论文标题
$ \ mathbb {f} _p $ -lie代数和有限$ p $ -groups的Zeta函数
Zeta functions of $\mathbb{F}_p$-Lie algebras and finite $p$-groups
论文作者
论文摘要
我们研究Zeta函数列举亚代词或lie代数的理想，这是Prime Order $ \ Mathbb {f} _p $的有限字段。我们首先开发了一种通用蓝图方法，用于计算$ \ mathbb {f} _p $ -lie代数的Zeta函数，并详细说明其实际应用，以获取有关任何已知文献中未涵盖的各种有趣的新示例的明确公式。对于Nilpotent情况，这还提供了计数Zeta功能，计算子组和有限的$ P $ - 组$ P $ P $的正常子组，几乎所有PURES都通过Lazard通信。我们研究了它们与有限$ p $ groups的研究的联系，并讨论可以从这些有限的dirichlet多项式中推论的内容。
We study zeta functions enumerating subalgebras or ideals of Lie algebras over finite field of prime order $\mathbb{F}_p$. We first develop a general blueprint method for computing zeta functions of $\mathbb{F}_p$-Lie algebras, and demonstrate its practical applications in detail to obtain explicit formulas for various interesting new examples that are not covered in any known literature yet. For nilpotent cases this also provides zeta functions counting subgroups and normal subgroups of finite $p$-groups of exponent $p$ for almost all primes via the Lazard correspondence. We investigate their connections to the study of finite $p$-groups, and discuss what can be deduced from these finite Dirichlet polynomials.
