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Journal of the London Mathematical Society 2001 64(1):29-43; doi:10.1017/S0024610701002113
© 2001 by London Mathematical Society
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© The London Mathematical Society

5-Torsion Points on Curves of Genus 2

John Boxall, David Grant and Franck Leprévost

Département de Mathématiques et de Mécanique, CNRS – FRE 2271, Universitéde Caen Esplanade de la Paix, 14032 Caen Cedex, France, boxall{at}math.unicaen.fr
Department of Mathematics, University of Colorado at Boulder Boulder, CO 80309-0395, USA, grant{at}boulder.colorado.edu
Universitéde Grenoble Institut Fourier BP 74, F-38402 St-Martin-d'Hères Cedex, France, franck.leprevost{at}ujf-grenoble.fr

Received 5 July 1999. Revision received 20 September 2000.

Let C be a smooth proper curve of genus 2 over an algebraically closed field k. Fix a Weierstrass point {infty}in C(k) and identify C with its image in its Jacobian J under the Albanese embedding that uses {infty} as base point. For any integer N≥1, we write JN for the group of points in J(k) of order dividing N and Formula for the subset of JN of points of order N. It follows from the Riemann–Roch theorem that C(k){cap}J2 consists of the Weierstrass points of C and that C(k){cap}Formula and C(k){cap}Formula are empty (see [3]). The purpose of this paper is to study curves C with C(k){cap}Formula non-empty.


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