There's a theorem by Denef (in:
"the rationality of the Poincaré series associated to the p-adic points on a variety") which states that a quite big class of local zeta functions (somehow defined within $\mathbb{Q}_p$) are rational functions. There also exists a version of the theorem describing how the zeta functions depend on $p$. This has in particular been applied to obtain results about various zeta functions associated to groups, counting e.g., subgroups of given indices or representations of given dimensions. The goal of this seminar is to understand these things. In particular, we want to understand the paper
by
Hrushovski-Martin-Rideau.
