On Entire Modular Forms of Half-Integral Weight on the Congruence Subgroup Γ0(4N)

2004 ◽  
Vol 11 (1) ◽  
pp. 111-123
Author(s):  
G. Lomadze
Keyword(s):  
Modular Forms ◽  
Quadratic Forms ◽  
Congruence Subgroup ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Positive Integers

Abstract Some entire modular forms of weight on the congruence subgroup Γ0(4N) are constructed if s is odd > 11. The constructed type of entire modular forms is useful for revealing the arithmetical meaning of additional terms in formulas for the number of representations of positive integers by positive quadratic forms with integral coefficients if the number s of variables is odd > 11.

On Some Entire Modular Forms of an Integral Weight for the Congruence Subgroup Γ0(4𝑁)

1999 ◽  
Vol 6 (5) ◽  
pp. 471-488
Author(s):  
G. Lomadze
Keyword(s):  
Modular Forms ◽  
Quadratic Forms ◽  
Congruence Subgroup ◽  
Integral Weight ◽  
Positive Integers

Abstract Four types of entire modular forms of weight are constructed for the congruence subgroup Γ0(4𝑁) when 𝑠 is even. One can find these forms helpful in revealing the arithmetical meaning of additional terms in the formulas for the number of representations of positive integers by positive quadratic forms with integral coefficients in an even number of variables.

scholarly journals On the divisibility of r2(n)

1977 ◽  
Vol 18 (1) ◽  
pp. 109-111 ◽  
Author(s):  
E. J. Scourfield
Keyword(s):  
Modular Form ◽  
Algebraic Number ◽  
Modular Forms ◽  
Congruence Subgroup ◽  
Algebraic Number Field ◽  
The Past ◽  
Divisibility Property ◽  
Integral Weight ◽  
Image Position ◽  
Positive Integers

During the past few years, some papers of P. Deligne and J.-P. Serre (see [2], [9], [10] and other references cited there) have included an investigation of certain properties of coefficients of modular forms, and in particular Serre [10] (see also [11]) obtained the divisibility property (1) below. Letbe a modular form of integral weight k ≧ 1 on a congruence subgroup of SL2(Z), and suppose that each cn belongs to the ring RK of integers of an algebraic number field K finite over Q. For c ∈ RK and m ≧ 1 an integer, write c ≡ 0 (mod m) if c ∈ m RK and c ≢ 0 (mod m) otherwise. Then Serre showed that there exists α > 0 such thatas x → ∞, where throughout this note N(n ≦ x: P) denotes the number of positive integers n ≦ x with the property P.

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 Modular forms arising from zeta functions in two variables attached to prehomogeneous vector spaces related to quadratic forms

2004 ◽  
Vol 175 ◽  
pp. 1-37 ◽  
Author(s):  
Takahiko Ueno
Keyword(s):  
Modular Forms ◽  
Quadratic Forms ◽  
Zeta Functions ◽  
Vector Spaces ◽  
Parabolic Subgroups ◽  
Converse Theorem ◽  
Half Integral Weight ◽  
Prehomogeneous Vector Spaces ◽  
Integral Weight ◽  
Weil Type

AbstractIn this paper, we prove the functional equations for the zeta functions in two variables associated with prehomogeneous vector spaces acted on by maximal parabolic subgroups of orthogonal groups. Moreover, applying the converse theorem of Weil type, we show that elliptic modular forms of integral or half integral weight can be obtained from the zeta functions.

scholarly journals Ternary quadratic forms and Shimura’s correspondence

1981 ◽  
Vol 81 ◽  
pp. 123-151 ◽  
Author(s):  
Paul Ponomarev
Keyword(s):  
Cusp Form ◽  
Modular Forms ◽  
Quadratic Forms ◽  
Ternary Quadratic Forms ◽  
Half Integral Weight ◽  
Integral Weight

In his paper [11] Shimura defined a correspondence between modular forms of half integral weight and modular forms of integral weight. To each pair (t, f(z)), consisting of a square-free integer t ≥ 1 and a cusp form of weight k/2 (k odd, ≥ 3), level N (divisible by 4) and character ϰ, he associated a certain function f(t)(z) (Ft(z) in Shimura’s notation).

