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.

scholarly journals On the Fourier expansions of Jacobi forms of half-integral weight

2006 ◽  
Vol 2006 ◽  
pp. 1-11
Author(s):  
B. Ramakrishnan ◽  
Brundaban Sahu
Keyword(s):  
Modular Forms ◽  
Jacobi Form ◽  
Jacobi Forms ◽  
Number Of Components ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Fourier Expansions ◽  
The Given ◽  
The Relationship ◽  
Vector Valued

Using the relationship between Jacobi forms of half-integral weight and vector valued modular forms, we obtain the number of components which determine the given Jacobi form of indexp,p2orpq, wherepandqare odd primes.

Fourier Coefficients of Modular Forms of Half-Integral Weight in Arithmetic Progressions

2020 ◽  
Author(s):  
Corentin Darreye
Keyword(s):  
Gaussian Distribution ◽  
Modular Forms ◽  
Prime Power ◽  
Fourier Coefficients ◽  
Arithmetic Progressions ◽  
Cusp Forms ◽  
Half Integral Weight ◽  
Mixed Distribution ◽  
Integral Weight ◽  
Holomorphic Cusp Forms

Abstract We study the probabilistic behavior of sums of Fourier coefficients in arithmetic progressions. We prove a result analogous to previous work of Fouvry–Ganguly–Kowalski–Michel and Kowalski–Ricotta in the context of half-integral weight holomorphic cusp forms and for prime power modulus. We actually show that these sums follow in a suitable range a mixed Gaussian distribution that comes from the asymptotic mixed distribution of Salié sums.

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 On the Fourier expansions of Jacobi forms

2004 ◽  
Vol 2004 (48) ◽  
pp. 2583-2594 ◽  
Author(s):  
Howard Skogman
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Jacobi Form ◽  
The Other ◽  
Jacobi Forms ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Fourier Expansions ◽  
The Relationship ◽  
Vector Valued

We use the relationship between Jacobi forms and vector-valued modular forms to study the Fourier expansions of Jacobi forms of indexesp,p2, andpqfor distinct odd primesp,q. Specifically, we show that for such indexes, a Jacobi form is uniquely determined by one of the associated components of the vector-valued modular form. However, in the case of indexes of the formpqorp2, there are restrictions on which of the components will uniquely determine the form. Moreover, for indexes of the formp, this note gives an explicit reconstruction of the entire Jacobi form from a single associated vector-valued modular form component. That is, we show how to start with a single associated vector component and use specific matrices fromSl2(ℤ)to find the other components and hence the entire Jacobi form. These results are used to discuss the possible modular forms of half-integral weight associated to the Jacobi form for different subgroups.

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 the Ramanujan–Petersson conjecture for modular forms of half-integral weight

2019 ◽  
Vol 31 (3) ◽  
pp. 703-711
Author(s):  
Sanoli Gun ◽  
Winfried Kohnen
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Half Integral Weight ◽  
Integral Weight

Abstract We investigate the (still unknown) Ramanujan–Petersson conjecture about the growth of the Fourier coefficients of cusp forms of half-integral weight and prove that it is optimal, at least for newforms in the plus space.

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).

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