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

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]).

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]).

Published online:
DOI: 10.5802/jtnb.477
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
@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/}
}
TY  - JOUR
TI  - On equations defining fake elliptic curves
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2005
DA  - 2005///
SP  - 57
EP  - 67
VL  - 17
IS  - 1
PB  - Université Bordeaux 1
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.477/
UR  - https://zbmath.org/?q=an%3A1093.11042
UR  - https://www.ams.org/mathscinet-getitem?mr=2152211
UR  - https://doi.org/10.5802/jtnb.477
DO  - 10.5802/jtnb.477
LA  - en
ID  - JTNB_2005__17_1_57_0
ER  - 
%0 Journal Article
%T On equations defining fake elliptic curves
%J Journal de Théorie des Nombres de Bordeaux
%D 2005
%P 57-67
%V 17
%N 1
%I Université Bordeaux 1
%U https://doi.org/10.5802/jtnb.477
%R 10.5802/jtnb.477
%G en
%F JTNB_2005__17_1_57_0
Pilar Bayer; Jordi Guàrdia. On equations defining fake elliptic curves. Journal de Théorie des Nombres de Bordeaux, Volume 17 (2005) no. 1, pp. 57-67. doi : 10.5802/jtnb.477. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.477/

[Al04] M. Alsina, Binary quadratic forms and Eichler orders. Journées Arithmétiques Graz 2003, in this volume. | Numdam | Zbl: 1079.11022

[AlBa04] M. Alsina, P. Bayer, Quaternion orders, quadratic forms and Shimura curves. CRM Monograph Series 22. AMS, 2004. | MR: 2038122 | Zbl: 1073.11040

[Ba02] P. Bayer, Uniformization of certain Shimura curves. In Differential Galois Theory, T. Crespo and Z. Hajto (eds.), Banach Center Publications 58 (2002), 13–26. | MR: 1972441 | Zbl: 1036.11026

[Bu96] K. Buzzard, Integral models of certain Shimura curves. Duke Math. J.  87 (1996), 591–612. | MR: 1446619 | Zbl: 0880.11048

[Ei55] M. Eichler, Zur Zahlentheorie der Quaternionen-Algebren. J. reine angew. Math. 195 (1955), 127–151. | MR: 80767 | Zbl: 0068.03303

[Gu02] J. Guàrdia, Jacobian nullwerte and algebraic equations. Journal of Algebra 253 (2002), 112–132. | MR: 1925010 | Zbl: 1054.14041

[Gu] J. Guàrdia, Jacobian Nullwerte, periods and symmetric equations for hyperelliptic curves. In preparation.

[HaMu95] K. Hashimoto, N. Murabayashi, Shimura curves as intersections of Humbert surfaces and defining equations of QM-curves of genus two. Tôhoku Math. J. 47 (1995), 271–296. | MR: 1329525 | Zbl: 0838.11044

[Jo81] B.W. Jordan, On the Diophantine Arithmetic of Shimura Curves. Thesis. Harvard University, 1981.

[Mi79] J.S. Milne, Points on Shimura varieties mod p. Proceed. of Symposia in Pure Mathematics 33, part 2 (1979), 165–184. | MR: 546616 | Zbl: 0418.14022

[Mo92] A. Mori, Explicit Period Matrices for Abelian Surfaces with Quaternionic Multiplications. Bollettino U. M. I. (7), 6-A (1992), 197–208. | MR: 1177921 | Zbl: 0767.14018

[Ro02] V. Rotger, Abelian varieties with quaternionic multiplication and their moduli. Thesis. Universitat de Barcelona, 2002.

[Ro03] V. Rotger, Quaternions, polarizations and class numbers. J. reine angew. Math. 561 (2003), 177–197. | MR: 1998611 | Zbl: 01969346

[Ro04] V. Rotger, Modular Shimura varieties and forgetful maps. Trans. Amer. Math. Soc. 356 (2004), 1535–1550. | MR: 2034317 | Zbl: 1049.11061

[RV00] F. Rodríguez-Villegas, Explicit models of genus 2 curves with split CM. Algorithmic number theory (Leiden, 2000). Lecture Notes in Compt. Sci. 1838, 505–513. Springer, 2000. | MR: 1850629 | Zbl: 1032.11026

[Sh67] G. Shimura , Construction of class fields and zeta functions of algebraic curves. Annals of Math. 85 (1967), 58–159. | MR: 204426 | Zbl: 0204.07201

[Sh77] G. Shimura , On the derivatives of theta functions and modular forms. Duke Math. J. 44 (1977), 365–387. | MR: 466028 | Zbl: 0371.14023

[Sh98] G. Shimura , Abelian varieties with complex multiplication and modular functions. Princeton Series, 46. Princeton University Press, 1998. | MR: 1492449 | Zbl: 0908.11023

[Vi80] M.-F. Vignéras, Arithmétique des algèbres de quaternions. LNM 800. Springer, 1980. | MR: 580949 | Zbl: 0422.12008

[We76] A. Weil, Sur les périodes des intégrales abéliennes. Comm. on Pure and Applied Math. 29 (1976), 813–819. | MR: 422164 | Zbl: 0342.14020

Cited by Sources: