Arithmetic-analytic properties of binary positive-definite quadratic forms

Completely p-primitive binary quadratic forms

We study the generalized Whittaker models for G = GSp(2;R) associated with indefinite binary quadratic forms when they arise from two standard representations of G: (i) a generalized principal series rep- resentation induced from the non-Siegel maximal parabolic subgroup and (ii) a (limit of) large discrete series representation. We prove the uniqueness of such models with moderate growth property. Moreover we express the values of the corresponding generalized Whittaker functions on a one-parameter subgroup of G in terms of the Meijer G-functions.

W-groups and values of binary forms

Polynomial sums over automorphs of a positive definite binary quadratic form

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.

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.

The introductory part contains definitions and basic properties of harmonic theta series, Siegel modular forms, and Hecke operators. Then the transformation formulas are recalled, related to the action of modular substitutions and regular Hecke operators on general harmonic theta series, including specialization to the case of Maass theta series. The following new results are obtained: construction of infinite sequences of eigenfunctions for all regular Hecke operators on spaces of Maass theta series; in the case of Maass theta series of genus $2$, all the eigenfunctions are constructed and the corresponding Andrianov zeta functions are expressed in the form of products of two $L$-functions of the relevant imaginary quadratic rings. The proofs are based on a combination of explicit formulas for the action of Hecke operators on theta series with Gauss composition of binary quadratic forms.

Composition of oriented binary quadratic form-classes over commutative rings

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.

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.

A quadratic $\mathbb {Z}$-lattice L of rank n is denned to be k-regular for a positive integer $k \leq n$ if L globally represents all quadratic $\mathbb {Z}$-lattices of rank k which are represented everywhere locally by L. It is shown that there exist only finitely many isometry classes of primitive positive definite quadratic $\mathbb {Z}$-lattices of rank 4 which are 2-regular.

Residue properties of certain quadratic units

We prove that if a polynomial with rational coefficients has a root mod p for every large prime p, then it has a real root. We show with examples how to use Chebotarev’s density theorem to study roots of polynomials mod p, leading up to our proof. As an application, we show that the primes can’t be covered by finitely many positive definite binary quadratic forms.

One of the most celebrated applications of Gauss’s 2 F 1 _2F_1 hypergeometric functions is in connection with the rapid convergence of sequences and special values that arise in the theory of arithmetic and geometric means. This theory was the inspiration for a recent paper (see M. J. Griffin, Ken Ono, N. Saikia, and W.-L. Tsai, [AGM and jellyfish swarms of elliptic curves, Amer. Math. Monthly 130 (2023), no. 4, 355-369]) in which a finite field analogue of A G M R \mathrm {AGM}_\mathbb {R} was defined and then studied using finite field hypergeometric functions. Instead of convergent sequences, one gets directed graphs that combine to form disjoint unions of graphs that individually resemble jellyfish. Echoing the connection of hypergeometric functions to periods of elliptic curves, these graphs organize elliptic curves over finite fields. Here we use such “jellyfish swarms” to prove new identities for Gauss’s class numbers of positive definite binary quadratic forms. Moreover, we prove that the sizes of jellyfish are in part dictated by the order of the prime above 2 in certain class groups.

We prove part of a conjecture of Borwein and Choi concerning an estimate on the square of the number of solutions to n=x 2+Ny 2 for a squarefree integer N.