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Items from 1 to 20 out of 46 results

article

One-line formula for automorphic differential operators on Siegel modular forms

Tomoyoshi Ibukiyama

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg > 2019 > 89 > 1 > 17-43

We consider the Siegel upper half space $$H_{2m}$$ H 2 m of degree 2m and a subset $$H_m\times H_m$$ H m × H m of $$H_{2m}$$ H 2 m consisting of two $$m\times m$$ m × m diagonal block matrices. We consider two actions of $$Sp(m,{\mathbb R})\times Sp(m,{\mathbb R}) \subset Sp(2m,{\mathbb R})$$ S p ( m , R ) × S p ( m , R ) ⊂ S p ( 2 m , R ) , one is the...

article

Multivariable Appell functions and nonholomorphic Jacobi forms

Sander Zwegers

Research in the Mathematical Sciences > 2019 > 6 > 1 > 1-15

Multivariable Appell functions show up in the work of Kac and Wakimoto in the computation of character formulas for certain $$s \ell (m,1)^\wedge $$ sℓ(m,1)∧ modules. Bringmann and Ono showed that the character formulas for the $$s \ell (m,1)^\wedge $$ sℓ(m,1)∧ modules $$L(\varLambda _{(s)})$$ L(Λ(s)) , where $$L(\varLambda _{(s)})$$ L(Λ(s)) is the irreducible $$s \ell (m,1)^\wedge $$ sℓ(m,1)∧...

article

Functional equations of real analytic Jacobi Eisenstein series

Shin-ichiro Mizumoto

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg > 2019 > 89 > 1 > 55-75

We prove the existence of meromorphic continuation and the functional equation of the real analytic Jacobi Eisenstein series of degree m and matrix index T in case T is a kernel form.

article

On newforms of half-integral weight and Jacobi forms

Balesh Kumar, Manickam Murugesan

Journal of Number Theory > 2018 > 193 > C > 128-153

Let N be an odd and square-free integer. Let α be a positive integer with α=2 or α=5. Let χ modulo N be a Dirichlet character and let χ0=(4χ(−1).)χ. Let(a)χ and χ2 are primitive characters mod N, if α=2;(b)χ is the principal character if α=5. In this paper, we set up the theory of newforms for the space of cusp forms of weight k+1/2 for Γ0(2αN) with character χ0.Moreover, we prove that the space of...

article

Modular forms for the $$A_{1}$$ A1 -tower

Martin Woitalla

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg > 2018 > 88 > 2 > 297-316

In the 1960s Igusa determined the graded ring of Siegel modular forms of genus two. He used theta series to construct $$\chi _{5}$$ χ5 , the cusp form of lowest weight for the group $${\text {Sp}}(2,\mathbb {Z})$$ Sp(2,Z) . In 2010 Gritsenko found three towers of orthogonal type modular forms which are connected with certain series of root lattices. In this setting Siegel modular forms can be identified...

article

Poincaré square series for the Weil representation

Brandon Williams

The Ramanujan Journal > 2018 > 47 > 3 > 605-650

We calculate the Jacobi Eisenstein series of weight $$k \ge 3$$ k≥3 for a certain representation of the Jacobi group, and evaluate these at $$z = 0$$ z=0 to give coefficient formulas for a family of modular forms $$Q_{k,m,\beta }$$ Qk,m,β of weight $$k \ge 5/2$$ k≥5/2 for the (dual) Weil representation on an even lattice. The forms we construct have rational coefficients and contain all cusp forms...

article

Taylor coefficients of non-holomorphic Jacobi forms and applications

Kathrin Bringmann

Research in the Mathematical Sciences > 2018 > 5 > 1 > 1-16

In this paper, we prove modularity results of Taylor coefficients of certain non-holomorphic Jacobi forms. It is well known that Taylor coefficients of holomorphic Jacobi forms are quasimodular forms. However, recently there has been a wide interest in Taylor coefficients of non-holomorphic Jacobi forms, for example, arising in combinatorics. In this paper, we show that such coefficients still inherit...

