Counting number fields in fibers (with an Appendix by Jean Gillibert)

Abstract

Let X be a projective curve over $${\mathbb Q}$$ and $${t\in {\mathbb Q}(X)}$$ a non-constant rational function of degree $${n\ge 2}$$ . For every $${\tau \in {\mathbb Z}}$$ pick $${P_\tau \in X(\bar{\mathbb Q})}$$ such that $${t(P_\tau )=\tau }$$ . Dvornicich and Zannier proved that, for large N, the field $${\mathbb Q}(P_1, \ldots , P_N)$$ is of degree at least $$e^{cN/\log N}$$ over $${\mathbb Q}$$ , where $${c>0}$$ depends only on X and t. In this paper we extend this result, replacing $${\mathbb Q}$$ by an arbitrary number field.