Exceptional Sets for Certain ₂F₁ Hypergeometric Functions

2026/07/15 by Archisman Bhattacharjee
Mathematics · #math.NT

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Abstract

For a,b,c ∈ ℚ, the exceptional set associated to the Gauss hypergeometric function ₂F₁(a,b,c;z) is defined by E(a,b,c) := \ z ∈ ℚ | ₂F₁(a,b,c;z) ∈ ℚ \. In this paper, the exceptional sets E(a,b,c) are determined explicitly for each ₂F₁(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.

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