Primitive algebraic points on curves (Q6554731)

Let \(C\) be a curve defined over \(\mathbb{Q}\), and \(\overline{\mathbb{Q}}\) an algebraic closure of \(\mathbb{Q}\). We say that a point \(P\in C(\overline{\mathbb{Q}})\) is \textit{primitive}, if the only subfields of \(\mathbb{Q}(P)\) are \(\mathbb{Q}\) and \(\mathbb{Q}(P)\). Also, if \([\mathbb{Q}(P):\mathbb{Q}]= d\), then we say that \(P\) has degree \(d\). We say that the curve a curve \(C\) defined over a field \(K\) has \(K\)-gonality \(m\), if \(m\) is the least degree of a non-constant morphism \(\pi : C \rightarrow \mathbb{P}^1\) defined over \(K\). In this paper, several sets of sufficient conditions for a curve \(C\) to have finitely many primitive points of a given degree \(d\) are given. A such result is the following: Let \(C\) be a curve defined over \(\mathbb{Q}\) with genus \(g\) and \(\mathbb{Q}\)-gonality \(m \geq 2\), and \(J\) its Jacobian. Let \(d \geq 2\) be an integer satisfying \(d\neq m\) and \(d <1+g/(m-1)\). Suppose either \(J(\mathbb{Q})\) is finite or \(d \leq g - 1\), and \(A(\mathbb{Q})\) is finite for every abelian subvariety \(A\) over \(\mathbb{Q}\) of \(J\) of dimension \(\leq d/2\). Then \(C\) has finitely many primitive degree \(d\) points. Moreover, if \(\gcd(d, m) = 1\) or \(d\) is prime then \(C\) has finitely many degree \(d\) points.