classes of ternary forms having a given level. We also give a formula for the number of genera of ternary forms with a given level and character.

On the representation of integers with large square factors by positive definite ternary quadratic forms

It is proved here that, if G is a positive definite integral ternary quadratic lattice of discriminant d and c is a squarefree integer which is primitively represented by the genus of G , then G primitively represents all sufficiently large integers of the type ct 2 , with g.c.d. ( t , 2 d ) = 1, which are primitively represented by the spinor genus of G .

Theta series of ternary and quaternary quadratic forms

In the theory of integral quadratic forms or in the allied subject of modular forms very little seems to be known about the linear independence of theta series associated to positive definite forms. For a system of inequivalent positive binary quadratic forms with a fixed fundamental discriminant, Hecke [4-1 showed that their theta series are independent. However, for positive forms in three or more variables, even in a fixed genus, there does not appear to be any non-trivial general result of a similar nature in that certain explicitly described forms in the genus yield linearly independent theta series. We consider here even positive definite ternary quadratic forms of discriminant 2p and quaternary quadratic forms of discriminant p where p is a prime congruent to 1 mod4. For a fixed such p these forms belong to a single genus G'.=G(3,2p), resp. G(4,p). Let G' denote the subset consisting of those forms in G which represent 2. In [6] Kitaoka observed that the number of form-classes in G'(4, p)

Asymptotic behavior of the number of representations of large integers by certain positive-definite ternary quadratic forms

Continuing the work in an earlier paper, the author uses an assumption concerning the location of zeros of Dirichlet L-series in order to derive an asymptotic formula for the number of representations of large integers by the ternary form f(x,y,z)=x2+2y2+Dz2, where D is of the form x2+2y2.