Primitive algebraic points on curves

Abstract

AbstractA number field K is primitive if K and $$\mathbb {Q}$$ Q are the only subextensions of K. Let C be a curve defined over $$\mathbb {Q}$$ Q . We call an algebraic point $$P\in C(\overline{\mathbb {Q}})$$ P ∈ C ( Q ¯ ) primitive if the number field $$\mathbb {Q}(P)$$ 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 $$C/\mathbb {Q}$$ C / Q be a hyperelliptic curve of genus g, and let $$3 \le d \le g-1$$ 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 $$S_d$$ S d or $$A_d$$ A d . However, for any even $$d \ge 4$$ d ≥ 4 , a hyperelliptic curve $$C/\mathbb {Q}$$ C / Q has infinitely many imprimitive degree d points whose Galois group is a subgroup of $$S_2 \wr S_{d/2}$$ S 2 ≀ S d / 2 .