CM liftings of supersingular elliptic curves

Ben Kane[1]

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

Journal de Théorie des Nombres de Bordeaux (2009)

  • Volume: 21, Issue: 3, page 635-663
  • ISSN: 1246-7405

Abstract

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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 .

How to cite

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Kane, Ben. "CM liftings of supersingular elliptic curves." Journal de Théorie des Nombres de Bordeaux 21.3 (2009): 635-663. <http://eudml.org/doc/10902>.

@article{Kane2009,
abstract = {Assuming GRH, we present an algorithm which inputs a prime $p$ and outputs the set of fundamental discriminants $D&lt;0$ such that the reduction map modulo a prime above $p$ from elliptic curves with CM by $\mathcal\{O\}_\{D\}$ to supersingular elliptic curves in characteristic $p$ is surjective. In the algorithm we first determine an explicit constant $D_p$ so that $|D|&gt; D_p$ implies that the map is necessarily surjective and then we compute explicitly the cases $|D|&lt;D_p$.},
affiliation = {Department of Mathematics Radboud Universiteit Nijmegen Heijendaalseweg 135, 6525 AJ Nijmegen, Netherlands},
author = {Kane, Ben},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Quaternion Algebra; Elliptic Curves; Maximal Orders; Half Integer Weight Modular Forms; Kohnen’s Plus Space; Shimura Lifts; quaternion algebra; elliptic curves; maximal orders; half integer weight modular forms; Kohnen's plus space; Shimura lifts},
language = {eng},
number = {3},
pages = {635-663},
publisher = {Université Bordeaux 1},
title = {CM liftings of supersingular elliptic curves},
url = {http://eudml.org/doc/10902},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Kane, Ben
TI - CM liftings of supersingular elliptic curves
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 3
SP - 635
EP - 663
AB - Assuming GRH, we present an algorithm which inputs a prime $p$ and outputs the set of fundamental discriminants $D&lt;0$ such that the reduction map modulo a prime above $p$ from elliptic curves with CM by $\mathcal{O}_{D}$ to supersingular elliptic curves in characteristic $p$ is surjective. In the algorithm we first determine an explicit constant $D_p$ so that $|D|&gt; D_p$ implies that the map is necessarily surjective and then we compute explicitly the cases $|D|&lt;D_p$.
LA - eng
KW - Quaternion Algebra; Elliptic Curves; Maximal Orders; Half Integer Weight Modular Forms; Kohnen’s Plus Space; Shimura Lifts; quaternion algebra; elliptic curves; maximal orders; half integer weight modular forms; Kohnen's plus space; Shimura lifts
UR - http://eudml.org/doc/10902
ER -

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