CM liftings of supersingular elliptic curves

Ben Kane

doi:10.5802/jtnb.692

Ben Kane ¹

¹Department of Mathematics Radboud Universiteit Nijmegen Heijendaalseweg 135, 6525 AJ Nijmegen, Netherlands

Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 3, pp. 635-663

Abstract (Original language)
Résumé

Assuming GRH, we present an algorithm which inputs a prime $p$ and outputs the set of fundamental discriminants $D < 0$ such that the reduction map modulo a prime above $p$ from elliptic curves with CM by $𝒪_{D}$ to supersingular elliptic curves in characteristic $p$ is surjective. In the algorithm we first determine an explicit constant $D_{p}$ so that $| D | > D_{p}$ implies that the map is necessarily surjective and then we compute explicitly the cases $| D | < D_{p}$ .

Sous GRH, nous présentons un algorithme qui, étant donné un nombre premier p, calcule l’ensemble des discriminants fondamentaux $D < 0$ , tels que l’application de réduction, modulo un premier aux dessus de $p$ , des courbes elliptiques avec multiplication complexe par $𝒪_{D}$ vers les courbes elliptiques supersingulières en caractéristique $p$ est surjective. Dans l’algorithme, nous déterminons d’abord une borne $D_{p}$ explicite telle que $| D | > D_{p}$ implique que l’application est nécessairement surjective et nous calculons ensuite explicitement les cas $| D | < D_{p}$ .

Article information
Cite this article

Received: 2007-09-16
Published online: 2010-03-22

Zbl MR

DOI: 10.5802/jtnb.692

Classification: 11G05, 11E20, 11E45, 11Y35, 11Y70
Keywords: Quaternion Algebra, Elliptic Curves, Maximal Orders, Half Integer Weight Modular Forms, Kohnen’s Plus Space, Shimura Lifts

Author's affiliations:

Ben Kane ¹

¹ Department of Mathematics Radboud Universiteit Nijmegen Heijendaalseweg 135, 6525 AJ Nijmegen, Netherlands

Ben Kane. CM liftings of supersingular elliptic curves. Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 3, pp. 635-663. doi: 10.5802/jtnb.692

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