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Explicit Chabauty over Number Fields

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Let C be a curve of genus at least 2 over a number field K of degree d. Let J be the Jacobian of C and r the rank of the Mordell-Weil group J(K). Chabauty is a practical method for explicitly computing C(K) provided r <= g – 1. In unpublished work, Wetherell suggested that Chabauty’s method should still be applicable provided the weaker bound r <= d (g – 1) is satisfied. We give details of this and use it to solve the Diophantine equation x2 + y3 = z10 by reducing the problem to determining the K-rational points on several genus 2 curves over K = Q(cube root of 2).

This talk is part of the Number Theory Seminar series.

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