# Vector bundles and modular forms for Fuchsian groups of genus zero

## TL;DR

This paper develops a geometric framework for studying modular forms associated with genus zero Fuchsian groups, revealing conditions for module freeness and constructing indecomposable vector bundles for complex cases.

## Contribution

It introduces a geometric approach to modular forms via vector bundles on orbifold curves, establishing freeness criteria and explicit constructions for indecomposable bundles.

## Key findings

- Modules are free for groups with up to two elliptic points.
- Explicit indecomposable vector bundles are constructed for groups with three or more elliptic points.
- The structure of modular forms is linked to vector bundle properties over orbifold curves.

## Abstract

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.