Let be a smooth projective absolutely irreducible curve of genus over a number field of degree , and let denote its Jacobian. Let denote the Mordell–Weil rank of . We give an explicit and practical Chabauty-style criterion for showing that a given subset is in fact equal to . This criterion is likely to be successful if . We also show that the only solution to the equation in coprime nonzero integers is . This is achieved by reducing the problem to the determination of -rational points on several genus- curves where or and applying the method of this paper.
"Explicit Chabauty over number fields." Algebra Number Theory 7 (4) 765 - 793, 2013. https://doi.org/10.2140/ant.2013.7.765