The congruence subgroup property for $S$-arithmetic subgroups of simple algebraic groups when $S$ has positive Dirichlet density

TL;DR

This paper proves the triviality of the congruence kernel for certain $S$-arithmetic subgroups of simple algebraic groups over number fields, supporting Serre's conjecture under specific density conditions.

Contribution

It establishes a new criterion involving positive Dirichlet density for the triviality of the congruence kernel in the context of $S$-arithmetic subgroups, without case-by-case analysis.

Findings

Proves the congruence kernel $C^S(G)$ is trivial under specified conditions.

Provides evidence supporting Serre's Congruence Subgroup Conjecture.

Uses recent results on almost strong approximation and previous congruence kernel studies.

Abstract

Let $G$ be an absolutely almost simple simply connected algebraic group defined over a number field $K$, and let $M/K$ be the minimal Galois extension over which $G$ becomes an inner form of a split group. Assume that $G$ satisfies the Margulis-Platonov conjecture over $K$. We prove that if $S$ is a set of valuations of $K$ that contains all archimedean ones but does not contain any nonarchimedean valuations $v$ for which $G$ is anisotropic over the completion $K_v$ such that its intersection $S \cap \mathrm{Spl}(M/K)$ with the set $\mathrm{Spl}(M/K)$ of nonarchimedean valuations of $K$ that split completely in $M$ has positive Dirichlet density, then the congruence kernel $C^S(G)$ is trivial. This result provides additional evidence for Serre's Congruence Subgroup Conjecture. The proof does not involve any case-by-case considerations and relies on previous results concerning the congruence kernel and recent results on almost strong approximation.