Abstract

This thesis consists of two parts. In the first part, we present a positive characteristic analogue of Shimura's theorem on the special values of modular forms at CM points. More precisely, we show using Hayes' theory of Drinfeld modules that the special value at a CM point of an arithmetic Drinfeld modular form of
arbitrary rank lies in the Hilbert class field of the CM field up to a period, independent of the chosen modular form.This is achieved via Pink's realization of Drinfeld modular forms as sections of a sheaf over the compactified Drinfeld modular curve.

In the second part of the thesis, we present various computational and algorithmic aspects both for the classical theory (over C) and function field theory. First, we implement the rings of quasimodular forms in SageMath and give some applications such as the symbolic calculation of the derivative of a classical modular form. Second, we explain how to compute objects associated with a Drinfeld modules such as the exponential, the logarithm, and Potemine's set of basic J-invariants. Lastly, we present a SageMath package for computing with Drinfeld modular forms and their expansion at infinity using the nonstandard A-expansion theory of López and Petrov.