Explicit Chabauty over number fields
By Samir Siksek. 2010
47 citations recorded by the source corpus.
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.
Subfields: Artificial Intelligence, Geometry and Topology, Information Systems
View Paper: https://www.semanticscholar.org/paper/d5337a151c28d258082fe397426bfc43f559bcd3
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Artificial Intelligence
In Geometry and Topology
In Information Systems
Citations in the corpus
- Generalized Fermat equations: A miscellany (2015, unknown): “(p, q, r) reference(s) (n, n, n), n ≥ 3 Wiles [68], Taylor-Wiles [66] (n, n, 2), n ≥ 4 Darmon-Merel [33], Poonen [57] (n, n, 3), n ≥ 3 Darmon-Merel [33], Poonen [57] (2n, 2n, 5), n ≥ 2 Bennett [1] (2, 4, n), n ≥ 4 Ellenberg [39], Bennett-Ellenberg-Ng [5], Bruin [15] (2, 6, n), n ≥ 3 Bennett-Chen [2]”
- Quadratic Chabauty and rational points, I: p-adic heights (2016, methodology): “Siksek [31] describes a variant of the Mordell-Weil sieve over number fields which is adapted to work well with his explicit Chabauty method over number fields introduced in loc.”