Journal de Th´
eorie des Nombres
de Bordeaux
16
(2004), 577–585
The cuspidal torsion packet on hyperelliptic
Fermat quotients
par
David GRANT
et
Delphy SHAULIS
R´esum´e. Soitℓ≥7 un nombre premier, soit Cla courbe projec-tive lisse d´efinie surQpar le mod`ele affiney(1−y) =xℓ
, soit∞le point `a l’infini de ce mod`ele deC, soitJ la jacobienne deCet soit φ:C →J le morphisme d’Abel-Jacobi associ´e `a∞. Soit Qune clˆoture alg´ebrique deQ. Nous traitons ici un cas non couvert dans [1], en montrant queφ(C)∩Jtors(Q) est compos´e de l’image parφ
des points de Weierstrass deC ainsi que les points (x, y) = (0,0) et (0,1) deC. Ici,Jtors d´esigne les points de torsion deJ.
Abstract. Letℓ≥7 be a prime,C be the non-singular projec-tive curve defined over Q by the affine modely(1−y) = xℓ, ∞ the point ofCat infinity on this model,J the Jacobian ofC, and φ:C→J the albanese embedding with∞as base point. LetQ be an algebraic closure ofQ. Taking care of a case not covered in [1], we show thatφ(C)∩Jtors(Q) consists only of the image under
φof the Weierstrass points ofC and the points (x, y) = (0,0) and (0,1), whereJtors denotes the torsion points ofJ.
References
[1] R. F. Coleman, A. Tamagawa, P. Tzermias,The cuspidal torsion packet on the Fermat curve. J. Reine Angew. Math496, (1998), 73–81.
DavidGrant
Department of Mathematics University of Colorado at Boulder Boulder, CO 80309-0395 USA E-mail:grant@boulder.colorado.edu
DelphyShaulis
Department of Mathematics University of Colorado at Boulder Boulder, CO 80309-0395 USA
E-mail:shaulis@euclid.colorado.edu
