Trivial points on towers of curves

TL;DR

This paper investigates trivial points on towers of algebraic curves over number fields, establishing their finiteness in certain cases and linking this to gonality growth, building on recent research in the area.

Contribution

It introduces a new framework for trivial points on towers of curves and connects their finiteness to gonality unboundedness under specific hypotheses.

Findings

Finiteness of trivial points in some cases.

Relation between trivial points and gonality growth.

Connection to recent results by Cadoret, Tamagawa, Ellenberg, Hall, and Kowalski.

Abstract

We define and study trivial points on towers of curves over number fields, and we show their finiteness in some cases. We relate these to the unboundeness of the gonality of the curves, which we show under some hypothesis. The problem is related to recent results of Cadoret and Tamagawa, and Ellenberg, Hall and Kowalski.