Chevalley-Weil Theorem and Subgroups of Class Groups

We prove, under some mild hypothesis, that an etale cover of curves defined over a number field has infinitely many specializations into an everywhere unramified extension of number fields. This constitutes an absolute version of the Chevalley-Weil theorem. Using this result, we are able to generalize the techniques of Mestre, Levin and the second author for constructing and counting number fields with large class group.