某些${}_2F_1$超几何函数的例外集

中文摘要 AI 辅助

对于\(a,b,c\in\mathbb{Q}\)，与高斯超几何函数\({}_2F_1(a,b,c;z)\)相关的例外集\(E(a,b,c)\)定义为\(E(a,b,c):=\{z\in\overline{\mathbb{Q}}\mid{}_2F_1(a,b,c;z)\in\overline{\mathbb{Q}}\}\)。本文针对单值群为竹内I类算术三角群的每个\({}_2F_1(a,b,c;z)\)，明确确定了例外集\(E(a,b,c)\)。通过超几何 - 模恒等式以及Wüstholz关于阿贝尔簇周期的超越性结果，还有Schneider关于具有复乘法的椭圆曲线\(j\)不变量代数值的经典结果得到了该描述。

英文摘要

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 $E(a,b,c) := \{ z \in \overline{\mathbb{Q}} \mid {}_2F_1(a,b,c;z) \in \overline{\mathbb{Q}} \}.$ 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 Wüstholz, and classical result of Schneider on algebraic values of $j$-invariant of elliptic curves with complex multiplication.

发表机构

• Louisiana State University（路易斯安那州立大学）

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