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.

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.

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.

Asymptotic formulas for coefficients of Kac–Wakimoto Characters

2013 ◽  
Vol 155 (1) ◽  
pp. 51-72 ◽  
Author(s):  
KATHRIN BRINGMANN ◽  
KARL MAHLBURG
Keyword(s):  
Modular Forms ◽  
Asymptotic Series ◽  
Circle Method ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Saddle Point Method ◽  
Asymptotic Formulas ◽  
Affine Lie Algebras ◽  
Maass Forms ◽  
Mock Modular Forms

AbstractWe study the coefficients of Kac and Wakimoto's character formulas for the affine Lie superalgebrassℓ(n+1|1)∧. The coefficients of these characters are the weight multiplicities of irreducible modules of the Lie superalgebras, and their asymptotic study begins with Kac and Peterson's earlier use of modular forms to understand the coefficients of characters for affine Lie algebras. In the affine Lie superalgebra setting, the characters are products of weakly holomorphic modular forms and Appell-type sums, which have recently been studied using developments in the theory of mock modular forms and harmonic Maass forms. Using our previously developed extension of the Circle Method for products of mock modular forms along with the Saddle Point Method, we find asymptotic series expansions for the coefficients of the characters with polynomial error.

scholarly journals Generic differential operators on Siegel modular forms and special polynomials

2020 ◽  
Vol 26 (5) ◽  
Author(s):  
Tomoyoshi Ibukiyama
Keyword(s):  
Modular Forms ◽  
Differential Operators ◽  
Siegel Modular Forms ◽  
Diagonal Block ◽  
Jacobi Forms ◽  
Taylor Coefficients ◽  
Special Polynomials ◽  
Generating Series ◽  
Block Variables ◽  
Vector Valued

AbstractHolomorphic vector valued differential operators acting on Siegel modular forms and preserving automorphy under the restriction to diagonal blocks are important in many respects, including application to critical values of L functions. Such differential operators are associated with vectors of new special polynomials of several variables defined by certain harmonic conditions. They include the classical Gegenbauer polynomial as a prototype, and are interesting as themselves independently of Siegel modular forms. We will give formulas for all such polynomials in two different ways. One is to describe them using polynomials characterized by monomials in off-diagonal block variables. We will give an explicit and practical algorithm to give the vectors of polynomials through these. The other one is rather theoretical but seems much deeper. We construct an explicit generating series of polynomials mutually related under certain mixed Laplacians. Here substituting the variables of the polynomials to partial derivatives, we obtain the generic differential operator from which any other differential operators of this sort are obtained by certain projections. This process exhausts all the differential operators in question. This is also generic in the sense that for any number of variables and block partitions, it is given by a recursive unified expression. As an application, we prove that the Taylor coefficients of Siegel modular forms with respect to off-diagonal block variables, or of corresponding expansion of Jacobi forms, are essentially vector valued Siegel modular forms of lower degrees, which are obtained as images of the differential operators given above. We also show that the original forms are recovered by the images of our operators. This is an ultimate generalization of Eichler–Zagier’s results on Jacobi forms of degree one. Several more explicit results and practical construction are also given.

scholarly journals Quasi-Jacobi forms, elliptic genera and strings in four dimensions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Seung-Joo Lee ◽  
Wolfgang Lerche ◽  
Guglielmo Lockhart ◽  
Timo Weigand
Keyword(s):  
Modular Forms ◽  
Jacobi Form ◽  
Elliptic Genus ◽  
Usual Sense ◽  
Jacobi Forms ◽  
Anomaly Cancellation ◽  
Four Dimensions ◽  
Holomorphic Anomaly ◽  
Witten Theory ◽  
Elliptic Genera

