Exceptional Sets for Certain ${}_2{F}_1$ Hypergeometric Functions

Abstract

For $a,b,c\in \mathbb{Q}$, the exceptional set associated to the Gauss hypergeometric function ${}_2{F}_1(a,b,c;z)$ is defined by $E(a,b,c):=\{z\in \overline{\mathbb{Q}}\mid {}_2{F}_1(a,b,c;z)\in \overline{\mathbb{Q}}\}.$ In this paper, the exceptional sets $E(a,b,c)$ are determined explicitly for each ${}_2{F}_1(a,b,c;z)$ whose monodromy group is an arithmetic triangle group in Takeuchi's class I. The description is obtained via hypergeometric-modular identities together with transcendence results for periods of abelian varieties due to Wüstholz, and classical result of Schneider on algebraic values of $j$-invariant of elliptic curves with complex multiplication.