A single source theorem for primitive points on curves

Abstract

Let $C$ be a curve defined over a number field $K$ and write $g$ for the genus of $C$ and $J$ for the Jacobian of $C$. Let $n \ge 2$. We say that an algebraic point $P \in C(\overline{K})$ has degree $n$ if the extension $K(P)/K$ has degree $n$. By the Galois group of $P$ we mean the Galois group of the Galois closure of $K(P)/K$ which we identify as a transitive subgroup of $S_n$. We say that $P$ is primitive if its Galois group is primitive as a subgroup of $S_n$. We prove the following 'single source' theorem for primitive points. Suppose $g>(n-1)^2$ if $n \ge 3$ and $g \ge 3$ if $n=2$. Suppose that either $J$ is simple, or that $J(K)$ is finite. Suppose $C$ has infinitely many primitive degree $n$ points. Then there is a degree $n$ morphism $\varphi : C \rightarrow \mathbb{P}^1$ such that all but finitely many primitive degree $n$ points correspond to fibres $\varphi^{-1}(\alpha)$ with $\alpha \in \mathbb{P}^1(K)$. We prove moreover, under the same hypotheses, that if $C$ has infinitely many degree $n$ points with Galois group $S_n$ or $A_n$, then $C$ has only finitely many degree $n$ points of any other primitive Galois group. The proof makes essential use of recent results of Burness and Guralnick on fixed point ratios of faithful, primitive group actions.