On equations defining fake elliptic curves
Journal de Théorie des Nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 57-67.

Les courbes de Shimura associées à des algèbres de quaternions rationnelles et non décomposées forment des espaces de modules grossiers pour les surfaces abeliennes principalement polarisées munies d’une multiplication par les quaternions. Ces objets sont également connus sous le nom de fausses courbes elliptiques. Nous présentons une méthode pour calculer des équations de courbes de genre 2 dont la Jacobienne est une fausse courbe elliptique avec multiplication complexe. La méthode est basée sur la connaissance explicite des matrices de périodes normalisées et sur l’utilisation de fonctions theta avec caractéristiques. Comme dans le cas des points CM sur les courbes modulaires classiques, les fausses courbes elliptiques CM jouent un role clé dans la construction des corps de classes au moyen des valeurs spéciales des fonctions automorphes (cf. [Sh67]).

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]).

@article{JTNB_2005__17_1_57_0,
     author = {Pilar Bayer and Jordi Gu\`ardia},
     title = {On equations defining fake elliptic curves},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {57--67},
     publisher = {Universit\'e Bordeaux 1},
     volume = {17},
     number = {1},
     year = {2005},
     doi = {10.5802/jtnb.477},
     zbl = {1093.11042},
     mrnumber = {2152211},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.477/}
}
Pilar Bayer; Jordi Guàrdia. On equations defining fake elliptic curves. Journal de Théorie des Nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 57-67. doi : 10.5802/jtnb.477. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.477/

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