Number Fields in Fibers: the Geometrically Abelian Case with Rational Critical Values
Number Theory
2016-10-14 v3
Abstract
Let X be an algebraic curve over Q and t a non-constant Q-rational function on X such that Q(t) is a proper subfield of Q(X). For every integer n pick a point P_n on X such that t(P_n)=n. We conjecture that, for large N, among the number fields Q(P_1), ..., Q(P_N) there are at least cN distinct. We prove this conjecture in the special case when t defines a geometrically abelian covering of the projective line, and the critical values of t are all rational. This implies, in particular, that our conjecture follows from a famous conjecture of Schinzel.
Keywords
Cite
@article{arxiv.1606.09164,
title = {Number Fields in Fibers: the Geometrically Abelian Case with Rational Critical Values},
author = {Yuri Bilu and Florian Luca},
journal= {arXiv preprint arXiv:1606.09164},
year = {2016}
}
Comments
Some typos are corrected. The article is now accepted in Periodica Math. Hungarica