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ABSTRACT. This paper presents the theory of holomorphic vector valued modular forms from a geometric perspective. More precisely, we define certain holomorphic vector bundles on the modular orbifold of generalized elliptic curves whose sections are vector valued modular forms. This perspective simplifies the theory, and it clarifies the role that exponents of representations of SL 2 (Z) play in the holomorphic theory of vector valued modular forms. Further, it allows one to use standard techniques in algebraic geometry to deduce free-module theorems and dimension formulae (deduced previously by other authors using different techniques), by identifying the modular orbifold with the weighted projective line P(4, 6).

Abstract. We prove the following Theorem. Suppose that F = ( f 1 , f 2 ) is a 2-dimensional vector-valued modular form on SL 2 (Z) whose component functions f 1 , f 2 have rational Fourier coefficients with bounded denominators. Then f 1 and f 2 are classical modular forms on a congruence subgroup of the modular group.

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We describe algorithms that allow the computation of fundamental domains in the Bruhat-Tits tree for the action of discrete groups arising from quaternion algebras. These algorithms are used to compute spaces of rigid modular forms of arbitrary even weight, and we explain how to evaluate such forms to high precision using overconvergent methods. Finally, these algorithms are applied to the calculation of conjectural equations for the canonical embedding of p-adically uniformizable rational Shimura curves. We conclude with an example in the case of a genus 4 Shimura curve.