CM liftings of supersingular elliptic curves

Kane, Ben

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

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 $\left|D\right|>D_p$ implique que l’application est nécessairement surjective et nous calculons ensuite explicitement les cas $\left|D\right|<D_p$.

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 $\left|D\right|>D_p$ implies that the map is necessarily surjective and then we compute explicitly the cases $\left|D\right|<D_p$.

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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