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⊆ C^(d)(K) is in fact equal to C^(d)(K).

Keywords

Chabauty, Coleman, curves, Jacobians, symmetric powers, divisors, differentials, abelian integrals

Mathematical Subject Classification

Primary: 11G30

Secondary: 11G35, 14K20, 14C20

Authors