#### Vol. 3, No. 2, 2009

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Chabauty for symmetric powers of curves

### Samir Siksek

Vol. 3 (2009), No. 2, 209–236
##### Abstract

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

##### Keywords
Chabauty, Coleman, curves, Jacobians, symmetric powers, divisors, differentials, abelian integrals
##### Mathematical Subject Classification 2000
Primary: 11G30
Secondary: 11G35, 14K20, 14C20
##### Milestones
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/