Representation growth of maximal class groups: various exceptional cases

TL;DR

This paper extends the understanding of representation growth in maximal class groups by explicitly constructing irreducible representations for specific cases, providing a complete analysis of their zeta functions.

Contribution

It introduces a constructive method to compute exceptional cases of p-local representation zeta functions for certain nilpotent groups, completing the analysis for groups M_3 and M_4.

Findings

Constructed all irreducible representations of degree p^N for M_{p+1}.

Constructed all irreducible representations of degree 2^N for M_4.

Provided a complete understanding of the irreducible representations and zeta functions for M_3 and M_4.

Abstract

This paper is a sequel to "Representation growth of maximal class groups: non-exceptional primes". We use a constructive method to calculate some exceptional cases of $p$-local representation zeta functions of a family of finitely generated nilpotent groups $M_n$ with maximal nilpotency class. Using the machinery of the constructive method from the prequel paper we construct all irreducible representations of degree $p^N$ for all $N\in \mathbb{N}$ for the group $M_{p+1}$ for a fixed prime $p$. We also construct all irreducible representations of degree $2^N$ for the group $M_4$. Together with the main result from the prequel, this gives us a complete understanding of the irreducible representations of the groups $M_3$ and $M_4$, along with their global representation zeta functions.