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Comptes Rendus Mathématique
Volume 347, n° 23-24
pages 1325-1330 (décembre 2009)
Doi : 10.1016/j.crma.2009.09.025
Received : 19 June 2009 ;  accepted : 29 September 2009
Parametrization of Abelian K -surfaces with quaternionic multiplication
Paramétrisation des K -surfaces abéliennes à multiplication quaternionique
 

Xavier Guitart a, 1 , Santiago Molina b, 2
a Departament de Matemàtica aplicada II, Universitat Politècnica de Catalunya, Jordi Girona 1-3 (Edifici Omega), 08034 Barcelona, Spain 
b Departament de Matemàtica aplicada IV, Universitat Politècnica de Catalunya, Av. Víctor Balaguer, s/n. 08800 Vilanova i la Geltrú, Spain 

Abstract

We prove that the Abelian K -surfaces whose endomorphism algebra is a rational quaternion algebra are parametrized, up to isogeny, by the K -rational points of the quotient of certain Shimura curves by the group of their Atkin–Lehner involutions. To cite this article: X. Guitart, S. Molina, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The full text of this article is available in PDF format.
Résumé

Nous démontrons que les K -surfaces abéliennes dont l’algèbre d’endomorphismes est une algèbre de quaternions sont paramétrisées, à isogénie près, par les points K -rationnels du quotient de certaines courbes de Shimura par le groupe de leurs involutions d’Atkin–Lehner. Pour citer cet article : X. Guitart, S. Molina, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The full text of this article is available in PDF format.
1  Partially supported by grants: 2009 SGR 1220, MTM2009-13060-C02-01 and AP2006-02868.
2  Partially supported by grants: 2009 SGR 1220, MTM2009-13060-C02-02 and AP2005-1609.


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