Dihedral evaluations of hypergeometric functions with the Kleinian projective monodromy

TL;DR

This paper investigates specific hypergeometric functions with complex monodromy groups and demonstrates how pull-back transformations simplify their monodromy to dihedral groups, aiding in their algebraic evaluation.

Contribution

It introduces a method to reduce the projective monodromy group of certain hypergeometric functions to a dihedral group via degree 21 pull-back transformations.

Findings

Reduced monodromy from PSL(2,F7) to D4

Explicit transformations for 3F2-functions

Enhanced understanding of hypergeometric function monodromy

Abstract

Algebraic hypergeometric functions can be compactly expressed as radical or dihedral functions on pull-back curves where the monodromy group is much simpler. This article considers the classical 3F2-functions with the projective monodromy group PSL(2,F7) and their pull-back transformations of degree 21 that reduce the projective monodromy to the dihedral group D4 of 8 elements.