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

Abstract

We prove that for any twist rigid compact p p -adic analytic group G G , its twist representation zeta function is a finite sum of terms n i − s f i ( p − s ) n_{i}^{-s}f_{i}(p^{-s}) , where n i n_{i} are natural numbers and f i ( t ) ∈ Q ( t ) 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 G is moreover a pro- p p group, we prove that its twist representation zeta function is rational in p − s p^{-s} . To establish these results we develop a Clifford theory for twist isoclasses of representations, including a new cohomological invariant of a twist isoclass.