Classes Of Binary Quadratic Forms Research Articles

Composition of binary quadratic forms over number fields

AbstractIn this article, the standard correspondence between the ideal class group of a quadratic number field and the equivalence classes of binary quadratic forms of given discriminant is generalized to any base number field of narrow class number one. The article contains an explicit description of the correspondence. In the case of totally negative discriminants, equivalent conditions are given for a binary quadratic form to be totally positive definite.

Asymptotics for the Number of Simple (4a + 1)-knots of Genus 1

Abstract We investigate the asymptotics of the total number of simple $(4a+1)$-knots with Alexander polynomial of the form $mt^2 +(1-2m) t + m$ for some nonzero $m \in [-X, X]$. Using Kearton and Levine’s classification of simple knots, we give equivalent algebraic and arithmetic formulations of this counting question. In particular, this count is the same as the total number of ${\mathbb{Z}}[1/m]$-equivalence classes of binary quadratic forms of discriminant $1-4m$, for $m$ running through the same range. Our heuristics, based on the Cohen–Lenstra heuristics, suggest that this total is asymptotic to $X^{3/2}/\log X$ and the largest contribution comes from the values of $m$ that are positive primes. Using sieve methods, we prove that the contribution to the total coming from $m$ positive prime is bounded above by $O(X^{3/2}/\log X)$ and that the total itself is $o(X^{3/2})$.

Black holes and Bhargava’s invariant theory

Attractor black holes in type II string compactifications on K3 × T 2 are in correspondence with equivalence classes of binary quadratic forms. The discriminant of the quadratic form governs the black hole entropy, and the count of attractor black holes at a given entropy is given by a class number. Here, we show this tantalizing relationship between attractors and arithmetic can be generalized to a rich family, connecting black holes in supergravity and string models with analogous equivalence classes of more general forms under the action of arithmetic groups. Many of the physical theories involved have played an earlier role in the study of ‘magical’ supergravities, while their mathematical counterparts are directly related to geometry-of-numbers examples in the work of Bhargava et al. This paper is dedicated to the memory of Peter Freund. The last section is devoted to some of MG’s personal reminiscences of Peter Freund.

Representation of the norm of ideals by quadratic forms with congruence conditions

Using a correspondence between the narrow ray class group modulo m of a quadratic field and a certain set of equivalence classes of binary quadratic forms proved by Furuta and Kubota, we find a quadratic form f and a pair of integers (x1,y1) such that the norm of all integral ideals a in a ray class is represented by f(mx+x1,my+y1) with some integers (x,y).

A proof of Hecke’s formula for binary quadratic forms

In Mathematische Werke, Hecke defines the operator [Formula: see text] and describes their utility in conjunction with theta series of quadratic forms. In particular, he shows that the image of theta series associated to classes of binary quadratic forms in CL[Formula: see text] is again a theta series associated to a collection of forms in CL[Formula: see text]. We state and prove an explicit formula for the action of [Formula: see text] on a binary quadratic form of negative discriminant.

Pair correlation of angles between reciprocal geodesics on the modular surface

The existence of the limiting pair correlation for angles between reciprocal geodesics on the modular surface is established. An explicit formula is provided, which captures geometric information about the length of reciprocal geodesics, as well as arithmetic information about the associated reciprocal classes of binary quadratic forms. One striking feature is the absence of a gap beyond zero in the limiting distribution, contrasting with the analog Euclidean situation.

Representations of certain binary quadratic forms as Lambert series

Classical algebraic number theory allows us to write certain Lambert series as a sum of theta series associated to classes of binary quadratic forms. We will recall how this is done and then give an “inversion” process to isolate some of these theta series and represent them by a linear combination of Lambert series.

Lengths of the periods of the continued fraction expansion of quadratic irrationalities and on the class numbers of real quadratic fields

It is proved that the relation h(d)=2 is valid for at least Cx1/2 log−2 x values of d≤x. Here h(d) is the number of the classes of binary quadratic forms of determinant d, while C>0 is a constant. Further, it is shown that for almost all primes p≡3 (mod 4), p≤x, for ε(p), a fundamental unit of field ℚ and l(p), the length of the period of the continued fraction expansion of √p, we have estimates ε(p)≫p2 log−c p, l(p)≫log p, which improve a result of Hooley (J. Reine Angew. Math., Vol. 353, pp. 98–131, 1984; MR 86d:11032). In addition, a generalization is given to composite discriminants of the Hirzebruch-Zagier formula, relating h(−p), p ≡ 3 (mod 4), with the continued fraction expansion of √p (Astérisque, no. 24–25, pp. 81–97, 1975; MR 51 #10293).

Action of the group of classes on representations of numbers by a ternary quadratic form: The formulas of Siegel and Kneser-Hsia-Peters

We specify in explicit form the action of the group of classes of binary quadratic forms of determinant dm on the set of primitive representations of the number m by a ternary quadratic form. As a consequence, we obtain an elementary proof of Siegel's formulas for the weighted number of representations by a genus and formulas for the weighted number of representations by a spinor genus of ternary forms.

Series ?F(m)qm, where F(m) is the number of odd classes of binary quadratic forms of determinant ?m

Consideration of the analytic continuation of the Eisenstein series of weight 3/2 for the group Γ0(4) leads to a new proof of Mordell's formula connecting the values X(ω)=∑ m=1 ∞ F(m)eπimω, Imω > 0, and χ(−1/ω). The behavior of the function χ(ω) for Γ0(4) is examined by the same method.

On the number of conjugacy classes in SL(2, Z)

Let H(t) be the number of conjugacy classes of elements in SL(2, L ) with trace t, and let h(n) be the number of equivalence classes of binary quadratic forms with discriminant n. Then for t≠±2, H(t)= h(t 2−4) . For all real θ > 0 there is a T( θ) such that whenever | t|> T( θ), H(t)>|t| 1−θ . There is a c>0 such that for those t such that t 2−4 is squarefree, H(t)≤c|t| .

Kronecker's fundamental limit formula in the theory of numbers and elliptic functions, and similar theorems

1. Many important applications of analysis to number-theory require the study of a function f (s) of a complex variable s = σ + i τ near a singular point s 0 = σ 0 + i τ 0 . The functions f (s) is frequently defined for σ > σ 0 by an infinite series, really d Dirichlet's series, the general term of which is a function of the variables of summation, e. g ., a quadratic form, raised to the power s . Thus the question of finding the number of classes of binary quadratic forms of given determinant, or the number of classes of ideals in a given field, depends upon the residue, Say R, of an appropriate f (s) at a simple pole s 0 . A deeper question then suggested is that of finding lim s → s 0 ( f (s) — R/ s-s 0 ). In particular, Kronecker's fundamental formula arises when f (s) is a homogeneous binary quadratic form in the variables of summation. Thus, let a a (≠ 0), b, c be any constants real or complex which are such that the roots ω 1 , ω 2 of the quadratic form ϕ (x, y) = ax 2 + bxy + cy 2 = a ( x - ω 1 y ) ( x - ω 2 y ) are neither real nor equal. We need only distinguish the two cases (I) I (ω 1 ) > 0, I (ω 2 ) < 0, (II) I (ω 1 ) > 0, I (ω 2 ) > 0, as the others can be included by writing — y for y .