Eichler integrals, period relations and Jacobi forms

This paper contains three main results: the first one is to derive two “period relations” and the second one is a complete characterization of period functions of Jacobi forms in terms of period relations. These are done by introducing a concept of “Jacobi integrals” on the full Jacobi group. The last one is to show, for the given holomorphic function P(τ, z) having two period relations, there exists a unique Jacobi integral, up to Jacobi forms, with a given function P(τ, z) as its period function. This is done by constructing a generalized Jacobi Poincaré series explicitly. This is to say that every holomorphic function with “period relations” is coming from a Jacobi integral. It is an analogy of Eichler cohomology theory studied in Knopp (Bull Am Math Soc 80:607–632, 1974) for the functions with elliptic and modular variables. It explains the functional equations satisfied by the “Mordell integrals” associated with the Lerch sums (Zwegers in Mock theta functions, PhD thesis, Universiteit Utrecht, 2002) or, more generally, with the higher Appell functions (Semikhatov et?al. in Commun Math Phys 255(2):469–512, 2005). Developing theories of Jacobi integrals with elliptic and modular variables in this paper is a natural extension of the Eichler integral with modular variable. Period functions can be explained in terms of the parabolic cohomology group as well.     

Parabolic cohomology and multiple Hecke L-values

The Ramanujan Journal - We derive various identities among the special values of multiple Hecke L-series. We show that linear combinations of multiple Hecke L-values can be expressed as linear...     

A short remark on sign changes of Fourier coefficients of half-integral weight modular forms

We give an upper bound on the first sign change of the Fourier coefficients of a cusp form of half-integral weight.     

Linear relations of theta series of genera of quadratic forms

In this paper we study linear relations among theta series of genera of positive definite n-ary quadratic forms with given level D,2D,4D and 8D for square free D. We obtain a basis for the space generated by genus theta series. This forms a basis of Eisenstein space.     

On the Petersson Norm of Certain Siegel Modular Forms

We shall prove a rationality result for a quotient of scalar products involving the Ikeda lift of an elliptic cusp form.     

Half integral weight Jacobi forms and periods¶of modular forms

By deriving Bol's type result, we show how to construct a Jacobi form with different weight from the given Jacobi form. We also show how this analogous theory related to the classical theory of Bol[1] by using the theta-series expansion of the Jacobi form. Furthermore, the partial converse of Bol's result involving the periods of modular forms reveals a connection between Jacobi forms and periods of modular forms. Received: 28 February 2000 / Revised version: 24 October 2000     

Weight-dependent congruence properties of modular forms

In this paper, we study congruence properties of modular forms in various ways. By proving a weight-dependent congruence property of modular forms, we give some sufficient conditions, in terms of the weights of modular forms, for a modular form to be non-p-ordinary. As applications of our main theorem we derive a linear relation among coefficients of new forms. Furthermore, congruence relations among special values of Dedekind zeta functions of real quadratic fields are derived.     

The joint weight enumerators and Siegel modular forms

The weight enumerator of a binary doubly even self-dual code is an isobaric polynomial in the two generators of the ring of invariants of a certain group of order 192. The aim of this note is to study the ring of coefficients of that polynomial, both for standard and joint weight enumerators.