Independence of rational points on twists of a given curve

In this paper, we study bounds for the number of rational points on twists C' of a fixed curve C over a number field K, under the condition that the group of 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'(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 K_v, where v is a place of bad reduction for C'.