Gordon Pall

Spinor genera of binary quadratic forms

Abstract Spinor genera are defined for binary quadratic forms with integer coefficients in such a way that the theory fits in with the Gaussian theory of genera. It is shown that spinor generic characters exist which distinguish the various spinor genera in the principal genus, and how they can be determined. It is known that each ambiguous class contains exactly two forms of the type [a, 0, c] or [a, a, c], each with its associate [c, 0, a], [4c − a, 4c − a, c]. Since the principal class contains such a form with a = 1, it is an interesting question whether one can predict the second form (not counting associates). This question includes that of Dirichlet about the representability of −1 by the principal class. Methods are given for evaluating the spinor-generic characters of ambiguous forms in the principal genus for variable discriminants d, and are carried through in the eleven cases where d is fundamental, there are two or four genera, and two spinor genera in the principal genus. The problem of determining the “second form” is thus completely solved except when there is more than one ambiguous class in the principal spinor genus.

Discriminantal divisors of binary quadratic forms

Abstract It can be deduced from a result of Gauss that the principal class of discriminant d represents primitively exactly one divisor of d (besides 1) in a certain set of those divisors. Instead of restricting ones attention to criteria for representability of −1, it would seem desirable to ask what that other divisor is. A study is made of the case d = 8 p ( p prime), making use of a remarkable parametric solution of x 2 + y 2 = z 2 + 2 w 2 = u 2 − 2 v 2 and of an idea (hitherto overlooked?) in an early paper of Georg Cantor. This yields a complete answer except in certain cases with p ≡ 1 (mod 16).

Factorization of Cayley numbers

Abstract The paper studies the problem of finding and counting the factorizations of a Cayley number κ with integral coordinates of norm lm as a product of Cayley numbers of norms l and m . Let e be the greatest common divisor of the components of κ and d the greatest common divisor of e , l , m . Then the number of factorizations is equal to the number of representations of d , respectively d 2 , by certain definite octonary quadratic forms of determinants 1, 1 16 , 1 64 , 1 256 .

Modules and rings in the Cayley algebra

Abstract We derive all integral Z-modules and rings (in the Cayley algebra) which contain a module with the norm-form y02 + … + y72. The norm-forms of these modules comprise eight classes of quadratic forms each of which is uniquely characterized by the values of certain of its ordinal invariants.

A reconsideration of Legendre-Jacobi symbols

Abstract The classical definition of the Jacobi symbol (a:b) was badly conceived for negative values of b. Alternative useful definitions of (a:−1) are proposed here. This is an elaboration of a point in the article “Spinor genera of binary quadratic forms” in this issue.

Binary quadratic and cubic forms, and unipositive matrices

Abstract The classical method of reducing a positive binary quadratic form to a semi-reduced form applies translations alternately left and right to minimize the absolute value of the middle coefficient — and may therefore be called absolute reduction. There is an alternative method which keeps the sign of the middle coefficient constant before the end: we call this method positive reduction. Positive reduction seems to make possible an algorithm for finding the representations of 1 by a binary cubic form with real linear factors, and has various properties somewhat simpler than those of absolute reduction. Some of these properties involve unipositive matrices (with nonnegative integer elements and determinant 1). Certain semigroups of unipositive matrices with unique factorization into primes are described. Two of these semigroups give a neat approach to the reduction of indefinite binary quadratic forms—which may generalize. Some remarks on unimodular automorphs occur in Section 6.