On the modularity of solutions of certain differential equations of hypergeometric type

TL;DR

This paper investigates the modularity of solutions to hypergeometric differential equations, providing a number-theoretic explanation for when modularity occurs and confirming a conjecture about the completeness of modular solutions.

Contribution

It offers a number-theoretic framework explaining the modularity of hypergeometric solutions and proves the conjecture on the complete classification of modular solutions.

Findings

Number-theoretic explanation for modularity occurrence

Confirmation of the conjecture on the list of modular solutions

Identification of missing cases in the classification

Abstract

The purpose of this paper is to provide answers to some questions raised in a paper by Kaneko and Koike about the modularity of the solutions of a differential equations of hypergeometric type. In particular, we provide a number-theoretic explanation of why the modularity of the solutions occurs in some cases and does not occur in other cases. This also proves their conjecture on the completeness of the list of modular solutions after adding some missing cases.