Related papers: Functional Equation for Theta Series

In this paper we are interested in developments of elliptic functions of Jacobi. In particular a trigonometric expansion of the classical theta functions introduced by the author (Algebraic methods and q-special functions, Editors: C.R.M.…

We define Jacobi forms with complex multiplication. Analogous to modular forms with complex multiplication, they are constructed from Hecke characters of the associated imaginary quadratic field. From this construction we obtain a Jacobi…

We describe the construction of vector valued modular forms transforming under a given congruence representation of the modular group SL(2,Z) in terms of theta series. We apply this general setup to obtain closed and easily computable…

In this article, we obtain a transformation formula for the higher power of odd zeta values, which generalizes Ramanujan's formula for odd zeta values. We have also investigated many important applications, which in turn provide…

Liu established an addition formula for the Jacobian theta function by using the theory of elliptic functions. From this addition formula he obtained the Ramanujan cubic theta function identity, Winquist's identity, a theta function…

The formulas that relate Jacobi's Epsilon and Zeta function with real moduli in the interval (1,inf) or with pure imaginary moduli to elliptic functions with moduli in the interval [0,1] are derived.

In this paper, we generalize Rademacher's proof of the transformation law of the eta function to the Jacobi theta function using Residue calculus.

Using deformed or twisted Eisenstein Series, we construct a Jacobi-Serre derivative on even-weight Jacobi forms that generalizes the classical Serre derivative on modular forms. As an application, we obtain Ramanujan equations for the index…

We derive explicit formulas for the action of the Hecke operator $T(p)$ on the genus theta series of a positive definite integral quadratic form and prove a theorem on the generation of spaces of Eisenstein series by genus theta series. We…

We prove some differential equations for the Riemann theta function associated to the Jacobian of a Riemann surface. The proof is based on some variants of a formula by Fay for the theta function, which are motivated by their analogues in…

The Weil representation discovered by Andre Weil plays an important role in the study of the tranformation properties of theta series. In this paper, we define the Weil-Schroedinger representation of the Jacobi group and prove that the…

Let $X$ be an irreducible singular Riemann surface, with desingularisation $\widetilde X$. The generalised Jacobian $J(X)$ of $X$ fibers over the Jacobian $J(\widetilde{X})$ of $\widetilde X$, and there is an Abel map $A$ of $\widetilde X$…

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…

We study the reduction of Picard modular surfaces modulo an inert prime, mod p and p-adic modular forms. We construct a theta operator on such modular forms and study its poles and its effect on Fourier-Jacobi expansions.

A proper choice of parameters of the Jacobi modular identity (Jacobi Imaginary transformation) implies that the summation of Gaussian shifts on infinity periodic grids can be represented as the Jacobi's third Theta function. As such,…

The main step in numerical evaluation of classical Sl2 (Z) modular forms and elliptic functions is to compute the sum of the first N nonzero terms in the sparse q-series belonging to the Dedekind eta function or the Jacobi theta constants.…

Jacobi-Forms can be decomposed as a linear combination of Thetafunctions with modular forms as coefficients. It is shown that the space of these coefficient modular forms of Fourier-Jacobi-Forms, which come from Siegel cusp forms, has full…

In this paper we construct the quantum group, at roots of unity, of abelian Chern-Simons theory. We then use it to model classical theta functions and the actions of the Heisenberg and modular groups on them.

In this paper, we give some extensions for Ramanujan's circular summation formula with the mixed products of two Jacobi's theta functions. As some applications, we also obtain many interesting identities of Jacobi's theta functions.

In this paper we consider Jacobi forms of half-integral index for any positive definite lattice L (classical Jacobi forms from the book of Eichler and Zagier correspond to the lattice A_1=<2>). We give a lot of examples of Jacobi forms of…