Completely p-primitive binary quadratic forms

W-groups and values of binary forms

Polynomial sums over automorphs of a positive definite binary quadratic form

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.

We obtain estimates for the number of integral solutions in large balls, of inequalities of the form $|Q(x, y)| < \epsilon$, where $Q$ is an indefinite binary quadratic form, in terms of the Hurwitz continued fraction expansions of the slopes of the lines on which $Q$ vanishes. The method is based on a coding of geodesics on the modular surface via Hurwitz expansions of the endpoints of their lifts in the Poincare half-plane.

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.

In the Transactions of the American Math. Society, vol. I, 1900, and vol. IV, 1904, Mr E. Kasner treats exhaustively the (2, 2) double binary form and discusses the theory connected with double binary forms and multiple binary forms in general terms. He shows the relations between the systems of multiple binary forms with digredient variables and the forms with cogredient variables. Hitherto nothing seems to have been written on systems of triple binary forms (with regard to higher forms see a paper on the (1, 1, 1, 1) form by C. Segre); so here I propose to discuss the complete system of a (1, 1, 1) binary form which consists of six forms connected by one syzygy. When two of the variables are the same we naturally get the (2, 1) form and when the three are the same We get the (3) or cubic binary form.

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.

Dedicated to the occasion of John Friedlander's 80th birthdayWe prove an asymptotic formula for primes of the shape f (a, b 2 ) with a, b integers and of the shape f (a, p 2 ) with p prime.Here f is a binary quadratic form with integer coefficients, irreducible over and has no local obstructions.This refines the seminal work of Friedlander and Iwaniec on primes of the form x 2 + y 4 and of Heath-Brown and Li on primes of the form a 2 + p 4 , as well as earlier work of the author with Lam and Schindler on primes of the form f (a, p) with f a positive definite form.MSC2020: primary 11N32

Composition of oriented binary quadratic form-classes over commutative rings

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.

Sufficient conditions of representability of a given integer by a fixed binary quadratic form are presented. It is shown that almost all integers satisfy these conditions. Bibliography: 2 titles.

Chapter 25 Binary Quartic Forms

For a binary quadratic form $Q$, we consider the action of $\mathrm{SO}_Q$ on a two-dimensional vector space. This representation yields perhaps the simplest nontrivial example of a prehomogeneous vector space that is not irreducible, and of a coregular space whose underlying group is not semisimple. We show that the nondegenerate integer orbits of this representation are in natural bijection with orders in cubic fields having a fixed "lattice shape". Moreover, this correspondence is discriminant-preserving: the value of the invariant polynomial of an element in this representation agrees with the discriminant of the corresponding cubic order. We use this interpretation of the integral orbits to solve three classical-style counting problems related to cubic orders and fields. First, we give an asymptotic formula for the number of cubic orders having bounded discriminant and nontrivial automorphism group. More generally, we give an asymptotic formula for the number of cubic orders that have bounded discriminant and any given lattice shape (i.e., reduced trace form, up to scaling). Via a sieve, we also count cubic fields of bounded discriminant whose rings of integers have a given lattice shape. We find, in particular, that among cubic orders (resp. fields) having lattice shape of given discriminant $D$, the shape is equidistributed in the class group $\mathrm{Cl}_D$ of binary quadratic forms of discriminant $D$. As a by-product, we also obtain an asymptotic formula for the number of cubic fields of bounded discriminant having any given quadratic resolvent field.

We study the ``imaginary" binary quadratic form equations ax^2+bxy+cy^2+g=0 over k[t] in rational function fields, showing that a condition with respect to the Artin reciprocity map, is the only obstruction to the local-global principle for integral solutions of the equation.

There is a classical geometric construction which uses a binary quadratic form to define an involution on the space of binary d-ics. We give a complete characterization of a general class of such involutions which are definable using compound transvectant formulae. We also study the associated varieties of forms which are preserved by such involutions. Along the way we prove a recoupling formula for transvectants, which is used to deduce a system of equations satisfied by the coefficients in these involutions.