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.

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.

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 $$k\ge 4$$.

The hyperbolic geometry of Markov’s theorem on Diophantine approximation and quadratic forms

Markov's theorem classifies the worst irrational numbers with respect to rational approximation and the indefinite binary quadratic forms whose values for integer arguments stay farthest away from zero. The main purpose of this paper is to present a new proof of Markov's theorem using hyperbolic geometry. The main ingredients are a dictionary to translate between hyperbolic geometry and algebra/number theory, and some very basic tools borrowed from modern geometric Teichm\"uller theory. Simple closed geodesics and ideal triangulations of the modular torus play an important role, and so do the problems: How far can a straight line crossing a triangle stay away from the vertices? How far can it stay away from the vertices of the tessellation generated by the triangle? Definite binary quadratic forms are briefly discussed in the last section.

Completely p-primitive binary quadratic forms

Let f(x,y)=ax2+bxy+cy2 be a binary quadratic form with integer coefficients. For a prime p not dividing the discriminant of f, we say f is completely p-primitive if for any non-zero integer N, the diophantine equation f(x,y)=N always has an integer solution (x,y)=(m,n) with (m,n,p)=1 whenever it has an integer solution. In this article, we study various properties of completely p-primitive binary quadratic forms. In particular, we give a necessary and sufficient condition for a definite binary quadratic form f to be completely p-primitive.

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.

On Representation of Integers by Binary Quadratic Forms

Given a negative D > −(logX)log2−δ , we give a new upper bound on the number of square free integers X which are represented by some but not all forms of the genus of a primitive positive definite binary quadratic form of discriminant D. We also give an analogous upper bound for square free integers of the form q+a X where q is prime and a ∈ Z is fixed. Combined with the 1/2-dimensional sieve of Iwaniec, this yields a lower bound on the number of such integers q+a X represented by a binary quadratic form of discriminant D, where D is allowed to grow with X as above. An immediate consequence of this, coming from recent work of the authors in [3], is a lower bound on the number of primes which come up as curvatures in a given primitive integer Apollonian circle packing.