Dennis R. Estes

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.

Breaking the Ong-Schnorr-Shamir signature scheme for quadratic number fields

Recently Ong, Schnorr, and Shamir [OSS1, OSS2] have presented new public key signature schemes based on quadratic equations. We will refer to these as the OSS schemes. The security of the schemes rest in part on the difficulty of finding solutions to

Minimal polynomials of integral symmetric matrices

X^2 - KY^2 \equiv M(mod{\mathbf{ }}n),

Representations under ring extensions: Latimer-MacDuffee and Taussky correspondences

(1) where n is the product of two large rational primes. In the original OSS scheme [OSS1], K, M, X, and Y were to be rational integers. However, when this version succumbed to an attack by Pollard [PS,S1], a new version was introduced [OSS2], where M, X, and Y were to be quadratic integers, i. e. elements of the ring \( Z[\sqrt d ] \). In this paper we will show that the OSS system in \( Z[\sqrt d ] \) is also breakable The method by which we do this is to reduce the problem of solving the congruence over the ring \( Z[\sqrt d ] \) to the problem of solving the congruence over the integers, for which we can use Pollard’s algorithm.

Matrix factorizations, exterior powers, and complete intersections

Abstract The set E(Z) of eigenvalues of symmetric matrices over Z is shown to be the set of all totally real algebraic integers.

A stable range for quadratic forms over commutative rings

Abstract We investigate what the possible minimal polynomials are for integral symmetric matrices. We show that the obvious necessary conditions are sufficient for polynomials of degree at most 4. We show that necessary conditions are sufficient for minimal polynomials of self-adjoint operators on positive definite unimodular lattices. We also give a relatively elementary proof of the result of Estes that any total real algebraic integer is the eigenvalue of an integral symmetric matrix. This question was asked by Alan Hoffman, who also showed that this result implies that any totally real algebraic integer is the eigenvalue of the adjacency matrix of some graph.

Determinants of Galois automorphisms of maximal commutative rings of 2×2 matrices

Hurwitzs proof of Lagranges theorem that every positive integer is a sum of four squares of integers is but one of the extensive list of articles which relate quaternion arithmetics to quaternary (and ternary) quadratic forms (see [Marie-France Vigneras, Arithmetique des algebres the quaternions, in “Lecture Notes in Mathematics,” Vol. 800, Springer-Verlag, New York, 1980] for a partial bibliography). Hurwitzs argument uses the property that a quaternion in the ring of Hurwitz quaternions with norm divisible by a prime p has a factor of norm p. The purpose of this article is to determine the connections between factorization in general quaternion orders over the integers, the ideal theory of the order and the genus structure of the corresponding quadratic lattice; and to list the positive definite quaternion orders which admit various types of factorization. Our work can be viewed as a continuation of Gordon Palls initial investigations in [Trans. Amer. Math. Soc. 59 (1946), 503–513; Duke Math. J. 4 (1938), 696–704].

Factorization in hereditary orders

C. G. Latimer and C. C. MacDuffee defined in [8] a bijective correspondence between similarity classes of nonsingular rational integral n x n matrices having a common separable characteristic polynomialf(x) and the classes of faithful ideals in the ring Z[x]/f(x) Z[x], Z the ring of rational integers. Several authors have used this simple duality with the ideas indigenous to one theory to obtain valuable consequences in the other (see [14] and [15] f or a survey), thereby motivating further analysis of the correspondence (as in [ 1, 2, 4, 17 and 181). Nevertheless, the theory has generally been catagorized as a subset of module theory of orders in semisimple algebras following Zassenhaus’s proof of Jordan’s theorem on the finiteness of such similarity classes (see [ 7, 14 and 21 I). The purpose of this note is to replace the semisimple assumption present in these previous studies with the weaker condition that the modules considered become trivial under a suitable extension of the base ring. The ease in which matrix and ideal theory are transferable under the Latimer-MacDuffee duality prompted our adherence to their approach, when possible, in this development over arbitrary commutative rings. We review prior matrix/ideal correspondences in Section 2 preliminary to the general framework in Section 3. There an n x II matrix class is replaced by the isomorphism class of a rank n projective R module M which is also a faithful module over a rank n projective R algebra A. The ideal theory is obtained by extending the base ring R of scalars to a commutative ring S for which M becomes trivial (S @ll MN S OR A as S OR A modules), and M is then identified with a fractional A ideal in S OR A. In the classical case, S is the field of quotients of a domain R. Theorem 1 reduces to the Latimer-MacDuffee duality in the latter case. Theorem 2 shows that the theory is equally adaptable to any commutative rank IZ projective R algebra A which is isomorphic to its dual over a suitable extension of R. As an application, we show that a nonderogatory matrix is similar to the companion matrix of its