A construction of nonabelian simple e´tale fundamental groups

Under the assumption of the generalized Riemann hypothesis (GRH), we show that there is a real quadratic field K such that the e ´ tale fundamental group π 1 et ´ ( Spec O K ) of the spectrum of the ring of integers 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 π 1 et ´ ( Spec O K ) is finite, nonabelian and simple under the assumption of the GRH.