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

top
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

top

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

top
  1. M. Alsina, Binary quadratic forms and Eichler orders. Journées Arithmétiques Graz 2003, in this volume. Zbl1079.11022
  2. M. Alsina, P. Bayer, Quaternion orders, quadratic forms and Shimura curves. CRM Monograph Series 22. AMS, 2004. Zbl1073.11040MR2038122
  3. P. Bayer, Uniformization of certain Shimura curves. In Differential Galois Theory, T. Crespo and Z. Hajto (eds.), Banach Center Publications 58 (2002), 13–26. Zbl1036.11026MR1972441
  4. K. Buzzard, Integral models of certain Shimura curves. Duke Math. J.  87 (1996), 591–612. Zbl0880.11048MR1446619
  5. J. Guàrdia, Jacobian nullwerte and algebraic equations. Journal of Algebra 253 (2002), 112–132. Zbl1054.14041MR1925010
  6. J. Guàrdia, Jacobian Nullwerte, periods and symmetric equations for hyperelliptic curves. In preparation. Zbl1177.11052
  7. M. Eichler, Zur Zahlentheorie der Quaternionen-Algebren. J. reine angew. Math. 195 (1955), 127–151. Zbl0068.03303MR80767
  8. 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. Zbl0838.11044MR1329525
  9. B.W. Jordan, On the Diophantine Arithmetic of Shimura Curves. Thesis. Harvard University, 1981. 
  10. J.S. Milne, Points on Shimura varieties mod p. Proceed. of Symposia in Pure Mathematics 33, part 2 (1979), 165–184. Zbl0418.14022MR546616
  11. A. Mori, Explicit Period Matrices for Abelian Surfaces with Quaternionic Multiplications. Bollettino U. M. I. (7), 6-A (1992), 197–208. Zbl0767.14018MR1177921
  12. 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. Zbl1032.11026MR1850629
  13. V. Rotger, Abelian varieties with quaternionic multiplication and their moduli. Thesis. Universitat de Barcelona, 2002. 
  14. V. Rotger, Quaternions, polarizations and class numbers. J. reine angew. Math. 561 (2003), 177–197. Zbl1094.11022MR1998611
  15. V. Rotger, Modular Shimura varieties and forgetful maps. Trans. Amer. Math. Soc. 356 (2004), 1535–1550. Zbl1049.11061MR2034317
  16. G. Shimura , Construction of class fields and zeta functions of algebraic curves. Annals of Math. 85 (1967), 58–159. Zbl0204.07201MR204426
  17. G. Shimura , On the derivatives of theta functions and modular forms. Duke Math. J. 44 (1977), 365–387. Zbl0371.14023MR466028
  18. G. Shimura , Abelian varieties with complex multiplication and modular functions. Princeton Series, 46. Princeton University Press, 1998. Zbl0908.11023MR1492449
  19. M.-F. Vignéras, Arithmétique des algèbres de quaternions. LNM 800. Springer, 1980. Zbl0422.12008MR580949
  20. A. Weil, Sur les périodes des intégrales abéliennes. Comm. on Pure and Applied Math. 29 (1976), 813–819. Zbl0342.14020MR422164

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.