Composition of oriented binary quadratic form-classes over commutative rings

One considers the arithmetic and analytic properties of positive-definite binary quadratic forms of discriminant −Dn2. The arithmetic structure of the set of these forms is described by forms of discriminant −D and by the Hecke operators T(n). One gives some arithmetic and analytic consequences.

(1)THIS is a systematic text-book on the elements of the theory of numbers, intended for beginners. The greater part of it deals with the usual elementary topics: divisibility, the H.C.F. process, congruences, and binary quadratic forms (but without the classical theory of reduction and equivalence). The proofs of the theorems are given in full and ample detail, and there are frequent numerical illustrations, and sets of examples for solution by the student. Most of these are quite straightforward, but there are several which strike one as distinctly difficult in relation to the text. (1) Elementary Number Theory By Prof. J. V. Uspensky Prof. M. A. Heaslet. Pp. x + 484. (New York and London: McGraw-Hill Book Co., Inc., 1939.) 26s. (2) Modern Elementary Theory of Numbers By Prof. Leonard Eugene Dickson. Pp. vii + 309. (Chicago: University of Chicago Press; London: Cambridge University Press, 1939.) 18s. net.

W-groups and values of binary forms

Completely p-primitive binary quadratic forms

Abstract.In this paper we give a lower bound with respect to block length for the trace of non-elliptic conjugacy classes of the Hecke groups. One consequence of our bound is that there are finitely many conjugacy classes of a given trace in anyHecke group. We show that another consequence of our bound is that class numbers are finite for related hyperbolic ℤ[λ]-binary quadratic forms. We give canonical class representatives and calculate class numbers for some classes of hyperbolic ℤ[λ]-binary quadratic forms.

A treatment by integral matrices is given for composition of pairs of integral quadratic forms with the same discriminant. The forms are associated with a pair of similar 2 × 2 matrices A B with irrational eigen values which generate the maximal order. The most general integral similarity between A B is given by a matrix whose entries are linear forms in two indeterminates with integral coefficients. This matrix is a compositum of two factors of the same nature. By equating determinants a composition of two quadratic forms results. The method can be generalized to n × n matrices.

We consider special maps from a symbolic compactum to a two-dimensional torus that connect a bidirectional translation with a given hyperbolic automorphism of a two-dimensional torus and preserve the maximum entropy measure. In contrast to the known geometric coding method with the aid of Markovian partitions (see references in [2] and [9]), the coding under consideration uses the arithmetic and group theoretic structure. This makes it possible to connect the arithmetic of the corresponding algebraic extension of Q and the properties of a certain binary quadratic form with the dynamics of the automorphism. The transformations can be parametrized by the units of the quadratic field or by the homoclinic points of some specific orbit. Let T be a hyperbolic automorphism ofT2 given by the matrixM = a b c d ∈ GL(2,Z). We put r = TrM,σ = detM,D = r2 − 4σ. Hyperbolicity implies that r 6= 0 for σ = −1, and |r| ≥ 3 for σ = +1. For simplicity, we shall assume that r > 0. Let λ = (r+ √ D)/2 be the greatest eigenvalue of M . As a symbolic compactum X we choose either a stationary Markov compactum Xr with state space 0,1, . . . , r and pairwise prohibitions {en = r ⇒ en+1 = 0, n ∈Z} for σ = −1, or a sofic compactumYr = {{en}−∞ : 0 ≤ en ≤ r−1, (en . . . en+s) 6= (r−1)(r−2)s−2(r−1) for any n ∈Z and any s ≥ 2} for σ = +1. Therefore, each of the two compacta is fully determined by the trace and is a β-compactum for β = λ (see [7]). Let τ denote a bidirectional translation on X, that is, (τe)n = en+1, and let μ be a measure of maximum entropy. We define the normalization operation as a map n : Q∞ −∞{0,1, . . . , 2r} → X such that if {xk} is a finite sequence, then n({xk}) = {ek} ′ l is the sequence of coefficients of a Markovian (sofic) β-decomposition with β = λ of the number x = P k xkλ −k. More precisely, l = [− logλ x] + 1, el = [λlx], ek = [λSk−l(λlx)], k > l, where Sx = {λx}. For an arbitrary sequence the normalization is defined as the weak limit of normalizations of finite approximations. On the well-posedness of this definition, see [4] and [9]. The sum of two sequences in X is defined to be the normalization of their coordinatewise sum. Similarly, one can define the inverse element with respect to addition. It can be shown that the sum and difference are defined for a.a. pairs of sequences under the measure μ×μ. After the natural arithmetic factorization mod 0, the maximum entropy measure p : X → X′ defines a group structure on the quotient space X′, with respect to which the translation τ is a group automorphism. We recall that a point s ∈T2 is said to be homoclinic to zero (briefly: homoclinic) if it belongs to the intersection of the leaves of the expanding and the contracting foliations passing through 0. Each homoclinic point s can be obtained in a unique way as the projection of a lattice point z(s) ∈ Z2 onto the one-dimensional eigenspace Lu corresponding to the eigenvalue λ along the other one-dimensional eigenspace Ls, followed by projection of the resulting point t = t(s) onto the torus (see [1]). The point t = t(s) will be called the plane coordinate of the homoclinic point s.

