Positive Definite Binary Quadratic Forms Research Articles

Cycle integrals of meromorphic modular forms and Siegel theta functions

We study meromorphic modular forms associated with positive definite binary quadratic forms and their cycle integrals along closed geodesics in the modular curve. We show that suitable linear combinations of these meromorphic modular forms have rational cycle integrals. Along the way we evaluate the cycle integrals of the Siegel theta function associated with an even lattice of signature (1, 2) in terms of Hecke’s indefinite theta functions.

Oscillations of Fourier coefficients of product of L-functions at integers in a sparse set

Let [Formula: see text] be a normalized Hecke eigenform of weight [Formula: see text] for the full modular group [Formula: see text]. In this paper, we obtain the asymptotic of higher moments of general divisor functions associated to the Fourier coefficients of Rankin–Selberg [Formula: see text]-functions [Formula: see text], supported at the integers represented by primitive integral positive-definite binary quadratic forms (reduced forms) of a fixed discriminant [Formula: see text] We improve previous results in the case when the reduced form is given by [Formula: see text]

Deterministic comparison of Hecke eigenvalues at the primes represented by a binary quadratic form

Let [Formula: see text] and [Formula: see text] be the normalized [Formula: see text]-Fourier coefficient of distinct Hecke eigenforms [Formula: see text] and [Formula: see text] of integral weight for the full modular group, respectively, and [Formula: see text] is a primitive integral positive-definite binary quadratic form of fixed discriminant [Formula: see text] with the class number [Formula: see text]. We establish a lower bound for the analytic density of the set [Formula: see text] for each fixed [Formula: see text] With the above notation, we also obtain a lower bound for the analytic density of the set [Formula: see text] for each fixed [Formula: see text] As a consequence, we obtain a deterministic criterion for the equality of two Hecke eigenforms.

On the number of integral binary n$n$‐ic forms having bounded Julia invariant

In 1848, Hermite introduced a reduction theory for binary forms of degree n $n$ which was developed more fully in the seminal 1917 treatise of Julia. This canonical method of reduction made use of a new, fundamental, but irrational SL 2 ${\rm SL}_2$ -invariant of binary n $n$ -ic forms defined over R $\mathbb {R}$ , which is now known as the Julia invariant. In this paper, for each n $n$ and k $k$ with n + k ⩾ 3 $n+k\geqslant 3$ , we determine the asymptotic behavior of the number of SL 2 ( Z ) ${\rm SL}_2(\mathbb {Z})$ -equivalence classes of binary n $n$ -ic forms, with k $k$ pairs of complex roots, having bounded Julia invariant. Specializing to ( n , k ) = ( 2 , 1 ) $(n,k)=(2,1)$ and (3,0), respectively, recovers the asymptotic results of Gauss and Davenport on positive definite binary quadratic forms and positive discriminant binary cubic forms, respectively.

On a class number formula of Hurwitz

In a little-known paper Hurwitz gave an infinite series representation of the class number for positive definite binary quadratic forms. In this paper we give a similar formula in the indefinite case. We also give a simple proof of Hurwitz’s formula and indicate some extensions.

Integers of the form $$ax^2+bxy+cy^2$$

We provide a characterization of integers represented by the positive definite binary quadratic form $$ax^2+bxy+cy^2$$ . Suppose that $$D=b^2-4ac$$ and $$d_K$$ is the discriminant of the imaginary quadratic field $$K={\mathbb {Q}}(\sqrt{D})$$ . We call $$f=\sqrt{D/d_K}$$ the conductor of $$ax^2+bxy+cy^2$$ . In order to prove the main results, we define the “relative conductor” of two orders in an imaginary quadratic field. We provide a characterization of decomposition of proper ideals of orders in imaginary quadratic fields. Next, we provide characterizations of prime powers $$l^h$$ , where l divides the conductor, represented by the positive definite binary quadratic form $$ax^2+bxy+cy^2$$ . Some interesting applications of the main results are also presented. For example, we provide an equivalent condition for when the equation $$m=4x^2+2xy+7y^2$$ has an integer solution. Note that its discriminant and conductor are $$-\,108$$ and 6 and we do not assume that m is prime to 2 or 3.

Meromorphic Modular Forms with Rational Cycle Integrals

AbstractWe study rationality properties of geodesic cycle integrals of meromorphic modular forms associated to positive definite binary quadratic forms. In particular, we obtain finite rational formulas for the cycle integrals of suitable linear combinations of these meromorphic modular forms.

On prime values of binary quadratic forms with a thin variable

In this paper, we generalize the result of Fouvry and Iwaniec dealing with prime values of the quadratic form x 2 + y 2 with one input restricted to a thin subset of the integers. We prove the same result with an arbitrary primitive positive definite binary quadratic form. In particular, for any positive definite binary quadratic form F and binary linear form G, there exist infinitely many ℓ , m ∈ Z such that both F ( ℓ , m ) and G ( ℓ , m ) are primes as long as there are no local obstructions.

Estimates for representation numbers of binary quadratic forms and Apollonian circle packings

Fix a primitive, positive definite binary quadratic form g with integer coefficients. We prove asymptotic formulas for sums of the form ∑rg(n)β and ∑rg⁎(n)β, where β≥0 and rg(n), resp. rg⁎(n), denote the number of inequivalent representations, resp. proper inequivalent representations, of n by g. These estimates generalize a previous result by Blomer and Granville (2006) by allowing for non-fundamental discriminants and also clarify some details in the proof of the positive density conjecture for integral Apollonian circle packings by Bourgain and Fuchs (2011).

