Unramified extensions of quadratic number fields with Galois group $SL_2(7)$

Abstract

We provide an infinite family of quadratic number fields with everywhere unramified Galois extensions of Galois group $SL_2(7)$. To my knowledge, this is the first instance of infinitely many such realizations for a perfect group which is not generated by involutions, a property which makes it difficult to approach for the problem in question and leads to somewhat delicate local-global problems in inverse Galois theory.