CM liftings of supersingular elliptic curves
Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 3, p. 635-663

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 .

DOI : https://doi.org/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
@article{JTNB_2009__21_3_635_0,
     author = {Kane, Ben},
     title = {CM liftings of supersingular elliptic curves},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {3},
     year = {2009},
     pages = {635-663},
     doi = {10.5802/jtnb.692},
     mrnumber = {2605537},
     zbl = {1214.11142},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2009__21_3_635_0}
}
Kane, Ben. 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. http://www.numdam.org/item/JTNB_2009__21_3_635_0/

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