Explicit Chabauty over number fields

Abstract

Let C be a smooth projective absolutely irreducible curve of genus g ≥ 2 over a number field K of degree d, and let J denote its Jacobian. Let r denote the Mordell-Weil rank of J(K). 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 solution to the equation x2 + y3 = z10 in coprime nonzero 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. © 2013 Mathematical Sciences Publishers.