Vol. 1, No. 1, 2013

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Success and challenges in determining the rational points on curves

Nils Bruin

Vol. 1 (2013), No. 1, 187–212
Abstract

We give an overview of current computational methods for determining the rational points on algebraic curves. We discuss how two methods, based on embedding a curve in an abelian variety, provide a practical method for deciding whether the curve has rational points and, if some additional technical condition is met, for the determination of these points.

While we cannot prove the methods are always successful, we do have a heuristic that makes us expect so. This means that the main problem becomes the determination of rational points on abelian varieties, in particular the determination of the free rank of the finitely generated group they form. We discuss some methods that provide bounds on this rank.

Finally, we report on some recent progress on applying these methods to nonhyperelliptic curves of genus 3.

Keywords
Selmer group, descent, Mordell-Weil sieving, rational points, curves, Chabauty, coverings
Mathematical Subject Classification 2010
Primary: 11G30
Secondary: 11G10, 14G25, 14H45
Milestones
Published: 14 November 2013
Authors
Nils Bruin
Department of Mathematics
Simon Fraser University
Burnaby, BC V5A 1S6
Canada