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.

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.

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

1996 ◽  
Vol 3 (5) ◽  
pp. 485-500
Author(s):  
G. Lomadze
Keyword(s):  
Modular Forms ◽  
Quadratic Forms ◽  
Congruence Group ◽  
Positive Integers

Abstract Entire modular forms of weights and for the congruence group 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 7 and 9 variables.

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).

scholarly journals Shimura correspondence for level p2 and the central values of L-series, II

2014 ◽  
Vol 10 (07) ◽  
pp. 1595-1635
Author(s):  
Ariel Pacetti ◽  
Gonzalo Tornaría
Keyword(s):  
Linear Combination ◽  
Modular Forms ◽  
Quadratic Forms ◽  
Theta Series ◽  
Geometric Interpretation ◽  
Integral Weight ◽  
Free Level ◽  
Hecke Eigenform ◽  
Special Points ◽  
Shimura Correspondence

Given a Hecke eigenform f of weight 2 and square-free level N, by the work of Kohnen, there is a unique weight 3/2 modular form of level 4N mapping to f under the Shimura correspondence. Furthermore, by the work of Waldspurger the Fourier coefficients of such a form are related to the quadratic twists of the form f. Gross gave a construction of the half integral weight form when N is prime, and such construction was later generalized to square-free levels. However, in the non-square free case, the situation is more complicated since the natural construction is vacuous. The problem being that there are too many special points so that there is cancellation while trying to encode the information as a linear combination of theta series. In this paper, we concentrate in the case of level p2, for p > 2 a prime number, and show how the set of special points can be split into subsets (indexed by bilateral ideals for an order of reduced discriminant p2) which gives two weight 3/2 modular forms mapping to f under the Shimura correspondence. Moreover, the splitting has a geometric interpretation which allows to prove that the forms are indeed a linear combination of theta series associated to ternary quadratic forms. Once such interpretation is given, we extend the method of Gross–Zagier to the case where the level and the discriminant are not prime to each other to prove a Gross-type formula in this situation.

Modular Properties of Theta-Functions and Representation of Numbers by Positive Quadratic Forms

1997 ◽  
Vol 4 (4) ◽  
pp. 385-400
Author(s):  
T. Vepkhvadze
Keyword(s):  
Modular Forms ◽  
Quadratic Forms ◽  
Theta Functions ◽  
Positive Integers

Abstract By means of the theory of modular forms the formulas for a number of representations of positive integers by two positive quaternary quadratic forms of steps 36 and 60 and by all positive diagonal quadratic forms with seven variables of step 8 are obtain.

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

1995 ◽  
Vol 2 (2) ◽  
pp. 189-199
Author(s):  
G. Lomadze
Keyword(s):  
Modular Forms ◽  
Quadratic Forms ◽  
Congruence Subgroup ◽  
Congruence Group

Abstract Two classes of entire modular forms of weight 5 and two of weight 6 are constructed for the congruence subgroup Γ0(4N). The constructed modular forms as well as the modular forms from [Lomadze, Georgian Math. J. 1: 53-76, 1994] will be helpful in the theory of representation of numbers by the quadratic forms in 10 and 12 variables.

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