Heun equations coming from geometry

TL;DR

This paper classifies Heun equations arising from geometric contexts, specifically from algebraic varieties and elliptic curves, linking them to Picard-Fuchs equations and hypergeometric functions.

Contribution

It provides a classification of Heun equations associated with algebraic varieties and shows their connection to hypergeometric functions via Belyi pullbacks.

Findings

List of Heun equations from algebraic varieties

Connection between Heun equations and hypergeometric functions

Classification based on elliptic curve families

Abstract

We give a list of Heun equations which are Picard-Fuchs associated to families of algebraic varieties. Our list is based on the classification of families of elliptic curves with four singular fibers done by Herfurtner. We also show that pullbacks of hypergeometric functions by rational Belyi functions with restricted ramification data give rise to Heun equations.