article

Trace formula for Jacobi forms of odd squarefree level

Hiroshi Sakata

Journal of Number Theory > 2018 > 182 > C > 57-82

We study the traces of Hecke operators acting on the space of Jacobi forms and give the trace formula in the case of general odd squarefree level N by using results of Skoruppa and Zagier. As an application, we obtain some trace relations of them, and construct an isomorphism map from the space of all Jacobi cusp new forms of level N and index 1 having the eigenvalue +1 with respect to any Atkin–Lehner...

article

The Fourier coefficients of the McKay–Thompson series and the traces of CM values

Toshiki Matsusaka

Research in Number Theory > 2017 > 3 > 1 > 1-16

The elliptic modular function $$j(\tau )$$ j ( τ ) enjoys many beautiful properties. Its Fourier coefficients are related to the Monster group, and its CM values generate abelian extensions over imaginary quadratic fields. Kaneko gave an arithmetic formula for the Fourier coefficients expressed in terms of the traces of the CM values. In this article, we are concerned with analogues of...

article

Powers of Jacobi triple product, Cohen’s numbers and the Ramanujan $${\Delta }$$ Δ -function

Valery Gritsenko, Haowu Wang

European Journal of Mathematics > 2018 > 4 > 2 > 561-584

We show that the eighth power of the Jacobi triple product is a Jacobi–Eisenstein series of weight 4 and index 4 and we calculate its Fourier coefficients. As applications we obtain explicit formulas for the eighth powers of theta-constants of arbitrary order and the Fourier coefficients of the Ramanujan $${\Delta }$$ Δ -function $${\Delta }(\tau )=\eta ^{24}(\tau ),\eta ^{12}(\tau )$$ Δ(τ)=η24(τ),η12(τ)...

article

Jacobi forms and differential operators: Odd weights

Soumya Das, Ritwik Pal

Journal of Number Theory > 2017 > 179 > C > 113-125

We show that it is possible to remove two differential operators from the standard collection of m of them used to embed the space of Jacobi forms of odd weight k and index m into several pieces of elliptic modular forms. This complements the previous work of one of the authors in the case of even weights.

article

Characters of topological N=2 vertex algebras are Jacobi forms on the moduli space of elliptic supercurves

R. Heluani, J. Van Ekeren

Advances in Mathematics > 2016 > 302 > C > 551-627

We show that trace functions on modules of topological N=2 super vertex algebras give rise to conformal blocks on elliptic supercurves. We show that they satisfy a system of linear partial differential equations with respect to the modular parameters of the supercurves. Under some finiteness condition on the vertex algebra these differential equations can be interpreted as a connection on the vector...

article

W-algebras, higher rank false theta functions, and quantum dimensions

Kathrin Bringmann, Antun Milas

Selecta Mathematica > 2017 > 23 > 2 > 1249-1278

Motivated by appearances of Rogers’ false theta functions in the representation theory of the singlet vertex operator algebra, for each finite-dimensional simple Lie algebra of ADE type, we introduce higher rank false theta functions as characters of atypical modules of certain W-algebras and compute asymptotics of irreducible characters which allows us to determine quantum dimensions of the corresponding...

article

Quantum Jacobi forms and finite evaluations of unimodal rank generating functions

Kathrin Bringmann, Amanda Folsom

Archiv der Mathematik > 2016 > 107 > 4 > 367-378

In this paper, we introduce the notion of a quantum Jacobi form, and offer the two-variable combinatorial generating function for ranks of strongly unimodal sequences as an example. We then use its quantum Jacobi properties to establish a new, simpler expression for this function as a two-variable Laurent polynomial when evaluated at pairs of rational numbers. Our results also yield a new expression...

article

The Kohnen plus space for Hilbert–Siegel modular forms

Ren-He Su

Journal of Number Theory > 2016 > 163 > C > 267-295

The Kohnen plus space, roughly speaking, is a space consisting of modular forms of half integral weight with some property in Fourier coefficients. For example, the n-th coefficient of a normal modular form of weight k+1/2 in the plus space is 0 unless (−1)kn is congruent to some square modulo 4. The concept of plus space was initially introduced by Kohnen in 1980. Eichler and Zagier showed that the...

