论文标题
代表性的合理性zeta函数紧凑型$ p $ -ADIC分析组
Rationality of representation zeta functions of compact $p$-adic analytic groups
论文作者
论文摘要
我们证明，对于任何Fab Compact $ p $ -Adic Analytic $ g $，其表示Zeta Zeta函数是$ n_ {i}^{ - s} f_ {i} f_ {i}（p^{ - s}）$的有限总和Zeta功能的横坐标的Meromorthic延续和合理性随着推论。如果$ g $也是Pro-P $组，我们证明其表示Zeta函数在$ p^{ - s} $中是合理的。 Jaikin-Zapirain证明了这些结果，分别以$ P> 2 $或$ G $制服和Pro-2 $的价格证明。我们提供了一个新的证明，该证明避免了Kirillov轨道方法，并适用于所有$ P $。 Arxiv的第一部分：2007.10694，第二部分上传为单独的纸。
We prove that for any FAb compact $p$-adic analytic group $G$, its representation zeta function is a finite sum of terms $n_{i}^{-s}f_{i}(p^{-s})$, where $n_{i}$ are natural numbers and $f_{i}(t)\in\mathbb{Q}(t)$ are rational functions. Meromorphic continuation and rationality of the abscissa of the zeta function follow as corollaries. If $G$ is moreover a pro-$p$ group, we prove that its representation zeta function is rational in $p^{-s}$. These results were proved by Jaikin-Zapirain for $p>2$ or for $G$ uniform and pro-$2$, respectively. We give a new proof which avoids the Kirillov orbit method and works for all $p$. First part of arXiv:2007.10694, second part uploaded as a separate paper.
