Explicit Chabauty over Number Fields

Let C be a smooth projective absolutely irreducible curve of genus g ≥ 2 over a number field K of degree d, and denote its Jacobian by J . Denote the Mordell–Weil rank of J(K) by r. We give an explicit and practical Chabauty-style criterion for showing that a given subset K ⊆ C(K) is in fact equal to C(K). This criterion is likely to be successful if r ≤ d(g − 1). We also show that the only solutions to the equation x2 + y3 = z10 in coprime non-zero integers is (x, y, z) = (±3,−2,±1). This is achieved by reducing the problem to the determination of K-rational points on several genus 2 curves where K = Q or Q( 3 √ 2), and applying the method of this paper.