论文标题
六学位的符号超测
Symplectic Hypergeometric Groups of Degree Six
论文作者
论文摘要
我们的计算表明，总计$ 40 $的六级副本多项式$ f，g $其中$ f（x）=（x-1）^6 $，$ g $是循环多项式的产品，$ g（0）= 1 $和$ f，$ f，$ f，g $ g $形式a primitive a Primitive a Primitive a Primitive a Primitive a Primitive a。本文的目的是确定具有最大单位单作的相应$ 40 $符号超测量组是否遵循算术和薄度之间的相同二分法，而$ 14 $ simplectic的超角度组则符合$ 14 $ f，g $ f，$ f，$ f（$ f（$ f（x）$ f（x）= $ f（x）=（x）=（x）=（x）=（x-x）=（x-x）=（x-x）=（x-x）=（x-x）=（x-x）=（x-x）=（x）=（x）^4，结果，我们证明，这些$ 40 $的$ \ mathrm {sp}（6）$中至少至少$ 18 $是算术。 此外，我们将搜索扩展到所有程度的六个符号超测量组。我们发现，与此类组相对应的多项式$ 458 $对多项式（最多到标量偏移）。以$ 211美元的价格，多项式$ f-g $的领先系数的绝对值最多为$ 2 $，相应组的算术性来自Singh和Venkataramana，而另外一个超地球仪的算术则来自Detinko，Flannery和Hulpke。 在本文中，我们显示剩余$ 246 $超几何组的算术性为$ 160 $。
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.
