Exceptional Sets for Certain ${}_2F_1$ Hypergeometric Functions

Archisman Bhattacharjee

Exceptional Sets for Certain ${}_2F_1$ Hypergeometric Functions

Number Theory 2026-07-15 v1

Authors: Archisman Bhattacharjee

Abstract

For $a,b,c \in \mathbb{Q}$ , the exceptional set associated to the Gauss hypergeometric function $_2F_1(a,b,c;z)$ is defined by $E(a,b,c) := \{ z \in \overline{\Q} \mid {}_2F_1(a,b,c;z) \in \overline{\mathbb{Q}} \}.$ In this paper, the exceptional sets $E(a,b,c)$ are determined explicitly for each $_2F_1(a,b,c;z)$ 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 W\"ustholz, and classical result of Schneider on algebraic values of $j$ -invariant of elliptic curves with complex multiplication.

Cite

@article{arxiv.2607.16331,
  title  = {Exceptional Sets for Certain ${}_2F_1$ Hypergeometric Functions},
  author = {Archisman Bhattacharjee},
  journal= {arXiv preprint arXiv:2607.16331},
  year   = {2026}
}