For $n=2$ the statement in the title is a theorem of B. Poonen (2009). He uses a one-parameter family of varieties together with a theorem of Coray, Sansuc and one of the authors (1980), on the Brauer–Manin obstruction for rational points on these varieties. For $n=p$, $p$ any prime number, A. Várilly-Alvarado and B. Viray (2012) considered analogous families of varieties. Replacing this family by its $(2p+1)$th symmetric power, we prove the statement in the title using a theorem on the Brauer–Manin obstruction for rational points on such symmetric powers. The latter theorem is based on work of one of the authors with Swinnerton-Dyer (1994) and with Skorobogatov and Swinnerton-Dyer (1998), work generalising results of Salberger (1988).

In this paper, we study bounds for the number of rational points on twists $C'$ of a fixed curve $C$ over a number field ${\mathcal K}$, under the condition that the group of ${\mathcal K}$-rational points on the Jacobian $J'$ of $C'$ has rank smaller than the genus of $C'$. The main result is that with some explicitly given finitely many possible exceptions, we have a bound of the form $2r + c$, where $r$ is the rank of $J'({\mathcal K})$ and $c$ is a constant depending on $C$. For the proof, we use a refinement of the method of Chabauty–Coleman: the main new ingredient is to use it for an extension field of ${\mathcal K}_v$, where $v$ is a place of bad reduction for $C'$.