AI 中文总结
本文利用Venkataramana的思想构造了不满足Singh-Venkataramana算术性判据的算术辛超几何群无限族，并证明特定多项式对在Sp(18m+4)中算术。
AI 中文摘要
利用Venkataramana构造算术正交超几何群无限族的思想，我们构造了不满足Singh和Venkataramana算术性判据的算术辛超几何群无限族的最早的一些例子。例如，我们证明了与多项式对$(x-1)^4P_m(x^{9})$和$(x^4+x^3+2x^2+x+1)Q_m(x^{9})$相关联的超几何群，其中$P_m$和$Q_m$是次数为$2m$的整系数多项式，且对应的对决定一个Zariski稠密的辛超几何群，对于任意整数$m\in\mathbb{N}$，在$\operatorname{Sp}(18m+4)$中是算术的。
英文摘要
Using the idea of Venkataramana's construction of infinite families of arithmetic orthogonal hypergeometric groups, we construct some of the very first examples of infinite families of arithmetic symplectic hypergeometric groups that do NOT satisfy the arithmeticity criterion of Singh and Venkataramana. For example, we show that the hypergeometric groups associated to the pairs of polynomials $(x-1)^4P_m(x^{9})$ and $(x^4+x^3+2x^2+x+1)Q_m(x^{9})$, where $P_m$ and $Q_m$ are integral polynomials of degree $2m$ such that the corresponding pair determines a Zariski dense symplectic hypergeometric group, are arithmetic in $\operatorname{Sp}(18m+4)$ for any integer $m\in\mathbb{N}$.
Comments12 pages. Comments are welcome
