scholarly journals Computing Jacobi forms

2016 ◽  
Vol 19 (A) ◽  
pp. 205-219 ◽  
Author(s):  
Nathan C. Ryan ◽  
Nicolás Sirolli ◽  
Nils-Peter Skoruppa ◽  
Gonzalo Tornaría
Keyword(s):  
Modular Forms ◽  
Modular Group ◽  
Ambient Space ◽  
Jacobi Forms ◽  
Modular Symbols ◽  
Hecke Eigenforms ◽  
Large Index ◽  
Integral Weight

We describe an implementation for computing holomorphic and skew-holomorphic Jacobi forms of integral weight and scalar index on the full modular group. This implementation is based on formulas derived by one of the authors which express Jacobi forms in terms of modular symbols of elliptic modular forms. Since this method allows a Jacobi eigenform to be generated directly from a given modular eigensymbol without reference to the whole ambient space of Jacobi forms, it makes it possible to compute Jacobi Hecke eigenforms of large index. We illustrate our method with several examples.

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.

On Hecke eigenvalues of Siegel modular forms in the Maass space

2018 ◽  
Vol 30 (3) ◽  
pp. 775-783 ◽  
Author(s):  
Sanoli Gun ◽  
Biplab Paul ◽  
Jyoti Sengupta
Keyword(s):  
Modular Forms ◽  
Absolute Constant ◽  
Modular Group ◽  
Siegel Modular Forms ◽  
Maass Forms ◽  
Formula One ◽  
Hecke Eigenvalues ◽  
Hecke Eigenforms ◽  
The Third ◽  
Limit Points

AbstractIn this article, we prove an Omega result for the Hecke eigenvalues {\lambda_{F}(n)} of Maass forms F which are Hecke eigenforms in the space of Siegel modular forms of weight k, genus two for the Siegel modular group {Sp_{2}({\mathbb{Z}})}. In particular, we prove\lambda_{F}(n)=\Omega\biggl{(}n^{k-1}\exp\biggl{(}c\frac{\sqrt{\log n}}{\log% \log n}\biggr{)}\biggr{)},when {c>0} is an absolute constant. This improves the earlier result\lambda_{F}(n)=\Omega\biggl{(}n^{k-1}\biggl{(}\frac{\sqrt{\log n}}{\log\log n}% \biggr{)}\biggr{)}of Das and the third author. We also show that for any {n\geq 3}, one has\lambda_{F}(n)\leq n^{k-1}\exp\biggl{(}c_{1}\sqrt{\frac{\log n}{\log\log n}}% \biggr{)},where {c_{1}>0} is an absolute constant. This improves an earlier result of Pitale and Schmidt. Further, we investigate the limit points of the sequence {\{\lambda_{F}(n)/n^{k-1}\}_{n\in{\mathbb{N}}}} and show that it has infinitely many limit points. Finally, we show that {\lambda_{F}(n)>0} for all n, a result proved earlier by Breulmann by a different technique.

scholarly journals On Generalized Modular forms and their Applications

2008 ◽  
Vol 192 ◽  
pp. 119-136 ◽  
Author(s):  
Winfried Kohnen ◽  
Geoffrey Mason
Keyword(s):  
Finite Index ◽  
Modular Forms ◽  
Vertex Operator ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Operator Algebras ◽  
Vertex Operator Algebras ◽  
Integral Weight

AbstractWe study the Fourier coefficients of generalized modular forms f(τ) of integral weight k on subgroups Γ of finite index in the modular group. We establish two Theorems asserting that f(τ) is constant if k = 0, f(τ) has empty divisor, and the Fourier coefficients have certain rationality properties. (The result is false if the rationality assumptions are dropped.) These results are applied to the case that f(τ) has a cuspidal divisor, k is arbitrary, and Γ = Γ0(N), where we show that f(τ) is modular, indeed an eta-quotient, under natural rationality assumptions on the Fourier coefficients. We also explain how these results apply to the theory of orbifold vertex operator algebras.

scholarly journals On the Kohnen plus space for Hilbert modular forms of half-integral weight I

2013 ◽  
Vol 149 (12) ◽  
pp. 1963-2010 ◽  
Author(s):  
Kaoru Hiraga ◽  
Tamotsu Ikeda
Keyword(s):  
Modular Form ◽  
Fourier Coefficient ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Hilbert Modular Forms ◽  
Jacobi Forms ◽  
Hecke Operator ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Hilbert Modular

AbstractIn this paper, we construct a generalization of the Kohnen plus space for Hilbert modular forms of half-integral weight. The Kohnen plus space can be characterized by the eigenspace of a certain Hecke operator. It can be also characterized by the behavior of the Fourier coefficients. For example, in the parallel weight case, a modular form of weight $\kappa + (1/ 2)$ with $\xi \mathrm{th} $ Fourier coefficient $c(\xi )$ belongs to the Kohnen plus space if and only if $c(\xi )= 0$ unless $\mathop{(- 1)}\nolimits ^{\kappa } \xi $ is congruent to a square modulo $4$. The Kohnen subspace is isomorphic to a certain space of Jacobi forms. We also prove a generalization of the Kohnen–Zagier formula.

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