Heun functions and diagonals of rational functions (unabridged version)

TL;DR

This paper explores diagonals of rational functions in three and four variables, revealing they are often squares of Heun functions linked to hypergeometric functions and modular forms, with some related to Shimura curves.

Contribution

It demonstrates that diagonals of certain rational functions can be expressed as squares of Heun functions, including those related to classical modular forms and Shimura curves, through creative telescoping.

Findings

Diagonals are squares of Heun functions.

Heun functions are related to hypergeometric functions and modular forms.

Some solutions correspond to Shimura curves.

Abstract

We provide a set of diagonals of simple rational functions of three and four variables that are squares of Heun functions. These Heun functions obtained through creative telescoping, turn out to be either pullbacked $_2F_1$ hypergeometric functions and in fact classical modular forms. We also obtain Heun functions that are Shimura curves as solutions of telescopers of rational functions.