Grigorchuk–Lubotzky–Pyber conjecture on congruence subgroup growth

Let $G$) be a simple, simply connected, connected algebraic group defined over a number field $k$, let $S$ be a finite set of valuations containing the archimedean ones, and set $\Gamma=G(\mathcal{O}_S)$. For each nonzero ideal $I$ of the ring of $S$-integers, let $\Gamma(I)$ be the principal congruence subgroup, and let $C_n(\Gamma)$ count congruence subgroups of index at most $n$. Define

$$\alpha_+(\Gamma)=\limsup_{n\to\infty}\frac{\log C_n(\Gamma)}{(\log n)^2/\log\log n},\qquad \alpha_-(\Gamma)=\liminf_{n\to\infty}\frac{\log C_n(\Gamma)}{(\log n)^2/\log\log n}.$$

Writing $R=|\Phi_+|/l$, where $\Phi_+$ is the set of positive roots of the root system of $G$ and $l=\operatorname{rank}(G)$, set

$$\gamma(G)=\frac{(\sqrt{R(R+1)}-R)^2}{4R^2}.$$

Grigorchuk–Lubotzky–Pyber conjecture. The two asymptotic constants coincide and equal

$$\alpha_+(\Gamma)=\alpha_-(\Gamma)=\gamma(G).$$

The lower bound was previously known under the Generalized Riemann Hypothesis for Artin $L$-functions, and unconditionally in certain arithmetic cases. The paper proves the upper bound in general and extends the lower-bound results, so the conjecture is proved modulo GRH in the stated generality and unconditionally in specified cases.