Diversity in Parametric Families of Number Fields

TL;DR

This paper improves lower bounds on the number of distinct number fields generated by rational points on a curve, refining previous results by Dvornicich and Zannier with a more precise estimate.

Contribution

It establishes a stronger lower bound on the count of distinct number fields associated with rational points on a curve, enhancing prior asymptotic estimates.

Findings

New lower bound of N/( ext{log} N)^{1-c} for distinct number fields

Improved asymptotic estimate over previous results

Quantitative measure of diversity in parametric families of number fields

Abstract

Let X be a projective curve defined over Q and t a non-constant Q-rational function on X of degree at least 2. For every integer n pick a point P_n on X such that t(P_n)=n. A result of Dvornicich and Zannier implies that, for large N, among the number fields Q(P_1),...,Q(P_N) there are at least cN/\log N distinct, where c>0. We prove that there are at least N/(\log N)^{1-c} distinct fields, where c>0.