Chabauty for symmetric powers of curves

Siksek, Samir

doi:10.2140/ant.2009.3.209

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Chabauty for symmetric powers of curves

Samir Siksek

Vol. 3 (2009), No. 2, 209–236

DOI: 10.2140/ant.2009.3.209

Abstract

Let $C$ be a smooth projective absolutely irreducible curve of genus $g \geq 2$ over a number field $K$ , and denote its Jacobian by $J$ . Let $d \geq 1$ be an integer and denote the $d$ -th symmetric power of $C$ by $C^{(d)}$ . In this paper we adapt the classic Chabauty–Coleman method to study the $K$ -rational points of $C^{(d)}$ . Suppose that $J (K)$ has Mordell–Weil rank at most $g - d$ . We give an explicit and practical criterion for showing that a given subset $ℒ \subseteq C^{(d)} (K)$ is in fact equal to $C^{(d)} (K)$ .

Keywords

Chabauty, Coleman, curves, Jacobians, symmetric powers, divisors, differentials, abelian integrals

Mathematical Subject Classification 2000

Primary: 11G30

Secondary: 11G35, 14K20, 14C20

Milestones

Received: 2 April 2008

Revised: 20 January 2009

Accepted: 17 February 2009

Published: 15 March 2009

Authors

Samir Siksek
Institute of Mathematics University of Warwick Coventry CV4 7AL United Kingdom
http://www.warwick.ac.uk/~maseap/

Vol. 3, No. 2, 2009