  This article lays the foundations for the study of modular forms transforming
with respect to representations of Fuchsian groups of genus zero. More
precisely, we define geometrically weighted graded modules of such modular
forms, where the graded structure comes from twisting with all isomorphism
classes of line bundles on the corresponding compactified modular curve, and we
study their structure by relating it to the structure of vector bundles over
orbifold curves of genus zero. We prove that these modules are free whenever
the Fuchsian group has at most two elliptic points. For three or more elliptic
points, we give explicit constructions of indecomposable vector bundles of rank
two over modular orbifold curves, which give rise to non-free modules of
geometrically weighted modular forms.
