Rationality of representation zeta functions of compact $p$-adic analytic groups

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.