no. 23-24
Number Theory
Parametrization of Abelian K-surfaces with quaternionic multiplication
[Paramétrisation des K-surfaces abéliennes à multiplication quaternionique]

Présenté par : Jean-Pierre Serre

Guitart, Xavier1 ; Molina, Santiago2
1 Departament de Matemàtica aplicada II, Universitat Politècnica de Catalunya, Jordi Girona 1-3 (Edifici Omega), 08034 Barcelona, Spain
2 Departament de Matemàtica aplicada IV, Universitat Politècnica de Catalunya, Av. Víctor Balaguer, s/n. 08800 Vilanova i la Geltrú, Spain
Comptes Rendus. Mathématique, Tome 347 (2009) no. 23-24, pp. 1325-1330.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.09.025
Guitart, Xavier 1 ; Molina, Santiago 2

1 Departament de Matemàtica aplicada II, Universitat Politècnica de Catalunya, Jordi Girona 1-3 (Edifici Omega), 08034 Barcelona, Spain
2 Departament de Matemàtica aplicada IV, Universitat Politècnica de Catalunya, Av. Víctor Balaguer, s/n. 08800 Vilanova i la Geltrú, Spain 
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Guitart, Xavier; Molina, Santiago. Parametrization of Abelian K-surfaces with quaternionic multiplication. Comptes Rendus. Mathématique, Tome 347 (2009) no. 23-24, pp. 1325-1330. doi : 10.1016/j.crma.2009.09.025. http://www.numdam.org/articles/10.1016/j.crma.2009.09.025/

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