Eulerian series as modular forms

In 1988, Hickerson proved the celebrated ``mock theta conjectures' in a collection of ten identities from Ramanujan's ``lost notebook' which express certain modular forms as linear combinations of mock theta functions. In the context of Maass forms, these identities arise from the peculiar phenomenon that two different harmonic Maass forms may have the same non-holomorphic parts. Using this perspective, we construct several infinite families of modular forms which are differences of mock theta functions.

     

Theta Functions with Harmonic Coefficients over Number Fields

We investigate theta functions attached to quadratic forms over a number field K. We establish a functional equation by regarding the theta functions as specializations of symplectic theta functions. By applying a differential operator to the functional equation, we show how theta functions with harmonic coefficients over K behave under modular transformations.     

Jacobi Theta Functions over Number Fields

We use Jacobi theta functions to construct examples of Jacobi forms over number fields. We determine the behavior under modular transformations by regarding certain coefficients of the Jacobi theta functions as specializations of symplectic theta functions. In addition, we show how sums of those Jacobi theta functions appear as a single coefficient of a symplectic theta function.     

On theta functions associated to indefinite quadratic forms

We introduce a family of theta functions associated to an indefinite quadratic form, and prove a modular transformation formulas by regarding each such function as a specialization of a symplectic theta function. An eighth rott of unity arises in these formulas, and it is expressly given in all cases. The theta functions feature many “translation variables,” which are useful for the study of the liftings of modular forms.     

Shimura lifts of half-integral weight modular forms arising from theta functions

In 1973, Shimura (Ann. Math. (2) 97:440–481, 1973) introduced a family of correspondences between modular forms of half-integral weight and modular forms of even integral weight. Earlier, in unpublished work, Selberg explicitly computed a simple case of this correspondence pertaining to those half-integral weight forms which are products of Jacobi’s theta function and level one Hecke eigenforms. Cipra (J. Number Theory 32(1):58–64, 1989) generalized Selberg’s work to cover the Shimura lifts where the Jacobi theta function may be replaced by theta functions attached to Dirichlet characters of prime power modulus, and where the level one Hecke eigenforms are replaced by more generic newforms. Here we generalize Cipra’s results further to cover theta functions of arbitrary Dirichlet characters multiplied by Hecke eigenforms.      

Tenth order mock theta functions in Ramanujan's lost notebook (IV)

Ramanujan's lost notebook contains many results on mock theta functions. In particular, the lost notebook contains eight identities for tenth order mock theta functions. Previously the author proved the first six of Ramanujan's tenth order mock theta function identities. It is the purpose of this paper to prove the seventh and eighth identities of Ramanujan's tenth order mock theta function identities which are expressed by mock theta functions and a definite integral. L. J. Mordell's transformation formula for the definite integral plays a key role in the proofs of these identities. Also, the properties of modular forms are used for the proofs of theta function identities.

     

Partial theta functions and mock modular forms as q-hypergeometric series

Ramanujan studied the analytic properties of many q-hypergeometric series. Of those, mock theta functions have been particularly intriguing, and by work of Zwegers, we now know how these curious q-series fit into the theory of automorphic forms. The analytic theory of partial theta functions however, which have q-expansions resembling modular theta functions, is not well understood. Here we consider families of q-hypergeometric series which converge in two disjoint domains. In one domain, we show that these series are often equal to one another, and define mock theta functions, including the classical mock theta functions of Ramanujan, as well as certain combinatorial generating functions, as special cases. In the other domain, we prove that these series are typically not equal to one another, but instead are related by partial theta functions.     

Tenth order mock theta functions in Ramanujan's lost notebook III

Ramanujan's lost notebook contains many results on mock thetafunctions. In particular, the lost notebook contains eight identitiesfor tenth order mock theta functions. Previously, the authorproved six of the eight tenth order mock theta function identities.It is the purpose of this paper to prove the fifth and sixthidentities of Ramanujan's tenth order mock theta functions.The properties of modular forms are used for the proofs of thetafunction identities.     

Differential equations for septic theta functions

We demonstrate that quotients of septic theta functions appearing in Ramanujan’s Notebooks and in Klein’s work satisfy a new coupled system of nonlinear differential equations with symmetric form. This differential system bears a close resemblance to an analogous system for quintic theta functions. The proof extends an elementary technique used by Ramanujan to prove the classical differential system for normalized Eisenstein series on the full modular group. In the course of our work, we show that Klein’s quartic relation induces symmetric representations for low-weight Eisenstein series in terms of weight one modular forms of level seven.     

Isogeny formulas for the Picard modular form and a three terms arithmetic geometric mean

In this paper we study the Picard modular forms and show a new three terms arithmetic geometric mean (AGM) system. This AGM system is expressed via the Appell hypergeometric function of two variables. The Picard modular forms are expressed via the theta constants, and they give the modular function for the family of Picard curves. Our theta constants are “Neben type” modular forms of weight 1 defined on the complex 2-dimensional hyperball with respect to an index finite subgroup of the Picard modular group. We define a simultaneous 3-isogeny for the family of Jacobian varieties of Picard curves. Our main theorem shows the explicit relations between two systems of theta constants which are corresponding to isogenous Jacobian varieties. This relation induces a new three terms AGM which is a generalization of Borweins' cubic AGM.     