article

On p-adic modular forms and the Bloch–Okounkov theorem

Michael J. Griffin, Marie Jameson, Sarah Trebat-Leder

Research in the Mathematical Sciences > 2016 > 3 > 1 > 1-14

Bloch–Okounkov studied certain functions on partitions f called shifted symmetric polynomials. They showed that certain q-series arising from these functions (the so-called q-brackets $$\left<f\right>_q$$ f q ) are quasimodular forms. We revisit a family of such functions, denoted $$Q_k$$ Q k , and study the p-adic properties of their q-brackets. To do this, we define regularized...

article

Nonvanishing modulo ℓ of Fourier coefficients of Jacobi forms

Markus Schwagenscheidt

Research in Number Theory > 2016 > 2 > 1 > 1-18

Let ϕ = ∑ r 2 ≤ 4 mn c ( n , r ) q n ζ r $\phi = \sum _{r^{2} \leq 4mn}c(n,r)q^{n}\zeta ^{r}$ be a Jacobi form of weight k (with k>2 if ϕ is not a cusp form) and index m with integral algebraic coefficients which is an eigenfunction of all Hecke operators T _p,(p,m)=1, and which has at least one nonvanishing coefficient c(n ...

article

Kaneko–Zagier type equation for Jacobi forms of index 1

Tomoya Kiyuna

The Ramanujan Journal > 2016 > 39 > 2 > 347-362

We introduce a certain fourth-order partial differential equation containing one parameter $$k$$ k , related to Jacobi forms of index 1. We show several properties of the equation and give explicit Jacobi-form solutions for suitable $$k$$ k .

article

Rankin–Cohen brackets on Jacobi forms and the adjoint of some linear maps

Abhash Kumar Jha, Brundaban Sahu

The Ramanujan Journal > 2016 > 39 > 3 > 533-544

Given a fixed Jacobi cusp form, we consider a family of linear maps between the spaces of Jacobi cusp forms using the Rankin–Cohen brackets, and then we compute the adjoint maps of these linear maps with respect to the Petersson scalar product. The Fourier coefficients of the Jacobi cusp forms constructed using this method involve special values of certain Dirichlet series associated to Jacobi cusp...

article

Identities for Generalized Appell Functions and the Blow-up Formula

Kathrin Bringmann, Jan Manschot, Larry Rolen

Letters in Mathematical Physics > 2016 > 106 > 10 > 1379-1395

In this paper, we prove identities for a class of generalized Appell functions which are based on the $${{\rm A}_2}$$ A 2 root lattice. The identities are reminiscent of periodicity relations for the classical Appell function and are proven using only analytical properties of the functions. Moreover, they are a consequence of the blow-up formula for generating functions of invariants of...

