Holomorphic orbispheres in elliptic curve quotients and Diophantine equations

Abstract

We compute quantum cohomology rings of elliptic ${\mathbb{P}}^1$ orbifolds via orbicurve counting. The main technique is the classification theorem which relates holomorphic orbicurves with certain orbifold coverings. The countings of orbicurves are related to the integer solutions of Diophantine equations. This reproduces the computation of Satake and Takahashi in the case of ${\mathbb{P}}_{3,3,3}^1$ via a different method.