Geodesic
Abelian Surfaces with an Automorphism and Quaternionic Multiplication
Canadian journal of mathematics, Tome 68 (2016) no. 1, pp. 24-43

Voir la notice de l'article provenant de la source Cambridge University Press

We construct one-dimensional families of Abelian surfaces with quaternionic multiplication, which also have an automorphism of order three or four. Using Barth's description of the moduli space of (2,4)-polarized Abelian surfaces, we find the Shimura curve parametrizing these Abelian surfaces in a specific case. We explicitly relate these surfaces to the Jacobians of genus two curves studied by Hashimoto and Murabayashi. We also describe a (Humbert) surface in Barth's moduli space that parametrizes Abelian surfaces with real multiplication by $\mathbf{Z}\left[ \sqrt{2} \right]$ .
DOI : 10.4153/CJM-2014-045-4
Mots-clés : 14K10, 11G10, 14K20, abelian surfaces, moduli, shimura curves 
Bonfanti, Matteo Alfonso; Geemen, Bert van. Abelian Surfaces with an Automorphism and Quaternionic Multiplication. Canadian journal of mathematics, Tome 68 (2016) no. 1, pp. 24-43. doi: 10.4153/CJM-2014-045-4
@article{10_4153_CJM_2014_045_4,
     author = {Bonfanti, Matteo Alfonso and Geemen, Bert van},
     title = {Abelian {Surfaces} with an {Automorphism} and {Quaternionic} {Multiplication}},
     journal = {Canadian journal of mathematics},
     pages = {24--43},
     year = {2016},
     volume = {68},
     number = {1},
     doi = {10.4153/CJM-2014-045-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2014-045-4/}
}
TY  - JOUR
AU  - Bonfanti, Matteo Alfonso
AU  - Geemen, Bert van
TI  - Abelian Surfaces with an Automorphism and Quaternionic Multiplication
JO  - Canadian journal of mathematics
PY  - 2016
SP  - 24
EP  - 43
VL  - 68
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-2014-045-4/
DO  - 10.4153/CJM-2014-045-4
ID  - 10_4153_CJM_2014_045_4
ER  - 
%0 Journal Article
%A Bonfanti, Matteo Alfonso
%A Geemen, Bert van
%T Abelian Surfaces with an Automorphism and Quaternionic Multiplication
%J Canadian journal of mathematics
%D 2016
%P 24-43
%V 68
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-2014-045-4/
%R 10.4153/CJM-2014-045-4
%F 10_4153_CJM_2014_045_4

[B] [B] Barth, W., Abelian surfaces with (1, 2)-polarization. In: Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math. 10, North-Holland, Amsterdam, 1987, pp. 41–84. Google Scholar

[BL] [BL] Birkenhake, C. and Lange, H., Complex Abelian varieties. Second ed., Grundlehren der Mathematischen Wissenschaften, 302, Springer-Berlag, Berlin, 2004. Google Scholar

[BW] [BW] Birkenhake, C. and Wilhelm, H., Humbert surfaces and the Kummer plane. Trans. Amer. Math. Soc. 335(2003), no. 5, 1819–1841. Google Scholar | DOI

[M] [M] Bosma, W., Cannon, J., and Playoust, C., The Magma algebra system I. The user language. Computational algebra and number theory (London, 1993). J. Symbolic Comput. 24(1997),no. 3-4, 235–265. Google Scholar | DOI

[CQ] [CQ] Cardona, G. and Quer, J., Field of moduli and field of definition for curves of genus 2. In: Computational aspects of algebraic curves, Lecture Notes Ser. Comput., 13, World Sci. Publ., Hackensack, NJ, 2005, pp. 71–83. Google Scholar

[E] [E] Elkies, N. D., Shimura curve computations via K3 surfaces of Neron-Severi rank at least 19. In: Algorithmic number theory, Lecture Notes in Comput. Sci., 5011, Springer, Berlin, 2008, pp. 196–211. Google Scholar

[F] [F] Freitag, E., Siegelsche Modulfunktionen. Grundlehren der Mathematischen Wissenschaften, 254, Springer-Verlag, Berlin, 1983. Google Scholar

[GS] [GS] Garbagnati, A. and Sarti, A., Kummer surfaces and K3 surfaces with (Z/2Z)4 symplectic action. arxiv:1305.3514 Google Scholar

[vG] [vG] van Geemen, B., Projective models of Picard modular varieties. In: Classification of irregular varieties, Lecture Notes in Mathematics, 1515, Springer, Berlin, 1992, pp. 68–99. Google Scholar

[GP1] [GP1] Gross, M. and Popescu, S., Equations of(l, d)-polarized Abelian surfaces. Math. Ann. 310(1998), no. 2, 333–377. Google Scholar | DOI

[GP2] [GP2] Gross, M. and Popescu, S., Calabi-Yau three-folds and moduli of Abelian surfaces II. Trans. Amer. Math. Soc. 363(2011), 3573–3599. Google Scholar | DOI

[HM] [HM] Hashimoto, K. and Murabayashi, N., Shimura curves as intersections of Humbert surfaces and defining equations of QM-curves of genus two. Tohoku Math. J. 47(1995), no. 2, 271–296. Google Scholar | DOI

[HKW] [HKW] Hulek, K., Kahn, C., and Weintraub, S. H., Moduli spaces of Abelian surfaces: compactification, degenerations, and Theta functions, de Gruyter Expositions in Mathematics, 12, Walter de Gruyter, Berlin, 1993. Google Scholar

[I] [I] Igusa, J., Arithmetic variety of moduli for genus two. Ann. of Math. 72(1960), 612–649. http://dx.doi.Org/10.2307/1970233 Google Scholar

[12] [12] Igusa, J., Theta functions. Die Grundlehren der mathematischen Wissenschaften, 194, Springer-Verlag, New York-Heidelberg, 1972. Google Scholar

[Me] [Me] Mestre, J-F., Construction de courbes de genre 2 à partir de leurs modules., In: Effective methods in algebraic geometry, Progr. Math., 94, Birkhäuser Boston, Boston, MA, 1991, pp. 313–334. Google Scholar

[PS] [PS] Petkova, M. and Shiga, H., A new interpretation of the Shimura curve with discriminant 6 in terms of Picard modular forms. Arch. Math. (Basel) 96(2011), no. 4, 335–348. Google Scholar

[R] [R] Rotger, V., Shimura curves embedded in Igusa's threefold. In: Modular curves and Abelian varieties, Progr. Math., 224, Birkhäuser, Basel, 2004, pp. 263–276. Google Scholar

Cité par Sources :