Composition law and complex multiplication

Let K be an imaginary quadratic field of discriminant dK, and let n be a nontrivial integral ideal of K in which N is the smallest positive integer. Let QN(dK) be the set of primitive positive definite binary quadratic forms of discriminant dK whose leading coefficients are relatively prime to N. We adopt an equivalence relation ∼n on QN(dK) so that the set of equivalence classes QN(dK)/∼n can be regarded as a group isomorphic to the ray class group of K modulo n. We further establish an explicit isomorphism of QN(dK)/∼n onto Gal(Kn/K) in terms of Fricke invariants, where Kn denotes the ray class field of K modulo n. This would be a certain extension of the classical composition theory of binary quadratic forms, originated and developed by Gauss and Dirichlet.

Intersection numbers of modular correspondences for genus zero modular curves

In this paper, we introduce modular polynomials for the congruence subgroup Γ0(M) when X0(M) has genus zero and therefore the polynomials are defined by a Hauptmodul of X0(M). We show that the intersection number of two curves defined by two modular polynomials can be expressed as the sum of the numbers of SL2(Z)-equivalence classes of positive definite binary quadratic forms over Z. We also show that the intersection numbers can be also combinatorially written by Fourier coefficients of the Siegel Eisenstein series of degree 2, weight 2 with respect to Sp2(Z).

Binary quadratic forms represented by sums of squares in an essentially unique way

In this article, we determine all positive definite integral binary quadratic forms that are represented by the sum of k integer squares in an essentially unique way for each integer $$k\ge 4$$.

Primes represented by positive definite binary quadratic forms

Let $f$ be a primitive positive definite integral binary quadratic form of discriminant $-D$ and let $\pi_f(x)$ be the number of primes up to $x$ which are represented by $f$. We prove several types of upper bounds for $\pi_f(x)$ within a constant factor of its asymptotic size: unconditional, conditional on the Generalized Riemann Hypothesis (GRH), and for almost all discriminants. The key feature of these estimates is that they hold whenever $x$ exceeds a small power of $D$ and, in some cases, this range of $x$ is essentially best possible. In particular, if $f$ is reduced then this optimal range of $x$ is achieved for almost all discriminants or by assuming GRH. We also exhibit an upper bound for the number of primes represented by $f$ in a short interval and a lower bound for the number of small integers represented by $f$ which have few prime factors.

Some expressions for binary theta series by eta-quotients and their applications

In this paper, we show that some single theta series of positive definite binary quadratic forms can be expressed in terms of eta-quotients. We also give two applications of our theorem as follows; (1) to prove that a certain modular form, which is a product of theta series and eta-product, has complex multiplication, (2) an explicit description for the image of η(τ)2η(Nτ)2 under the Hecke operator T2.

Diagonal quadratic forms representing all binary diagonal quadratic forms

A (positive definite integral) quadratic form is called diagonally 2-universal if it represents all positive definite integral binary diagonal quadratic forms. In this article, we show that, up to equivalence, there are exactly 18 (positive definite integral) quinary diagonal quadratic forms that are diagonally 2-universal. Furthermore, we provide a “diagonally 2-universal criterion” for diagonal quadratic forms, which is similar to “15-Theorem” proved by Conway and Schneeberger.

Petersson inner products of weight-one modular forms

Abstract In this paper we study the regularized Petersson product between a holomorphic theta series associated to a positive definite binary quadratic form and a weakly holomorphic weight-one modular form with integral Fourier coefficients. In [18], we proved that these Petersson products posses remarkable arithmetic properties. Namely, such a Petersson product is equal to the logarithm of a certain algebraic number lying in a ring class field associated to the binary quadratic form. A similar result was obtained independently using a different method by W. Duke and Y. Li [5]. The main result of this paper is an explicit factorization formula for the algebraic number obtained by exponentiating a Petersson product.

Theta series associated with certain positive definite binary quadratic forms

Recently, P. C. Toh derived identities expressing theta series associated with fundamental idoneal discriminants in terms of generalized Lambert series. His method depends on the fact that the ring of integers of a number field is a Dedekind domain, and i

Note to the Paper ``A Positive Definite Binary Quadratic Form as a Sum of Five Squares of Linear Forms (Completion of Mordell's Proof)'' (Bull. Polish Acad. Sci. Math. 61 (2013), 23–26)

Note to the Paper ``A Positive Definite Binary Quadratic Form as a Sum of Five Squares of Linear Forms (Completion of Mordell's Proof)'' (Bull. Polish Acad. Sci. Math. 61 (2013), 23–26)

Correlations of representation functions of binary quadratic forms

This paper extends previous work on linear correlations of representation functions of positive definite binary quadratic forms to allow indefinite forms.

A Positive Definite Binary Quadratic Form as a Sum of Five Squares of Linear Forms (Completion of Mordell's Proof)

The paper completes an incomplete proof given by L. J. Mordell in 1930 of the following theorem: every positive definite classical binary quadratic form is the sum of five squares of linear forms with integral coefficients.