The cuspidal torsion packet on hyperelliptic Fermat quotients

Grant, David; Shaulis, Delphy

doi:10.5802/jtnb.462

Grant, David ¹ ; Shaulis, Delphy ¹

¹ Department of Mathematics University of Colorado at Boulder Boulder, CO 80309-0395 USA

Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 3, pp. 577-585.

Résumé
Abstract

Soit $ℓ \geq 7$ un nombre premier, soit $C$ la courbe projective lisse définie sur $ℚ$ par le modèle affine $y (1 - y) = x^{ℓ}$ , soit $\infty$ le point à l’infini de ce modèle de $C$ , soit $J$ la jacobienne de $C$ et soit $φ : C \to J$ le morphisme d’Abel-Jacobi associé à $\infty$ . Soit $\bar{ℚ}$ une clôture algébrique de $ℚ$ . Nous traitons ici un cas non couvert dans [12], en montrant que $φ (C) \cap J_{tors} (\bar{ℚ})$ est composé de l’image par $φ$ des points de Weierstrass de $C$ ainsi que les points $(x, y) = (0, 0)$ et $(0, 1)$ de $C$ . Ici, $J_{tors}$ désigne les points de torsion de $J$ .

Let $ℓ \geq 7$ be a prime, $C$ be the non-singular projective curve defined over $ℚ$ by the affine model $y (1 - y) = x^{ℓ}$ , $\infty$ the point of $C$ at infinity on this model, $J$ the Jacobian of $C$ , and $φ : C \to J$ the albanese embedding with $\infty$ as base point. Let $\bar{ℚ}$ be an algebraic closure of $ℚ$ . Taking care of a case not covered in [12], we show that $φ (C) \cap J_{tors} (\bar{ℚ})$ consists only of the image under $φ$ of the Weierstrass points of $C$ and the points $(x, y) = (0, 0)$ and $(0, 1)$ , where $J_{tors}$ denotes the torsion points of $J$ .

MR Zbl

DOI : 10.5802/jtnb.462

Affiliations des auteurs :

Grant, David ¹ ; Shaulis, Delphy ¹

¹ Department of Mathematics University of Colorado at Boulder Boulder, CO 80309-0395 USA

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     title = {The cuspidal torsion packet on hyperelliptic {Fermat} quotients},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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     publisher = {Universit\'e Bordeaux 1},
     volume = {16},
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     doi = {10.5802/jtnb.462},
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Grant, David; Shaulis, Delphy. The cuspidal torsion packet on hyperelliptic Fermat quotients. Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 3, pp. 577-585. doi : 10.5802/jtnb.462. http://www.numdam.org/articles/10.5802/jtnb.462/

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