Uniform Representation Zeta Functions of Congruence Groups

Definitions and mathematical statement

A ${\mathbb{Z}}$-Lie lattice $L$ is a finitely generated free ${\mathbb{Z}}$-module with a ${\mathbb{Z}}$-bilinear Lie bracket. Assume $\mathfrak g={\mathbb{Q}}\otimes_{\mathbb{Z}} L$ is a semisimple ${\mathbb{Q}}$-Lie algebra of dimension $d=\operatorname{rank}_{\mathbb{Z}} L$. For the good primes in the source, let $G_{p,k}$ denote its associated principal congruence group at level $k$. The algebraic group/integral model, passage from the Lie lattice to the group, and indexing of levels are not supplied.

For a compact group $G$, let $r_m(G)$ count equivalence classes of continuous irreducible complex representations of dimension $m$. When these counts are finite and the series is defined, its representation zeta function is \[ \zeta_G^{\mathrm{irr}}(s)=\sum_{m\geq1}r_m(G)m^{-s} =\sum_{\rho\in\operatorname{Irr}_{\mathrm{cont}}(G)}(\dim\rho)^{-s}. \] The source asks for ordinary irreducible-representation counting, not twist-isoclasses.

In the simultaneous uniform interpretation of the cleaned statement, the question is \[ \forall L\text{ as above}\ \exists W_L(X,Y)\in{\mathbb{Q}}(X,Y)\ \exists S_L\text{ a finite set of primes}\quad \forall p\notin S_L\ \forall k\geq0: \] \[ \zeta_{G_{p,k}}^{\mathrm{irr}}(s)=p^{dk}W_L(p,p^{-s}). \] The identity is understood first in a domain of convergence, and as a meromorphic identity wherever the appropriate continuation is established. The same $W_L$ is independent of both $p$ and $k$, and the level dependence is precisely $p^{dk}$. This is stronger than separate rationality in $p^{-s}$ for each prime or a finite collection of formulas selected by prime-dependent data. No splitting assumption on $\mathfrak g$ has been added.

There are two source-level issues. First, level zero is explicitly included; it must not be replaced without explanation by a positive BCH-admissible level, and the missing level convention can affect the prefactor. Second, the original wording "for every $k$ ... for almost all $p$" could allow an exceptional set depending on $k$, while the cleaned question naturally suggests one finite exceptional set working for all levels. These distinct quantifier orders require confirmation.

The supplied assessment dated 2026-08-21 reports uniform rationality and functional equations for broad compact $p$-adic analytic families but says these do not settle this exact all-lattice/all-level formula. The cited type-A work is not promoted to the full nonsplit semisimple scope.

Short English statement

The question asks whether one rational function of the prime and $p^{-s}$ gives the representation zeta functions at every congruence level, with all level dependence in the stated power $p^{dk}$. The construction and zero-level convention, and whether the excluded primes are uniform in the level, remain unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $30000984$, exact source code catalogue entry, proposed year 2008; no individual proposer is supplied.
• Profinite and Asymptotic Group Theory (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/28, pp. 1537--1588. https://doi.org/10.4171/owr/2008/28.
• Supplied reference: Avni--Klopsch--Onn--Voll, https://arxiv.org/abs/1007.2900.
• Supplied reference labelled "later type-A uniformity work": https://doi.org/10.1093/imrn/rnae270.
• The supplied corrected_verified note reports checking content against the official report. The partially solved label and assessment checked 2026-08-21 are dated supplied triage requiring independent verification of exact formula scope and current status.