On the Representation of Numbers by the Direct Sums of Some Binary Quadratic Forms

1998 ◽  
Vol 5 (1) ◽  
pp. 55-70
Author(s):  
N. Kachakhidze
Keyword(s):  
Positive Integer ◽  
Quadratic Forms ◽  
Cusp Forms ◽  
Binary Quadratic Forms ◽  
Direct Sums

Abstract The systems of bases are constructed for the spaces of cusp forms Sk (Γ0(3), χ) (k≥6), Sk (Γ0(7), χ) (k≥3) and Sk (Γ0(11), χ) (k≥3). Formulas are obtained for the number of representation of a positive integer by the sum of k binary quadratic forms of the kind , of the kind and of the kind .

scholarly journals CONGRUENCES CONCERNING LEGENDRE POLYNOMIALS III

2013 ◽  
Vol 09 (04) ◽  
pp. 965-999 ◽  
Author(s):  
ZHI-HONG SUN
Keyword(s):  
Positive Integer ◽  
Quadratic Forms ◽  
Legendre Polynomials ◽  
Binary Quadratic Forms ◽  
Modulo P ◽  
Mod P

Suppose that p is an odd prime and d is a positive integer. Let x and y be integers given by p = x2+dy2 or 4p = x2+dy2. In this paper we determine x( mod p) for many values of d. For example, [Formula: see text] where x is chosen so that x ≡ 1 ( mod 3). We also pose some conjectures on supercongruences modulo p2 concerning binary quadratic forms.

scholarly journals Cusp Forms in and the Number of Representations of Positive Integers by Some Direct Sum of Binary Quadratic Forms with Discriminant

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Barış Kendirli
Keyword(s):  
Quadratic Forms ◽  
Cusp Forms ◽  
Binary Quadratic Forms ◽  
Direct Sum ◽  
Positive Integers

A basis of is given and the formulas for the number of representations of positive integers by some direct sum of the quadratic forms , , are determined.

On the number of representations of a positive integer as a sum of two binary quadratic forms

2014 ◽  
Vol 10 (06) ◽  
pp. 1395-1420 ◽  
Author(s):  
Şaban Alaca ◽  
Lerna Pehlivan ◽  
Kenneth S. Williams
Keyword(s):  
Positive Integer ◽  
Linear Combination ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Definite Integral ◽  
Binary Quadratic Forms ◽  
Finite Linear Combination ◽  
Positive Integers ◽  
Finite Linear

Let ℕ denote the set of positive integers and ℤ the set of all integers. Let ℕ0 = ℕ ∪{0}. Let a1x2 + b1xy + c1y2 and a2z2 + b2zt + c2t2 be two positive-definite, integral, binary quadratic forms. The number of representations of n ∈ ℕ0 as a sum of these two binary quadratic forms is [Formula: see text] When (b1, b2) ≠ (0, 0) we prove under certain conditions on a1, b1, c1, a2, b2 and c2 that N(a1, b1, c1, a2, b2, c2; n) can be expressed as a finite linear combination of quantities of the type N(a, 0, b, c, 0, d; n) with a, b, c and d positive integers. Thus, when the quantities N(a, 0, b, c, 0, d; n) are known, we can determine N(a1, b1, c1, a2, b2, c2; n). This determination is carried out explicitly for a number of quaternary quadratic forms a1x2 + b1xy + c1y2 + a2z2 + b2zt + c2t2. For example, in Theorem 1.2 we show for n ∈ ℕ that [Formula: see text] where N is the largest odd integer dividing n and [Formula: see text]

ON A CORRESPONDENCE BETWEEN JACOBI CUSP FORMS AND ELLIPTIC CUSP FORMS

2013 ◽  
Vol 09 (04) ◽  
pp. 917-937 ◽  
Author(s):  
B. RAMAKRISHNAN ◽  
KARAM DEO SHANKHADHAR
Keyword(s):  
Modular Forms ◽  
Quadratic Forms ◽  
Cusp Forms ◽  
Jacobi Forms ◽  
Binary Quadratic Forms ◽  
Congruence Subgroups ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Forms Of Higher Degree ◽  
Appl Math

