Chabauty for symmetric powers of curves

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 L⊆ 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

Samir Siksek

Abstract

Keywords

Mathematical Subject Classification 2000

Milestones

Authors