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 2 over a number field K, and denote its Jacobian by J. Let d 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 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/