On equations defining fake elliptic curves

Pilar Bayer[1]; Jordi Guàrdia[2]

  • [1] Facultat de Matemàtiques Universitat de Barcelona Gran Via de les Corts Catalanes 585. E-08007, Barcelona
  • [2] Departament de Matemàtica Aplicada IV Escola Politècnica Superior d’Enginyeria de Vilanova i la Geltrú Avinguda Víctor Balaguer s/n E-08800, Vilanova i la Geltrú

Journal de Théorie des Nombres de Bordeaux (2005)

  • Volume: 17, Issue: 1, page 57-67
  • ISSN: 1246-7405

Abstract

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Shimura curves associated to rational nonsplit quaternion algebras are coarse moduli spaces for principally polarized abelian surfaces endowed with quaternionic multiplication. These objects are also known as fake elliptic curves. We present a method for computing equations for genus 2 curves whose Jacobian is a fake elliptic curve with complex multiplication. The method is based on the explicit knowledge of the normalized period matrices and on the use of theta functions with characteristics. As in the case of CM-points on classical modular curves, CM-fake elliptic curves play a key role in the construction of class fields by means of special values of automorphic functions (cf. [Sh67]).

How to cite

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Bayer, Pilar, and Guàrdia, Jordi. "On equations defining fake elliptic curves." Journal de Théorie des Nombres de Bordeaux 17.1 (2005): 57-67. <http://eudml.org/doc/249453>.

@article{Bayer2005,
abstract = {Shimura curves associated to rational nonsplit quaternion algebras are coarse moduli spaces for principally polarized abelian surfaces endowed with quaternionic multiplication. These objects are also known as fake elliptic curves. We present a method for computing equations for genus 2 curves whose Jacobian is a fake elliptic curve with complex multiplication. The method is based on the explicit knowledge of the normalized period matrices and on the use of theta functions with characteristics. As in the case of CM-points on classical modular curves, CM-fake elliptic curves play a key role in the construction of class fields by means of special values of automorphic functions (cf. [Sh67]).},
affiliation = {Facultat de Matemàtiques Universitat de Barcelona Gran Via de les Corts Catalanes 585. E-08007, Barcelona; Departament de Matemàtica Aplicada IV Escola Politècnica Superior d’Enginyeria de Vilanova i la Geltrú Avinguda Víctor Balaguer s/n E-08800, Vilanova i la Geltrú},
author = {Bayer, Pilar, Guàrdia, Jordi},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {abelian sufaces; Shimura curves; explicit equations of curves},
language = {eng},
number = {1},
pages = {57-67},
publisher = {Université Bordeaux 1},
title = {On equations defining fake elliptic curves},
url = {http://eudml.org/doc/249453},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Bayer, Pilar
AU - Guàrdia, Jordi
TI - On equations defining fake elliptic curves
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2005
PB - Université Bordeaux 1
VL - 17
IS - 1
SP - 57
EP - 67
AB - Shimura curves associated to rational nonsplit quaternion algebras are coarse moduli spaces for principally polarized abelian surfaces endowed with quaternionic multiplication. These objects are also known as fake elliptic curves. We present a method for computing equations for genus 2 curves whose Jacobian is a fake elliptic curve with complex multiplication. The method is based on the explicit knowledge of the normalized period matrices and on the use of theta functions with characteristics. As in the case of CM-points on classical modular curves, CM-fake elliptic curves play a key role in the construction of class fields by means of special values of automorphic functions (cf. [Sh67]).
LA - eng
KW - abelian sufaces; Shimura curves; explicit equations of curves
UR - http://eudml.org/doc/249453
ER -

References

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