Arithmeticity of Some Hypergeometric Groups

TL;DR

This paper proves that certain hypergeometric groups, defined by specific parameter pairs, are arithmetic, contributing to the understanding of their algebraic and geometric properties.

Contribution

It establishes the arithmeticity of hypergeometric groups for new parameter sets, expanding knowledge of their structure and classification.

Findings

Hypergeometric groups for given parameters are arithmetic.

Identification of new classes of arithmetic hypergeometric groups.

Advances understanding of the algebraic properties of these groups.

Abstract

We show that the hypergeometric groups associated to the pairs of the parameters $\left(0,0,\frac{1}{3}, \frac{2}{3}\right)$, $\left(\frac{1}{2},\frac{1}{2},\frac{1}{4},\frac{3}{4}\right)$; and $\left(0,\frac{1}{12}, \frac{5}{12},\frac{7}{12},\frac{11}{12}\right), \left(\frac{1}{2},\frac{1}{3},\frac{1}{3},\frac{2}{3},\frac{2}{3}\right)$ are arithmetic.