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

Stasinski, Alexander; Zordan, Michele

Abstract

We prove that for any FAb compact p-adic analytic group G, its representation zeta function is a finite sum of terms n −s i fi(p −s), where ni are natural numbers and fi(t) ∈ Q(t) are rational functions. Meromorphic continuation and rationality of the abscissa of the zeta function follow as corollaries. If G is moreover a prop 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.

Citation

Stasinski, A., & Zordan, M. (in press). Rationality of representation zeta functions of compact p-adic analytic groups. American Journal of Mathematics,

• Journal Article Type: Article
• Acceptance Date: Dec 6, 2023
• Deposit Date: Jun 17, 2024
• Publicly Available Date: Aug 21, 2024
• Journal: American Journal of Mathematics
• Print ISSN: 0002-9327
• Electronic ISSN: 1080-6377
• Publisher: Johns Hopkins University Press
• Peer Reviewed: Peer Reviewed
• Public URL: https://durham-repository.worktribe.com/output/2484540
• Publisher URL: https://muse.jhu.edu/journal/5