Ramanujan-type partial theta identities and conjugate Bailey pairs,II. Multisums

In the first paper of this series, we described how to find conjugate Bailey pairs from residual identities of Ramanujan-type partial theta identities. Here we carry this out for four multisum residual identities of Warnaar and two more due to the authors. Applying known Bailey pairs gives expressions in the algebra of modular forms and indefinite theta functions.     

Mock and mixed mock modular forms in the lower half-plane

We study mock and mixed mock modular forms in the lower half-plane. In particular, our results apply to Zwegers’ three-variable mock Jacobi form \({\mu(u,v;\tau)}\), three-variable generalizations of the universal mock modular partition rank generating function, and the quantum and mock modular strongly unimodal sequence rank generating function. We do not rely upon the analytic properties of these functions; we establish our results concisely using the theory of q-hypergeometric series and partial theta functions. We extend related results of Ramanujan, Hikami, and prior work of the author with Bringmann and Rhoades, and also incorporate more recent aspects of the theory pertaining to quantum modular forms and the behavior of these functions at rational numbers when viewed as functions of \({\tau}\) (or equivalently, at roots of unity when viewed as functions of q).     

Linear relations among Poincaré series via harmonic weak Maass forms

We discuss the problem of the vanishing of Poincaré series. This problem is known to be related to the existence of weakly holomorphic forms with prescribed principal part. The obstruction to the existence is related to the pseudomodularity of Ramanujan??s mock theta functions. We embed the space of weakly holomorphic modular forms into the larger space of harmonic weak Maass forms. From this perspective we discuss the linear relations between Poincaré series.     

Partitions with initial repetitions

A variety of interesting connections with modular forms, mock theta functions and Rogers- Ramanujan type identities arise in consideration of partitions in which the smaller integers are repeated as summands more often than the larger summands. In particular, this concept leads to new interpretations of the Rogers Selberg identities and Bailey＇s modulus 9 identities.     

Congruences for Ramanujan’s f and ω functions via generalized Borcherds products

Bruinier and Ono recently developed the theory of generalized Borcherds products, which uses coefficients of certain Maass forms as exponents in infinite product expansions of meromorphic modular forms. Using this, one can use classical results on congruences of modular forms to obtain congruences for Maass forms. In this note we work out the example of Ramanujan’s mock theta functions f and ω in detail.     

The modular properties and the integral representations of the multiple elliptic gamma functions

We show the modular properties of the multiple “elliptic” gamma functions, which are an extension of those of the theta function and the elliptic gamma function. The modular property of the theta function is known as Jacobi's transformation, and that of the elliptic gamma function was provided by Felder and Varchenko. In this paper, we deal with the multiple sine functions, since the modular properties of the multiple elliptic gamma functions result from the equivalence between two ways to represent the multiple sine functions as infinite products.We also derive integral representations of the multiple sine functions and the multiple elliptic gamma functions. We introduce correspondences between the multiple elliptic gamma functions and the multiple sine functions.     

A Dimensionally Continued Poisson Summation Formula

We generalize the standard Poisson summation formula for lattices so that it operates on the level of theta series, allowing us to introduce noninteger dimension parameters (using the dimensionally continued Fourier transform). When combined with one of the proofs of the Jacobi imaginary transformation of theta functions that does not use the Poisson summation formula, our proof of this generalized Poisson summation formula also provides a new proof of the standard Poisson summation formula for dimensions greater than 2 (with appropriate hypotheses on the function being summed). In general, our methods work to establish the (Voronoi) summation formulae associated with functions satisfying (modular) transformations of the Jacobi imaginary type by means of a density argument (as opposed to the usual Mellin transform approach). In particular, we construct a family of generalized theta series from Jacobi theta functions from which these summation formulae can be obtained. This family contains several families of modular forms, but is significantly more general than any of them. Our result also relaxes several of the hypotheses in the standard statements of these summation formulae. The density result we prove for Gaussians in the Schwartz space may be of independent interest.     

An infinite family of summation identities

Theta functions have historically played a prominent role in number theory. One such role is the construction of modular forms. In this work, a generalized theta function is used to construct an infinite family of summation identities. Our results grew out of some observations noted during a presentation given by the author at the 1992 AMS-MAA-SIAM Joint Meetings in Baltimore.

     

Quaternionic theta constants

We investigate the six quaternionic theta constants introduced by Freitag and Hermann. More precisely we investigate their restrictions to the Hermitian resp. Siegel half-space of degree 2. It turns out that these theta constants generate the graded ring of symmetric Hermitian modular forms for the principal congruence subgroup of level 1 + i over the Gaussian number field resp. of Siegel modular forms for the principal congruence subgroup of level 2 and even weight. As an application we obtain a simple construction of Igusa’s Siegel modular form of degree 2 and weight 30 with respect to the non-trivial character.     

Theta series and trace operators for Г<Subscript>n</Subscript>[q]

We consider the action of suitable trace operators on non homogeneous theta series that are Siegel modular forms for the principal congruence subgroups of the symplectic group of odd levelq: Г _n [q]. This is used for investigating whether modular forms forГ _n [N], withN|q, which are linear combination of such theta series, can be expressed as combination of theta series that are modular forms with respect toГ _n [N].