CM liftings of supersingular elliptic curves

Ben Kane

doi:10.5802/jtnb.692

Ben Kane

Journal de Théorie des Nombres de Bordeaux, Tome 21 (2009) no. 3, pp. 635-663.

Résumé
Abstract

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 $| D | < D_{p}$ .

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 $| D | < D_{p}$ .

Reçu le : 2007-09-16
Publié le : 2010-03-22

MR 2605537 | Zbl 1214.11142

DOI : https://doi.org/10.5802/jtnb.692
Classification : 11G05, 11E20, 11E45, 11Y35, 11Y70
Mots clés : Quaternion Algebra, Elliptic Curves, Maximal Orders, Half Integer Weight Modular Forms, Kohnen’s Plus Space, Shimura Lifts

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     author = {Ben Kane},
     title = {CM liftings of supersingular elliptic curves},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {635--663},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {3},
     year = {2009},
     doi = {10.5802/jtnb.692},
     zbl = {1214.11142},
     mrnumber = {2605537},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_2009__21_3_635_0/}
}

Ben Kane. CM liftings of supersingular elliptic curves. Journal de Théorie des Nombres de Bordeaux, Tome 21 (2009) no. 3, pp. 635-663. doi : 10.5802/jtnb.692. https://jtnb.centre-mersenne.org/item/JTNB_2009__21_3_635_0/

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