论文标题
有限字段的组表示Zeta Zeta函数
Weil zeta functions of group representations over finite fields
论文作者
论文摘要
在本文中，我们定义和研究Zeta函数$ζ_G$ - 类似于Hasse -Weil Zeta函数 - 该功能列举了（涂鸦）组$ g $的有限字段，列举了绝对不可约的表示。 Zeta函数在所有Uberg组的复杂半平面上收敛，并接受Euler产品分解。我们进行调查的动机是，在正整数$ k $上，相互价值$ζ_G（k）^{ - 1} $与$ k $随机元素产生完整的$ g $的组环的可能性相吻合。到目前为止获得的明确公式表明，$ζ_G$的行为相当良好。 本文的一个核心对象是$ζ_G$的收敛$ a（g）$。我们为免费的Abelian，免费的Abelian Pro-P $，免费Pro-P $，免费的pronilpotent和免费的Prosoluble群体计算了Ablcissae。更一般而言，我们获得了免费Pro-$ \ Mathfrak {C} $ groups的脓肿的界限（有时是显式值），其中$ \ Mathfrak {c} $是一类具有规定组成因子的有限组。我们证明，每个实际数字$ a \ geq 1 $都是某些涂鸦集团$ g $的Abscissa $ a（g）$。 此外，我们表明$ζ_G$的欧拉因子是$ p^{ - s} $中的有理功能，如果$ g $实际上是Abelian。对于有限组$ g $，我们使用$ g $的理性表示理论计算$ζ_G$。
In this article we define and study a zeta function $ζ_G$ - similar to the Hasse-Weil zeta function - which enumerates absolutely irreducible representations over finite fields of a (profinite) group $G$. The zeta function converges on a complex half-plane for all UBERG groups and admits an Euler product decomposition. Our motivation for this investigation is the observation that the reciprocal value $ζ_G(k)^{-1}$ at a positive integer $k$ coincides with the probability that $k$ random elements generate the completed group ring of $G$. The explicit formulas obtained so far suggest that $ζ_G$ is rather well-behaved. A central object of this article is the abscissa of convergence $a(G)$ of $ζ_G$. We calculate the abscissae for free abelian, free abelian pro-$p$, free pro-$p$, free pronilpotent and free prosoluble groups. More generally, we obtain bounds (and sometimes explicit values) for the abscissae of free pro-$\mathfrak{C}$ groups, where $\mathfrak{C}$ is a class of finite groups with prescribed composition factors. We prove that every real number $a \geq 1$ is the abscissa $a(G)$ of some profinite group $G$. In addition, we show that the Euler factors of $ζ_G$ are rational functions in $p^{-s}$ if $G$ is virtually abelian. For finite groups $G$ we calculate $ζ_G$ using the rational representation theory of $G$.