Notation Prologue Chapter I. Lattices 1. Introduction 2. Bases and sublattices 3. Lattices under linear transformation 4. Forms and lattices 5. The polar lattice Chapter II. Reduction 1. Introduction 2. The basic process 3. Definite quadratic forms 4. Indefinite quadratic forms 5. Binary cubic forms 6. Other forms Chapter III. Theorems of Blichfeldt and Minkowski 1. Introduction 2. Blichfeldt's and Mnowski's theorems 3. Generalisations to non-negative functions 4. Characterisation of lattices 5. Lattice constants 6. A method of Mordell 7. Representation of integers by quadratic forms Chapter IV. Distance functions 1. Introduction 2. General distance-functions 3. Convex sets 4. Distance functions and lattices Chapter V. Mahler's compactness theorem 1. Introduction 2. Linear transformations 3. Convergence of lattices 4. Compactness for lattices 5. Critical lattices 6. Bounded star-bodies 7. Reducibility 8. Convex bodies 9. Speres 10. Applications to diophantine approximation Chapter VI. The theorem of Minkowski-Hlawka 1. Introduction 2. Sublattices of prime index 3. The Minkowski-Hlawka theorem 4. Schmidt's theorems 5. A conjecture of Rogers 6. Unbounded star-bodies Chapter VII. The quotient space 1. Introduction 2. General properties 3. The sum theorem Chapter VIII. Successive minima 1. Introduction 2. Spheres 3. General distance-functions Chapter IX. Packings 1. Introduction 2. Sets with V(/varphi) =n^2/Delta(/varphi) 3. Voronoi's results 4. Preparatory lemmas 5. Fejes Toth's theorem 6. Cylinders 7. Packing of spheres 8. The proudctio of n linear forms Chapter X. Automorphs 1. Introduction 2. Special forms 3. A method of Mordell 4. Existence of automorphs 5. Isolation theorems 6. Applications of isolation 7. An infinity of solutions 8. Local methods Chapter XI. Ihomogeneous problems 1. Introduction 2. Convex sets 3. Transference theorems for convex sets 4. The producti of n linear forms Appendix References Index quotient space. successive minima. Packings. Automorphs. Inhomogeneous problems.

Let G be an abelian group and let R be a commutative ring with identity. Denote by R t G a commutative twisted group algebra (a commutative twisted group ring) of G over R, by ℬ(R) and ℬ(R t G) the nil radicals of R and R t G, respectively, by G p the p-component of G and by G 0 the torsion subgroup of G. We prove that: i. If R is a ring of prime characteristic p, the multiplicative group R* of R is p-divisible and ℬ(R) = 0, then there exists a twisted group algebra R t 1 (G/G p ) such that R t G/ℬ(R t G) ≅ R t 1 (G/G p ) as R-algebras; ii. If R is a ring of prime characterisitic p and R* is p-divisible, then ℬ(R t G) = 0 if and only if ℬ(R) = 0 and G p = 1; and iii. If B(R) = 0, the orders of the elements of G 0 are not zero divisors in R, H is any group and the commutative twisted group algebra R t G is isomorphic as R-algebra to some twisted group algebra R t 1 H, then R t G 0 ≅ R t 1 H 0 as R-algebras.

