G. L. Watson

Integral quadratic forms

1. Introductory 2. Reduction 3. The Rational Invariants 4. p-Adic Equivalence 5. The Congruence Class and the Genus 6. Rational Transformations 7. Equivalence and Spinor-Relatedness 8. The General Rational Automorph.

One-class genera of positive ternary quadratic forms

We consider positive-definite ternary quadratic forms with integer coefficients. Such a form, f , can be written in matrix notation as Here x′ is the transpose of the column vector x = { x 1 , x 2 , x 3 ) and a , ij = a ji is the coefficient of x i x j in f . Clearly det A is positive and even and so is a negative integer.

Determination of a binary quadratic form by its values at integer points

Let f = f ( x, y ) be a quadratic form with real coefficients in two integer variables x , y . Let V ( f ) be the set of values taken by f ( x, y ) at points ( x, y ) ≠ (0,0). Impose the same conditions on a second form f ′. Trivially, f equivalent to f ′ implies V ( f ) = V ( f ′). It will be shown that the converse implication holds in general for definite forms; the obvious exception f = x 2 + xy + y 2 , f′ = x 2 + 3 y 2 will be shown to be essentially the only one.

Cubic Diophantine equations: the necessary congruence condition

1. Let p be a prime, t a positive integer, and ϕ a cubic polynomial with integral coefficients, and an integral constant term. We study the congruence because of its obvious connection with the equation

Cubic Diophantine equations: a supplementary congruence condition

For the solubility of an inhomogeneous polynomial Diophantine equation, there is one well-known necessary, but not sufficient condition; namely the necessary congruence condition (NCC) explained in §2, below. Till recently, no progress had been made with the general cubic equation, because no one knew what else to assume. Examples given here, see (4.3), (5.4), indicate that some rather subtle hypothesis is needed. The first such hypothesis, see Davenport and Lewis [1], was very far from being necessary for the solubility of the equation. It would seem that any supplementary hypothesis which (loosely) is somewhere near necessary and also (together with the NCC) somewhere near sufficient deserves separate detailed investigation before one proceeds to use it.

Biquadratic congruences: an acknowledgement

Professor Kneser has pointed out to me that the results proved in my paper [3] are not new. To be precise, my Theorem 1 is a special case of the result proved in [2], while the routine argument by which I deduced my Theorem 2 is given in substance in [1; 241]. Then my Theorem 3 , though perhaps new, follows almost trivially.