Explicit Chabauty over number fields. (English) Zbl 1330.11043

Siksek, Samir

Algebra Number Theory 7, No. 4, 765-793 (2013).

Summary: Let \(C\) be a smooth projective absolutely irreducible curve of genus \(g\geq2\) 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 \(\mathcal K\subset C(K)\) is in fact equal to \(C(K)\). This criterion is likely to be successful if \(r\leq d(g-1)\). We also show that the only solution to the equation \(x^2+y^3=z^{10}\) in coprime nonzero integers is \((x,y,z)=(\pm3,-2,\pm1)\). This is achieved by reducing the problem to the determination of \(K\)-rational points on several genus-2 curves where \(K=\mathbb Q\) or \(\mathbb Q(\root{3}\of{2})\) and applying the method of this paper.

MSC:

11G30	Curves of arbitrary genus or genus \(\ne 1\) over global fields
14K20	Analytic theory of abelian varieties; abelian integrals and differentials
14C20	Divisors, linear systems, invertible sheaves

Keywords:

Chabauty; Coleman; Jacobian; divisor; abelian integral; Mordell-Weil sieve; generalized Fermat; rational points

Software:

Magma