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PreprintCC BY 4.0arXiv

Exceptional Sets for Certain 2F1 Hypergeometric Functions

  • Archisman Bhattacharjee
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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 ∈ ℚ̅ | 2F1(a,b,c;z) ∈ ℚ̅ }. 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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Cite

BibTeXRIS
BibTeX
@misc{bhattacharjee2026exceptional,
  title = {{Exceptional Sets for Certain $_2F_1$ Hypergeometric Functions}},
  author = {Archisman Bhattacharjee},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.16331},
  eprint = {2607.16331},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.16331},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Exceptional Sets for Certain _2F_1 Hypergeometric Functions
AU  - Archisman Bhattacharjee
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2607.16331
UR  - https://arxiv.org/abs/2607.16331
AB  - For a,b,c ∈ ℚ, the exceptional set associated to the Gauss hypergeometric function _2F_1(a,b,c;z) is defined by E(a,b,c) := { z ∈ ℚ̅ | _2F_1(a,b,c;z) ∈ ℚ̅ }. 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 Wüstholz, and classical result of Schneider on algebraic values of j-invariant of elliptic curves with complex multiplication.
ER  -