Independence of rational points on twists of a given curve

Abstract

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 $C'$.