六次辛超几何群
Symplectic Hypergeometric Groups of Degree Six
AI总结：
本文研究了六次辛超几何群的算术性，通过计算多项式对并扩展搜索范围，证明了在具有极大幂单单值群的40个群中至少18个、以及剩余246个群中160个群在Sp(6)中是算术的。
摘要：
我们的计算表明，共有 $40$ 对六次互质多项式 $f,g$，其中 $f(x)=(x-1)^6$，$g$ 是分圆多项式的乘积，$g(0)=1$，且 $f,g$ 构成本原对。本文旨在确定与这 $40$ 对多项式对应的、具有极大幂单单值群的辛超几何群，是否遵循与 $14$ 个辛超几何群相同的算术性与薄性二分法，后者对应于满足上述条件的四次多项式对 $f,g$，其中 $f(x)=(x-1)^4$。结果我们证明了这 $40$ 个群中至少有 $18$ 个在 $\mathrm{Sp}(6)$ 中是算术的。此外，我们将搜索范围扩展到所有六次辛超几何群。我们发现共有 $458$ 对多项式（至多相差标量移位）对应于此类群。对于其中 $211$ 对，差值多项式 $f-g$ 的首项系数绝对值至多为 $2$，相应群的算术性由 Singh 和 Venkataramana 的结论得出，而另一个超几何群的算术性由 Detinko、Flannery 和 Hulpke 的结论得出。在本文中，我们证明了剩余 $246$ 个超几何群中 $160$ 个群的算术性。
英文摘要：
Our computations show that there is a total of $40$ pairs of degree six coprime polynomials $f,g$ where $f(x)=(x-1)^6$, $g$ is a product of cyclotomic polynomials, $g(0)=1$ and $f,g$ form a primitive pair. The aim of this article is to determine whether the corresponding $40$ symplectic hypergeometric groups with a maximally unipotent monodromy follow the same dichotomy between arithmeticity and thinness that holds for the $14$ symplectic hypergeometric groups corresponding to the pairs of degree four polynomials $f,g$ where $f(x)=(x-1)^4$ and $g$ is as described above. As a result we prove that at least $18$ of these $40$ groups are arithmetic in $\mathrm{Sp}(6)$. In addition, we extend our search to all degree six symplectic hypergeometric groups. We find that there is a total of $458$ pairs of polynomials (up to scalar shifts) corresponding to such groups. For $211$ of them, the absolute values of the leading coefficients of the difference polynomials $f-g$ are at most $2$ and the arithmeticity of the corresponding groups follows from Singh and Venkataramana, while the arithmeticity of one more hypergeometric group follows from Detinko, Flannery and Hulpke. In this article, we show the arithmeticity of $160$ of the remaining $246$ hypergeometric groups.
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