Communications in Number Theory and Physics

Volume 13 (2019)

Number 3

Vector bundles and modular forms for Fuchsian groups of genus zero

Pages: 487 – 528



Luca Candelori (Department of Mathematics, Wayne State University, Detroit, Michigan, U.S.A.)

Cameron Franc (Department of Mathematics, University of Saskatchewan, Saskatoon, SK, Canada)


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.


modular forms, orbifold curves, graded modules, Fuchsian groups

2010 Mathematics Subject Classification

11F12, 11F23

Received 12 May 2017

Accepted 5 April 2019

Published 8 August 2022