Uniformization, accessory parameters and modular forms

Several topics related to modular forms and to the accessory parameter problem for the uniformization of hyperbolic Riemann surfaces are discussed. In the first part of the thesis we present an algorithm for the computation of the accessory parameters for the Fuchsian uniformization of certain punctured spheres. Then, via modular forms of rational weight, we show that the knowledge of the uniformizing differential equation leads to the complete knowledge of the ring of modular forms $M_*(Gamma)$ and of its Rankin-Cohen structure. In the second part of the thesis, a new operator $partial_ ho$ is defined on the space of quasimodular forms $widetilde{M}_*(Gamma)$ from an infinitesimal deformation of the uniformizing differential equation. It is shown that $partial_ ho$ can be described in terms of well-known derivations on $widetilde{M}_*(Gamma)$ and certain integrals of weight four-cusp forms; the relation between the operator $partial_ ho$ and a classical construction in Teichm"uller theory is discussed. The functions $partial_ ho{g},,ginwidetilde{M}_*(Gamma),$ motivate the study and the introduction of a new class of functions, called emph{extended modular forms}. Extended modular forms are defined as certain components of vector-valued modular forms with respect to symmetric tensor representations. Apart from the functions $partial_ ho{g},$ examples of extended modular forms are: Eichler integrals, more general iterated integrals of modular forms, and elliptic multiple zeta values.