Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Genus two curves with quaternionic multiplication and modular Jacobian

Authors: Josep González and Jordi Guàrdia
Journal: Math. Comp. 78 (2009), 575-589
MSC (2000): Primary 11G10, 11G18
Published electronically: June 18, 2008
MathSciNet review: 2448722
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We describe a method to determine all the isomorphism classes of principal polarizations of the modular abelian surfaces $ A_f$ with quaternionic multiplication attached to a normalized newform $ f$ without complex multiplication. We include an example of $ A_f$ with quaternionic multiplication for which we find numerically a curve $ C$ whose Jacobian is $ A_f$ up to numerical approximation, and we prove that it has quaternionic multiplication and is isogenous to $ A_f$.

References [Enhancements On Off] (What's this?)

  • [BCP97] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235-265.
  • [BoMe88] Jean-Benoît Bost and Jean-François Mestre, Moyenne arithmético-géométrique et périodes des courbes de genre 1 et 2, Gaz. Math. 38 (1988), 36–64 (French). MR 970659
  • [BFGR06] N. Bruin, V. Flynn, J. González, and V. Rotger, On finiteness conjectures for modular quaternion algebras, Math. Proc. Camb. Philos. Soc., 141 (2006), no. 3,409-468.
  • [Car86] Henri Carayol, Sur les représentations 𝑙-adiques associées aux formes modulaires de Hilbert, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 409–468 (French). MR 870690
  • [CaFl96] J. W. S. Cassels and E. V. Flynn, Prolegomena to a middlebrow arithmetic of curves of genus 2, London Mathematical Society Lecture Note Series, vol. 230, Cambridge University Press, Cambridge, 1996. MR 1406090
  • [GGR05] J. González, J. Guàrdia, and V. Rotger, Abelian surfaces of $ {\rm GL}\sb 2$-type as Jacobians of curves, Acta Arith. 116 (2005), no. 3, 263-287.
  • [GJGG02] E. González-Jiménez, J. González, and J. Guàrdia, Computations on modular Jacobian surfaces, Algorithmic number theory (Sydney, 2002), Lecture Notes in Comput. Sci., vol. 2369, Springer, Berlin, 2002, pp. 189-197.
  • [Gua02] Jordi Guàrdia, Jacobian nullwerte and algebraic equations, J. Algebra 253 (2002), no. 1, 112–132. MR 1925010,
  • [KW06] Chandrashekhar Khare, Serre’s modularity conjecture: the level one case, Duke Math. J. 134 (2006), no. 3, 557–589. MR 2254626,
  • [Mil72] J. S. Milne, On the arithmetic of abelian varieties, Invent. Math. 17 (1972), 177–190. MR 0330174,
  • [Rib80] K. A. Ribet, Twists of modular forms and endomorphisms of abelian varieties, Math. Ann. 253 (1980), no. 1, 43-62.
  • [Rot03] V. Rotger, Quaternions, polarization and class numbers, J. Reine Angew. Math. 561 (2003), 177-197.
  • [Shi71] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo, 1971, Kanô Memorial Lectures, No. 1.
  • [Shi73] G. Shimura, On the factors of the jacobian variety of a modular function field, J. Math. Soc. Japan 25 (1973), 523-544.
  • [Vig80] Marie-France Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980 (French). MR 580949
  • [Wei57] André Weil, Zum Beweis des Torellischen Satzes, Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. IIa. 1957 (1957), 33–53 (German). MR 0089483

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11G10, 11G18

Retrieve articles in all journals with MSC (2000): 11G10, 11G18

Additional Information

Josep González
Affiliation: Escola Politècnica Superior d’Engenyeria de Vilanova i la Geltrú, Avda Victor Balaguer s/n, 08800 Vilanova i la Geltrú, Spain

Jordi Guàrdia
Affiliation: Escola Politècnica Superior d’Engenyeria de Vilanova i la Geltrú, Avda Victor Balaguer s/n, 08800 Vilanova i la Geltrú, Spain

Keywords: Genus two curves, quaternionic multiplication, modular abelian surfaces
Received by editor(s): July 10, 2007
Published electronically: June 18, 2008
Additional Notes: The authors were partially supported by MTM2006-15038-C02-02.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.