The Margulis-Platonov conjecture on normal subgroups of groups of rational points

Theorem 1.1 (number fields): for $G$ absolutely almost simple and simply connected over a number field $k$, every noncentral normal subgroup $N\triangleleft G(k)$ equals $\delta_A^{-1}(W)$ for an open normal $W$ of $\prod_{v\in A}G(k_v)$; when $A$ is empty, $N=G(k)$. With Rapinchuk's theorem it gives the congruence subgroup property for $S$ of positive Dirichlet density in the split places. The October 5 companion proves the same over every global function field, including characteristic two. It does not settle Serre's congruence subgroup conjecture for finite $S$ or arbitrary infinite $S$.

Theorem 1.1 (number fields): for $G$ absolutely almost simple and simply connected over a number field $k$, every noncentral normal subgroup $N\triangleleft G(k)$ equals $\delta_A^{-1}(W)$ for an open normal $W$ of $\prod_{v\in A}G(k_v)$; when $A$ is empty, $N=G(k)$. With Rapinchuk's theorem it gives the congruence subgroup property for $S$ of positive Dirichlet density in the split places. The October 5 companion proves the same over every global function field, including characteristic two. It does not settle Serre's congruence subgroup conjecture for finite $S$ or arbitrary infinite $S$.