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Keywords

JACOBI FORMS (46)
MODULAR FORMS (7)
DIFFERENTIAL OPERATORS (5)
SIEGEL MODULAR FORMS (5)
EISENSTEIN SERIES (3)
FOURIER COEFFICIENTS (2)
FOURIER COEFFICIENTS OF AUTOMORPHIC FORMS (2)
FUNCTIONAL EQUATIONS (2)
MOCK MODULAR FORMS (2)
RANKIN–COHEN BRACKETS (2)
SELF-DUAL CODES (2)
THETA FUNCTIONS (2)
VERTEX ALGEBRAS (2)
[MATHEMATICAL SUBJECT CODES] 11F55 (2)
[MATHEMATICAL SUBJECT CODES] 53B50 (2)
P-ADIC MODULAR FORMS (1)
Q-DIFFERENCE EQUATIONS (1)
ADJOINT MAP (1)
APPELL FUNCTIONS (1)
AUTOMORPHIC FORMS (1)
AUTOMORPHIC PRODUCTS (1)
AUTOMORPHIC REPRESENTATION (1)
BINARY CODES (1)
BLOW-UP FORMULA (1)
BLOWUP-FORMULAS (1)
BORCHERDS LIFTS (1)
CHARACTERS (1)
CHIRAL ALGEBRAS (1)
COMPLETE M-SPOTTY WEIGHT ENUMERATORS (1)
COMPLEX MULTIPLICATION (1)
CONGRUENCES FOR MODULAR FORMS (1)
CRANKS (1)
DIMENSION FORMULA (1)
DIRICHLET SERIES (1)
DONALDSON INVARIANTS (1)
ELLIPTIC CURVES (1)
ELLIPTIC FUNCTIONS (1)
ELLIPTIC MODULAR FUNCTION (1)
FALSE THETA FUNCTION (1)
FALSE THETA FUNCTIONS (1)
FOURIER EXPANSION (1)
FROBENIUS MANIFOLDS (1)
GENERALIZED APPELL FUNCTIONS (1)
HALF-INTEGRAL WEIGHT (1)
HALF-INTEGRAL WEIGHT MODULAR FORMS (1)
HARMONIC MAASS FORMS (1)
HILBERT MODULAR FORMS (1)
HILBERT-SIEGEL MODULAR FORMS OVER Q(5) (1)
HILBERT–SIEGEL MODULAR FORMS (1)
HYPERGEOMETRIC SUMS (1)
INDEFINITE QUADRATIC SPACES (1)
INDIVISIBILITY (1)
INTEGRAL LATTICES (1)
INVARIANT SPACE (1)
JACOBI EISENSTEIN SERIES (1)
JACOBI GROUP (1)
JACOBI GROUPS (1)
JACOBI-LIKE FORMS (1)
JOYCE INVARIANTS (1)
KOHNEN PLUS SPACE (1)
LOCAL DENSITY (1)
LOWERING OPERATOR (1)
MAASS FORMS (1)
MAASS RELATION (1)
MAASS SPEZIALSCHAR (1)
MACWILLIAMS IDENTITY (1)
MOCK JACOBI FORM (1)
MOCK THETA FUNCTION (1)
MODULAR CORRESPONDENCES (1)
MODULAR FORMS. (1)
MODULAR FUNCTIONS (1)
MODULAR INVARIANCE (1)
MOMENTS (1)
MOONSHINE (1)
MORDELL INTEGRAL (1)
NEWFORM THEORY (1)
NON-HOLOMORPHIC MODULAR FORMS (1)
NONHOLOMORPHIC MODULAR FORMS (1)
NONVANISHING (1)
ORTHOGONAL GROUPS (1)
PARAMODULAR FORMS (1)
PARTIAL EPSTEIN ZETA FUNCTIONS (1)
PARTIAL JACOBI THETA FUNCTION (1)
PARTIAL THETA FUNCTION (1)
PARTITION CRANKS (1)
PARTITION RANKS (1)
PERIODS (1)
PSEUDODIFFERENTIAL OPERATORS (1)
QUANTUM JACOBI FORMS (1)
QUANTUM MODULAR FORMS (1)
RAMANUJAN (1)
RANKS (1)
REPRESENTATION PROBLEMS (1)
RESIDUE CALCULUS (1)
RESTRICTION MAP (1)
SELBERG INTEGRALS (1)
SHADOWS (1)
SINGULAR MODULI (1)
SKEW HOLOMORPHIC JACOBI FORMS (1)
SPECIAL VALUES OF L-FUNCTIONS (1)
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Search results

One-line formula for automorphic differential operators on Siegel modular forms

Multivariable Appell functions and nonholomorphic Jacobi forms

Functional equations of real analytic Jacobi Eisenstein series

On newforms of half-integral weight and Jacobi forms

Modular forms for the $$A_{1}$$ A1 -tower

Poincaré square series for the Weil representation

Taylor coefficients of non-holomorphic Jacobi forms and applications

Trace formula for Jacobi forms of odd squarefree level

The Fourier coefficients of the McKay–Thompson series and the traces of CM values

Powers of Jacobi triple product, Cohen’s numbers and the Ramanujan $${\Delta }$$ Δ -function

Jacobi forms and differential operators: Odd weights

Characters of topological N=2 vertex algebras are Jacobi forms on the moduli space of elliptic supercurves

W-algebras, higher rank false theta functions, and quantum dimensions

Quantum Jacobi forms and finite evaluations of unimodal rank generating functions

The Kohnen plus space for Hilbert–Siegel modular forms

On p-adic modular forms and the Bloch–Okounkov theorem

Nonvanishing modulo ℓ of Fourier coefficients of Jacobi forms

Kaneko–Zagier type equation for Jacobi forms of index 1

Rankin–Cohen brackets on Jacobi forms and the adjoint of some linear maps

Identities for Generalized Appell Functions and the Blow-up Formula

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