On the abscissae of Weil representation zeta functions for procyclic groups

2024/11/19 by Kionke, Steffen
#11M41 #20E18 #20P05 #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2411.12848

Abstract

A famous conjecture of Chowla on the least primes in arithmetic progressions implies that the abscissa of convergence of the Weil representation zeta function for a procyclic group G only depends on the set S of primes dividing the order of G and that it agrees with the abscissa of the Dedekind zeta function of ℤ[p^-1| p \not∈ S]. Here we show that these consequences hold unconditionally for random procyclic groups in a suitable model. As a corollary, every real number 1 ≤ β≤ 2 is the Weil abscissa of some procyclic group.

Related

• Weil zeta functions of group representations over finite fields
• On representation zeta functions of groups and a conjecture of Larsen and Lubotzky
• Representation Growth of Linear Groups
• Representation growth and representation zeta functions of groups
• Representation growth of compact linear groups