Number Fields in Fibers: the Geometrically Abelian Case with Rational Critical Values

TL;DR

This paper proves a special case of a conjecture relating the number of distinct number fields generated by points on a curve with a rational function, focusing on geometrically abelian covers with rational critical values.

Contribution

It establishes the conjecture for geometrically abelian coverings with rational critical values, linking it to Schinzel's conjecture.

Findings

Proves the conjecture in the geometrically abelian case with rational critical values.

Shows the conjecture follows from Schinzel's conjecture.

Provides a new connection between algebraic geometry and number field distribution.

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.