Primitive algebraic points on curves

TL;DR

This paper investigates conditions under which algebraic points on curves are primitive, establishing finiteness results for certain degrees and demonstrating infinite imprimitive points for specific cases, advancing understanding of algebraic points' Galois properties.

Contribution

It provides new criteria for finiteness of primitive points on curves and analyzes the Galois groups of these points, especially on hyperelliptic curves, highlighting differences between primitive and imprimitive points.

Findings

Finiteness of primitive degree d points on hyperelliptic curves with simple Jacobian for 3 ≤ d ≤ g-1.

Existence of infinitely many imprimitive degree d points with specific Galois groups for even d ≥ 4.

Conditions linking curve properties to the Galois groups of algebraic points.

Abstract

A number field $K$ is primitive if $K$ and $\mathbb{Q}$ are the only subextensions of $K$. Let $C$ be a curve defined over $\mathbb{Q}$. We call an algebraic point $P\in C(\overline{\mathbb{Q}})$ primitive if the number field $\mathbb{Q}(P)$ is primitive. We present several sets of sufficient conditions for a curve $C$ to have finitely many primitive points of a given degree $d$. For example, let $C/\mathbb{Q}$ be a hyperelliptic curve of genus $g$, and let $3 \le d \le g-1$. Suppose that the Jacobian $J$ of $C$ is simple. We show that $C$ has only finitely many primitive degree $d$ points, and in particular it has only finitely many degree $d$ points with Galois group $S_d$ or $A_d$. However, for any even $d \ge 4$, a hyperelliptic curve $C/\mathbb{Q}$ has infinitely many imprimitive degree $d$ points whose Galois group is a subgroup of $S_2 \wr S_{d/2}$.