Primitive algebraic points on curves

2023/06/30 by Khawaja, Maleeha, Siksek, Samir
#11G30 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2306.17772

Abstract

A number field K is primitive if K and ℚ are the only subextensions of K. Let C be a curve defined over ℚ. We call an algebraic point P∈ C(ℚ) primitive if the number field 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/ℚ be a hyperelliptic curve of genus g, and let 3 ≤ d ≤ 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 ≥ 4, a hyperelliptic curve C/ℚ has infinitely many imprimitive degree d points whose Galois group is a subgroup of S₂ \wr S_d/2.

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