Abstract We investigate the interplay between the enumerative geometry of Calabi-Yau fourfolds with fluxes and the modularity of elliptic genera in four-dimensional string theories. We argue that certain contributions to the elliptic genus are given by derivatives of modular or quasi-modular forms, which may encode BPS invariants of Calabi-Yau or non-Calabi-Yau threefolds that are embedded in the given fourfold. As a result, the elliptic genus is only a quasi-Jacobi form, rather than a modular or quasi-modular one in the usual sense. This manifests itself as a holomorphic anomaly of the spectral flow symmetry, and in an elliptic holomorphic anomaly equation that maps between different flux sectors. We support our general considerations by a detailed study of examples, including non-critical strings in four dimensions.For the critical heterotic string, we explain how anomaly cancellation is restored due to the properties of the derivative sector. Essentially, while the modular sector of the elliptic genus takes care of anomaly cancellation involving the universal B-field, the quasi-Jacobi one accounts for additional B-fields that can be present.Thus once again, diverse mathematical ingredients, namely here the algebraic geometry of fourfolds, relative Gromow-Witten theory pertaining to flux backgrounds, and the modular properties of (quasi-)Jacobi forms, conspire in an intriguing manner precisely as required by stringy consistency.

scholarly journals Higher pullbacks of modular forms on orthogonal groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Brandon Williams
Keyword(s):  
Modular Forms ◽  
Differential Operators ◽  
Jacobi Form ◽  
Siegel Modular Forms ◽  
Orthogonal Groups

Abstract We apply differential operators to modular forms on orthogonal groups O ⁢ ( 2 , ℓ ) {\mathrm{O}(2,\ell)} to construct infinite families of modular forms on special cycles. These operators generalize the quasi-pullback. The subspaces of theta lifts are preserved; in particular, the higher pullbacks of the lift of a (lattice-index) Jacobi form ϕ are theta lifts of partial development coefficients of ϕ. For certain lattices of signature ( 2 , 2 ) {(2,2)} and ( 2 , 3 ) {(2,3)} , for which there are interpretations as Hilbert–Siegel modular forms, we observe that the higher pullbacks coincide with differential operators introduced by Cohen and Ibukiyama.

A note on the moments of the final size of the general epidemic model

1980 ◽  
Vol 17 (2) ◽  
pp. 532-538 ◽  
Author(s):  
Ross Dunstan
Keyword(s):  
Population Size ◽  
Epidemic Model ◽  
Asymptotic Series ◽  
Algebraic Methods ◽  
Series Expansions ◽  
Final Size

In the general epidemic model we study the first two moments of the final size. Beginning with the backwards equation, algebraic methods are used to find their asymptotic series expansions as the population size increases.

scholarly journals VERTEX OPERATOR ALGEBRAS AND WEAK JACOBI FORMS

2012 ◽  
Vol 23 (06) ◽  
pp. 1250024 ◽  
Author(s):  
MATTHEW KRAUEL ◽  
GEOFFREY MASON
Keyword(s):  
Vertex Operator ◽  
Vertex Operator Algebra ◽  
Operator Algebras ◽  
Congruence Subgroup ◽  
Modular Function ◽  
Jacobi Form ◽  
Jacobi Forms ◽  
Order Automorphism ◽  
Strongly Regular ◽  
Vector Valued

Let V be a strongly regular vertex operator algebra. For a state h ∈ V1 satisfying appropriate integrality conditions, we prove that the space spanned by the trace functions Tr M qL(0)-c/24ζh(0) (M a V-module) is a vector-valued weak Jacobi form of weight 0 and a certain index 〈h, h〉/2. We discuss refinements and applications of this result when V is holomorphic, in particular we prove that if g = eh(0) is a finite-order automorphism then Tr V qL(0)-c/24g is a modular function of weight 0 on a congruence subgroup of SL 2(ℤ).

scholarly journals CONGRUENCES AMONG POWER SERIES COEFFICIENTS OF MODULAR FORMS

2013 ◽  
Vol 09 (06) ◽  
pp. 1447-1474
Author(s):  
RICHARD MOY
Keyword(s):  
Power Series ◽  
Modular Forms ◽  
Series Expansions ◽  
Modular Functions ◽  
Apéry Numbers ◽  
Congruence Relations ◽  
Power Series Expansions

Many authors have investigated the congruence relations among the coefficients of power series expansions of modular forms f in modular functions t. In a recent paper, R. Osburn and B. Sahu examine several power series expansions and prove that the coefficients exhibit congruence relations similar to the congruences satisfied by the Apéry numbers associated with the irrationality of ζ(3). We show that many of the examples of Osburn and Sahu are members of infinite families.

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