We discuss multiplicative properties of the binary quadratic form $a x^2 + b x y + c y^2$ by considering a ring of matrices which is closed under a triple product. We prove that the ring forms a ternary algebra in the sense of Hestenes, and then derive both multiplicative formulas for a large class of binary quadratic forms and a type of multiplication for points on a conic section which generalizes the algebra of rational points on the unit circle.

For any squarefree positive m there exists exactly one solvable antipellian equation, which can be used to construct a certain dihedral extension L/Q, cyclic of degree 4 above k=Q(√−m). We calculate the conductor of L/k and the value of the Artin character of L/k on the corresponding congruence ideal classes of order 2 of k. From this, we deduce results for the representations of powers of primes by binary quadratic forms, in the case where the norm of the fundamental unit of Q(√m) is +1.

Let [Formula: see text] denote the number of overpartitions of [Formula: see text]. In this paper, we will give a complete determination of [Formula: see text] modulo [Formula: see text] by relating it to some binary quadratic forms; this will generalize the result of Kim on [Formula: see text] modulo [Formula: see text]. Moreover, we give infinitely many new Ramanujan-like congruences for overpartition function modulo [Formula: see text].

Let G be an abelian group, R a commutative ring of prime characteristic p with identity and R t G a commutative twisted group ring of G over R. Suppose p is a fixed prime, G p and S(R t G) are the p-components of G and of the unit group U(R t G) of R t G, respectively. Let R* be the multiplicative group of R and let f α(S) be the α-th Ulm-Kaplansky invariant of S(R t G) where α is any ordinal. In the paper the invariants f n (S), n ∈ ℕ ∪ {0}, are calculated, provided G p = 1. Further, a commutative ring R with identity of prime characteristic p is said to be multiplicatively p-perfect if (R*)p = R*. For these rings the invariants f α(S) are calculated for any ordinal α and a description, up to an isomorphism, of the maximal divisible subgroup of S(R t G) is given.

Historically the theory of quadratic forms has its origins in number-theoretic questions of the following type: Which integers can be written in the form x 2 + 2y 2 , which are sums of three squares, or more generally, which integers can be represented by an arbitrary quadratic form Σ a ij x i x j integral coefficients? This general question is exceptionally difficult and we are still quite far from a complete solution. It is natural and considerably simpler to first investigate these questions over the field of rational numbers, that is, to ask for rational instead of integral solutions to the equation Σ a ij x i x j = a. This leads to the problem of classification of quadratic forms over \( \mathbb{Q} \), which was first solved by Minkowski. His solution appears in this chapter basically unaltered, except for a few simplifications and the use of modern terminology. The Gaussian sums of Gauss and Dirich-let play a significant role in the more formal algebraic part of the theory.

In [8], odd dimensional knot cobordism groups C2n^ (n'2>2) were studied and given a purely algebraic description. C2 Λ_j is the set of all cobordism classes of knotted (2n— l)-spheres in the (2n+l)-sphere, and it is an abelian group under connected sum. In this paper we extend this to knotted homology 3-spheres in the 5-sphere and consider their homology cobordism classes. The set of all such homology cobordism classes, denoted by Cy is also an abelian group under connected sum. We shall describe the group Cf using the usual homology cobordism group of homology 3-spheres and Levine's cobordism group of certain matrices (Theorem 1.1). Our argument heavily depends on that of J. Levine. In §3 we introduce the bounding genus and the Murasugi number of a knotted homology 3-sphere in S*. Using our techniques we can estimate these numerical invariants from above. In §4 we consider the case of algebraic 3knots which appear around isolated singularities of complex hypersurfaces in C (see [11]). For example, we shall show that two algebraic 3-knots which are homology 3-spheres and defined by weighted homogeneous polynomials are homology cobordant if and only if they are isotopic. Throughout the paper, we shall work in the C°° category. Homology groups will always be with integral coefficients. The author would like to thank Professor S. Fukuhara for suggesting the problem. He is also grateful to Professor F. Michel who was kind enough to send us a copy of S. Akbulut's note.