Abstract:
This paper deals with generalized elliptic integrals and generalized modular functions. Several new inequalities are given for these and related functions.

Abstract:
The theory of multivariable Bessel functions is exploited to establish further links with the elliptic functions. The starting point of the present investigations is the Fourier expansion of the theta functions, which is used to derive an analogous expansion for the Jacobi functions (sn,dn,cn...) in terms of multivariable Bessel functions, which play the role of Fourier coefficients. An important by product of the analysis is an unexpected link with the elliptic modular functions.

Abstract:
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 product. 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.

Abstract:
Let $p\ge 5$ be a prime. We show that the space of weight one Eisenstein series defines an embedding into $\PP^{(p-3)/2}$ of the modular curve $X_1(p)$ for the congruence group $\Gamma_1(p)$ that is scheme-theoretically cut out by explicit quadratic equations.

Abstract:
We consider the space of elliptic hypergeometric functions of the sl_2 type associated with elliptic curves with one marked point. This space represents conformal blocks in the sl_2 WZW model of CFT. The modular group acts on this space. We give formulas for the matrices of the action in terms of values at roots of unity of Macdonald polynomials of the sl_2 type.

Abstract:
A concise review of the notions of elliptic functions, modular forms, and theta-functions is provided, devoting most of the paper to applications to Conformal Field Theory (CFT), introduced within the axiomatic framework of quantum field theory. Many features, believed to be peculiar to chiral 2D (= two dimensional) CFT, are shown to have a counterpart in any (even dimensional) globally conformal invariant quantum field theory. The treatment is based on a recently introduced higher dimensional extension of the concept of vertex algebra.

Abstract:
Global conformal invariance (GCI) of quantum field theory (QFT) in two and higher space-time dimensions implies the Huygens' principle, and hence, rationality of correlation functions of observable fields (see Commun. Math. Phys. 218 (2001) 417-436; hep-th/0009004). The conformal Hamiltonian $H$ has discrete spectrum assumed here to be finitely degenerate. We then prove that thermal expectation values of field products on compactified Minkowski space can be represented as finite linear combinations of basic (doubly periodic) elliptic functions in the conformal time variables (of periods 1 and $\tau$) whose coefficients are, in general, formal power series in $q^{1/2}=e^{i\pi\tau}$ involving spherical functions of the "space-like" fields' arguments. As a corollary, if the resulting expansions converge to meromorphic functions, then the finite temperature correlation functions are elliptic. Thermal 2-point functions of free fields are computed and shown to display these features. We also study modular transformation properties of Gibbs energy mean values with respect to the (complex) inverse temperature $\tau$ ($Im(\tau)=\beta/(2\pi)>0$). The results are used to obtain the thermodynamic limit of thermal energy densities and correlation functions.

Abstract:
We prove that each counting function of the m-simple branched covers with a fixed genus of an elliptic curve is expressed as a polynomial of the Eisenstein series E_2, E_4 and E_6 . The special case m=2 is considered by Dijkgraaf.

Abstract:
We consider properties of modular graph functions, which are non-holomorphic modular functions associated with the Feynman graphs for a conformal scalar field theory on a two-dimensional torus. Such functions arise, for example, in the low energy expansion of genus-one Type II superstring amplitudes. We demonstrate that these functions are sums, with rational coefficients, of special values of single-valued elliptic multiple polylogarithms, which will be introduced in this paper. This insight suggests the many interrelations between these modular graph functions (a few of which were motivated in an earlier paper) may be obtained as a consequence of identities involving elliptic polylogarithms.

Abstract:
It is well-known that every elliptic curve over the rationals admits a parametrization by means of modular functions. In this short note, we show that only finitely many elliptic curves over $\mathbf{Q}$ can be parametrized by modular units. This answers a question raised by Zudilin in a recent work on Mahler measures. Further, we give the list of all elliptic curves $E$ of conductor up to $1000$ parametrized by modular units supported in the rational torsion subgroup of $E$. Finally, we raise several open questions.