In this paper, we prove a generalization of a correspondence between holomorphic Jacobi cusp forms of higher degree (matrix index) and elliptic cusp forms obtained by K. Bringmann [Lifting maps from a vector space of Jacobi cusp forms to a subspace of elliptic modular forms, Math. Z.253 (2006) 735–752], for forms of higher levels (for congruence subgroups). To achieve this, we make use of the method adopted by M. Manickam and the first author in Sec. 3 of [On Shimura, Shintani and Eichler–Zagier correspondences, Trans. Amer. Math. Soc.352 (2000) 2601–2617], who obtained similar correspondence in the degree one case. We also derive a similar correspondence in the case of skew-holomorphic Jacobi forms (matrix index and for congruence subgroups). Such results in the degree one case (for the full group) were obtained by N.-P. Skoruppa [Developments in the theory of Jacobi forms, in Automorphic Functions and Their Applications, Khabarovsk, 1988 (Acad. Sci. USSR, Inst. Appl. Math., Khabarovsk, 1990), pp. 168–185; Binary quadratic forms and the Fourier coefficients of elliptic and Jacobi modular forms, J. Reine Angew. Math.411 (1990) 66–95] and by M. Manickam [Newforms of half-integral weight and some problems on modular forms, Ph.D. thesis, University of Madras (1989)].

scholarly journals On a completed generating function of locally harmonic Maass forms

2014 ◽  
Vol 150 (5) ◽  
pp. 749-762 ◽  
Author(s):  
Kathrin Bringmann ◽  
Ben Kane ◽  
Sander Zwegers
Keyword(s):  
Differential Operator ◽  
Generating Function ◽  
Cusp Form ◽  
Quadratic Forms ◽  
Cusp Forms ◽  
Binary Quadratic Forms ◽  
Maass Forms ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Harmonic Maass Forms

AbstractWhile investigating the Doi–Naganuma lift, Zagier defined integral weight cusp forms $f_D$ which are naturally defined in terms of binary quadratic forms of discriminant $D$. It was later determined by Kohnen and Zagier that the generating function for the function $f_D$ is a half-integral weight cusp form. A natural preimage of $f_D$ under a differential operator at the heart of the theory of harmonic weak Maass forms was determined by the first two authors and Kohnen. In this paper, we consider the modularity properties of the generating function of these preimages. We prove that although the generating function is not itself modular, it can be naturally completed to obtain a half-integral weight modular object.

RATIONAL REPRESENTATIONS OF PRIMES BY BINARY QUADRATIC FORMS

2007 ◽  
Vol 03 (04) ◽  
pp. 513-528
Author(s):  
RONALD EVANS ◽  
MARK VAN VEEN
Keyword(s):  
Positive Integer ◽  
Quadratic Forms ◽  
Binary Quadratic Forms ◽  
Rational Solutions ◽  
Algebraic Integers ◽  
Squarefree Integer

Let q be a positive squarefree integer. A prime p is said to be q-admissible if the equation p = u2 + qv2 has rational solutions u, v. Equivalently, p is q-admissible if there is a positive integer k such that [Formula: see text], where [Formula: see text] is the set of norms of algebraic integers in [Formula: see text]. Let k(q) denote the smallest positive integer k such that [Formula: see text] for all q-admissible primes p. It is shown that k(q) has subexponential but suprapolynomial growth in q, as q → ∞.

On the Number of Representations of Positive Integers by a Direct Sum of Binary Quadratic Forms with Discriminant –23

1997 ◽  
Vol 4 (6) ◽  
pp. 523-532
Author(s):  
G. Lomadze
Keyword(s):  
Quadratic Forms ◽  
Binary Quadratic Forms ◽  
Direct Sums ◽  
Direct Sum ◽  
Positive Integers

Abstract Explicit exact formulas are obtained for the number of representations of positive integers by some direct sums of quadratic forms and .

On the Representation of Numbers by the Direct Sums of Some Quaternary Quadratic Forms

2001 ◽  
Vol 8 (1) ◽  
pp. 87-95
Author(s):  
N. Kachakhidze
Keyword(s):  
Quadratic Forms ◽  
Cusp Forms ◽  
Arbitrary Integer ◽  
Direct Sums ◽  
Positive Integers

Abstract The systems of bases are constructed for the spaces of cusp forms S 2m (Γ0(5), χ m ) and S 2m (Γ0(13), χ m ) for an arbitrary integer m ≥ 2. Formulas are obtained for the number of representations of a positive integers by the direct sums of some quaternary quadratic forms.

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