A construction of nonabelian simple ${\acute{\mathrm{e}}}$tale fundamental groups

Abstract

Under the assumption of the generalized Riemann hypothesis (GRH), we show that there is a real quadratic field $K$ such that the ${\acute{\mathrm{e}}}$tale fundamental group $\pi ^{\acute{\mathrm{et}}}_1(\mathrm {Spec}\;\mathcal {O}_K)$ of the spectrum of the ring of integers $\mathcal {O}_K$ of $K$ is isomorphic to $A_5$. The proof uses standard methods involving Odlyzko bounds, as well as the proof of Serre’s modularity conjecture. To the best of the author’s knowledge, this is the first example of a number field $K$ for which $\pi ^{\acute{\mathrm{et}}}_1(\mathrm {Spec}\;\mathcal {O}_K)$ is finite, nonabelian and simple under the assumption of the GRH.