Number fields in fibers: the geometrically abelian case with rational critical values

Let X be an algebraic curve over ({mathbb {Q}}) and ({tin {mathbb {Q}}(X)}) a non-constant rational function such that ({{mathbb {Q}}(X)ne {mathbb {Q}}(t)}). For every ({ n in {mathbb {Z}}}) pick ({P_ n in X(bar{{mathbb {Q}}})}) such that ({t(P_n)=n}). We conjecture that, for large N, among the number fields ({mathbb {Q}}(P_1), ldots , {mathbb {Q}}(P_N)) there are at least cN distinct. We prove this conjecture in the special case when (bar{{mathbb {Q}}}(X)/bar{{mathbb {Q}}}(t)) is an abelian field extension and the critical values of t are all rational. This implies, in particular, that our conjecture follows from a more famous conjecture of Schinzel.