#### 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 $\mathfrak{3}$.

##### 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