Kaplansky’s Ternary Quadratic Form
نویسنده
چکیده
This paper proves that if N is a nonnegative eligible integer, coprime to 7, which is not of the form x2+y2+7z2, thenN 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. 2000 Mathematics Subject Classification. Primary 11E25.
منابع مشابه
Isospectral Definite Ternary F Q [t]-lattices
We prove that the representations numbers of a ternary definite integral quadratic form defined over Fq[t], where Fq is a finite field of odd characteristic, determine its integral equivalence class when q is large enough with respect to its successive minima. Equivalently, such a quadratic form is determined up to integral isometry by its theta series.
متن کاملRamanujan’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 consi...
متن کاملRepresentation by Ternary Quadratic Forms
The problem of determining when an integral quadratic form represents every positive integer has received much attention in recent years, culminating in the 15 and 290 Theorems of Bhargava-Conway-Schneeberger and Bhargava-Hanke. For ternary quadratic forms, there are always local obstructions, but one may ask whether there are ternary quadratic forms which represent every locally represented in...
متن کاملOn almost universal ternary inhomogeneous quadratic polynomials
A fundamental question in the study of integral quadratic forms is the representation problem which asks for an effective determination of the set of integers represented by a given quadratic form. A related and equally interesting problem is the representation of integers by inhomogeneous quadratic polynomials. An inhomogeneous quadratic polynomial is a sum of a quadratic form and a linear for...
متن کاملGauss Sums & Representation by Ternary Quadratic Forms
This paper specifies some conditions as to when an integer m is locally represented by a positive definite diagonal integer-matrix ternary quadratic form Q at a prime p. We use quadratic Gauss sums and a version of Hensel’s Lemma to count how many solutions there are to the equivalence Q(~x) ≡ m (mod p) for any k ≥ 0. Given that m is coprime to the determinant of the Hessian matrix of Q, we can...
متن کامل