scholarly journals Heegner cycles, modular forms and jacobi forms

1991 ◽  
Vol 3 (1) ◽  
pp. 93-116 ◽  
Author(s):  
Nils-Peter Skoruppa
Keyword(s):  
Modular Forms ◽  
Jacobi Forms ◽  
Heegner Cycles

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.

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 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.

Taylor coefficients of mock-Jacobi forms and moments of partition statistics

2014 ◽  
Vol 157 (2) ◽  
pp. 231-251 ◽  
Author(s):  
KATHRIN BRINGMANN ◽  
KARL MAHLBURG ◽  
ROBERT C. RHOADES
Keyword(s):  
Modular Forms ◽  
Asymptotic Series ◽  
Circle Method ◽  
Jacobi Form ◽  
Series Expansions ◽  
Jacobi Forms ◽  
Transformation Theory ◽  
Taylor Coefficients ◽  
Partition Statistics ◽  
A New Technique

AbstractWe develop a new technique for deriving asymptotic series expansions for moments of combinatorial generating functions that uses the transformation theory of Jacobi forms and “mock” Jacobi forms, as well as the Hardy-Ramanujan Circle Method. The approach builds on a suggestion of Zagier, who observed that the moments of a combinatorial statistic can be simultaneously encoded as the Taylor coefficients of a function that transforms as a Jacobi form. Our use of Jacobi transformations is a novel development in the subject, as previous results on the asymptotic behavior of the Taylor coefficients of Jacobi forms have involved the study of each such coefficient individually using the theory of quasimodular forms and quasimock modular forms.As an application, we find asymptotic series for the moments of the partition rank and crank statistics. Although the coefficients are exponentially large, the error in the series expansions is polynomial, and have the same order as the coefficients of the residual Eisenstein series that are undetectable by the Circle Method. We also prove asymptotic series expansions for the symmetrized rank and crank moments introduced by Andrews and Garvan, respectively. Equivalently, the former gives asymptotic series for the enumeration of Andrews k-marked Durfee symbols.

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