On Some Entire Modular Forms of Weights 5 and 6 for the Congruence Group Γ0(4N)

1994 ◽  
Vol 1 (1) ◽  
pp. 53-76
Author(s):  
G. Lomadze
Keyword(s):  
Modular Forms ◽  
Quadratic Forms ◽  
Congruence Subgroup ◽  
Congruence Group ◽  
Positive Integers

Abstract Two entire modular forms of weight 5 and two of weight 6 for the congruence subgroup Γ0(4N) are constructed, which will be useful for revealing the arithmetical sense of additional terms in formulas for the number of representations of positive integers by quadratic forms in 10 and 12 variables.

ON THE FOURIER COEFFICIENTS OF MODULAR FORMS OF HALF-INTEGRAL WEIGHT

2013 ◽  
Vol 09 (08) ◽  
pp. 1879-1883 ◽  
Author(s):  
YOUNG JU CHOIE ◽  
WINFRIED KOHNEN
Keyword(s):  
Modular Form ◽  
Growth Condition ◽  
Cusp Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Congruence Subgroup ◽  
Half Integral Weight ◽  
Fundamental Discriminant ◽  
Integral Weight ◽  
Elliptic Modular Form

It is known that if the Fourier coefficients a(n)(n ≥ 1) of an elliptic modular form of even integral weight k ≥ 2 on the Hecke congruence subgroup Γ0(N)(N ∈ N) satisfy the bound a(n) ≪f nc for all n ≥ 1, where c > 0 is any number strictly less than k - 1, then f must be cuspidal. Here we investigate the case of half-integral weight modular forms. The main objective of this note is to show that to deduce that f is a cusp form, it is sufficient to impose a suitable growth condition only on the Fourier coefficients a(|D|) where D is a fundamental discriminant with (-1)kD > 0.

scholarly journals Ternary quadratic forms and half-integral weight modular forms

2012 ◽  
Vol 15 ◽  
pp. 418-435 ◽  
Author(s):  
Alia Hamieh
Keyword(s):  
Positive Integer ◽  
Modular Forms ◽  
Quadratic Forms ◽  
Dimensional Subspace ◽  
Two Dimensional ◽  
Ternary Quadratic Forms ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Shimura Correspondence

AbstractLet k be a positive integer such that k≡3 mod 4, and let N be a positive square-free integer. In this paper, we compute a basis for the two-dimensional subspace Sk/2(Γ0(4N),F) of half-integral weight modular forms associated, via the Shimura correspondence, to a newform F∈Sk−1(Γ0(N)), which satisfies $L(F,\frac {1}{2})\neq 0$. This is accomplished by using a result of Waldspurger, which allows one to produce a basis for the forms that correspond to a given F via local considerations, once a form in the Kohnen space has been determined.

ARITHMETIC TRACES OF NON-HOLOMORPHIC MODULAR INVARIANTS

2010 ◽  
Vol 06 (01) ◽  
pp. 69-87 ◽  
Author(s):  
ALISON MILLER ◽  
AARON PIXTON
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Algebraic Numbers ◽  
Heegner Points ◽  
Positive Weight ◽  
Poincare Series ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Modular Invariants

We extend results of Bringmann and Ono that relate certain generalized traces of Maass–Poincaré series to Fourier coefficients of modular forms of half-integral weight. By specializing to cases in which these traces are usual traces of algebraic numbers, we generalize results of Zagier describing arithmetic traces associated to modular forms. We define correspondences [Formula: see text] and [Formula: see text]. We show that if f is a modular form of non-positive weight 2 - 2 λ and odd level N, holomorphic away from the cusp at infinity, then the traces of values at Heegner points of a certain iterated non-holomorphic derivative of f are equal to Fourier coefficients of the half-integral weight modular forms [Formula: see text].

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