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

2020/07/21 by Alexander Stasinski, Stasinski, Alexander, Michele Zordan +1
Mathematics · #20J06 #22E35 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Group Theory (math.GR) #Primary 20E18 #Secondary 20C15 #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2007.10694

openalex publication_date 2020/07/21 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

We prove that for any FAb compact p-adic analytic group G, its\nrepresentation zeta function is a finite sum of terms\nn_i^-sf_i(p^-s), where n_i are natural numbers and\nf_i(t)\∈\ℚ(t) are rational functions. Meromorphic continuation and\nrationality of the abscissa of the zeta function follow as corollaries. If G\nis moreover a pro-p group, we prove that its representation zeta function is\nrational in p^-s. These results were proved by Jaikin-Zapirain for p>2 or\nfor G uniform and pro-2, respectively. We give a new proof which avoids the\nKirillov orbit method and works for all p. First part of arXiv:2007.10694,\nsecond part uploaded as a separate paper.\n

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