Computing the cardinality of CM elliptic curves using torsion points

François Morain[1]

  • [1] Laboratoire d’Informatique de l’École polytechnique (LIX) F-91128 Palaiseau Cedex France

Journal de Théorie des Nombres de Bordeaux (2007)

  • Volume: 19, Issue: 3, page 663-681
  • ISSN: 1246-7405

Abstract

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Let be an elliptic curve having complex multiplication by a given quadratic order of an imaginary quadratic field . The field of definition of is the ring class field of the order. If the prime splits completely in , then we can reduce modulo one the factors of and get a curve defined over . The trace of the Frobenius of is known up to sign and we need a fast way to find this sign, in the context of the Elliptic Curve Primality Proving algorithm (ECPP). For this purpose, we propose to use the action of the Frobenius on torsion points of small order built with class invariants generalizing the classical Weber functions.

How to cite

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Morain, François. "Computing the cardinality of CM elliptic curves using torsion points." Journal de Théorie des Nombres de Bordeaux 19.3 (2007): 663-681. <http://eudml.org/doc/249967>.

@article{Morain2007,
abstract = {Let $\mathcal\{E\}/\overline\{\mathbb\{Q\}\}$ be an elliptic curve having complex multiplication by a given quadratic order of an imaginary quadratic field $\mathbb\{K\}$. The field of definition of $\mathcal\{E\}$ is the ring class field $\Omega $ of the order. If the prime $p$ splits completely in $\Omega $, then we can reduce $\mathcal\{E\}$ modulo one the factors of $p$ and get a curve $E$ defined over $\mathbb\{F\}_\{p\}$. The trace of the Frobenius of $E$ is known up to sign and we need a fast way to find this sign, in the context of the Elliptic Curve Primality Proving algorithm (ECPP). For this purpose, we propose to use the action of the Frobenius on torsion points of small order built with class invariants generalizing the classical Weber functions.},
affiliation = {Laboratoire d’Informatique de l’École polytechnique (LIX) F-91128 Palaiseau Cedex France},
author = {Morain, François},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Elliptic curves; complex multiplication; modular curves; class invariants; ECPP algorithm; SEA algorithm; elliptic curves},
language = {eng},
number = {3},
pages = {663-681},
publisher = {Université Bordeaux 1},
title = {Computing the cardinality of CM elliptic curves using torsion points},
url = {http://eudml.org/doc/249967},
volume = {19},
year = {2007},
}

TY - JOUR
AU - Morain, François
TI - Computing the cardinality of CM elliptic curves using torsion points
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2007
PB - Université Bordeaux 1
VL - 19
IS - 3
SP - 663
EP - 681
AB - Let $\mathcal{E}/\overline{\mathbb{Q}}$ be an elliptic curve having complex multiplication by a given quadratic order of an imaginary quadratic field $\mathbb{K}$. The field of definition of $\mathcal{E}$ is the ring class field $\Omega $ of the order. If the prime $p$ splits completely in $\Omega $, then we can reduce $\mathcal{E}$ modulo one the factors of $p$ and get a curve $E$ defined over $\mathbb{F}_{p}$. The trace of the Frobenius of $E$ is known up to sign and we need a fast way to find this sign, in the context of the Elliptic Curve Primality Proving algorithm (ECPP). For this purpose, we propose to use the action of the Frobenius on torsion points of small order built with class invariants generalizing the classical Weber functions.
LA - eng
KW - Elliptic curves; complex multiplication; modular curves; class invariants; ECPP algorithm; SEA algorithm; elliptic curves
UR - http://eudml.org/doc/249967
ER -

References

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