Kane: CM liftings of supersingular elliptic curves

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Ben Kane
CM liftings of supersingular elliptic curves
Journal de théorie des nombres de Bordeaux, 21 no. 3 (2009), p. 635-663
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Class. Math.: 11G05, 11E20, 11E45, 11Y35, 11Y70
Keywords: Quaternion Algebra, Elliptic Curves, Maximal Orders, Half Integer Weight Modular Forms, Kohnen’s Plus Space, Shimura Lifts

Résumé - Abstract

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 $| D | < D_{p}$ .

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