Chabauty for symmetric powers of curves

Abstract

Let C be a smooth projective absolutely irreducible curve of genus g >= 2 over a number field K, and denote its Jacobian by J. Let d >= 1 be an integer and denote the d-th symmetric power of C by C-(d). In this paper we adapt the classic Chabauty-Coleman method to study the K-rational points of C-(d). Suppose that J(K) has Mordell-Weil rank at most g - d . We give an explicit and practical criterion for showing that a given subset L subset of C-(d) (K) is in fact equal to C-(d) (K).