Fibrations with few rational points

TL;DR

This paper investigates conditions under which families of algebraic varieties have very few members with rational points, generalizing previous results and providing a unified framework for understanding rational point distribution.

Contribution

It introduces new conditions on singular fibers that quantitatively limit the number of varieties with rational points within a family.

Findings

Identifies specific singular fiber conditions that restrict rational points

Provides a unifying framework for existing results

Quantifies the scarcity of rational points in certain families

Abstract

We study the problem of counting the number of varieties in families which have a rational point. We give conditions on the singular fibres that force very few of the varieties in the family to contain a rational point, in a precise quantitative sense. This generalises and unifies existing results in the literature by Serre, Browning-Dietmann, Bright-Browning-Loughran, Graber-Harris-Mazur-Starr, et al.