Valeurs d'une fonction de Picard en des points algébriques |
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Authors: | P.A. Desrousseaux |
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Affiliation: | CNRS - UMR 8524, Mathématiques, bâtiment M2, Université des Sciences et Technologies de Lille, 59655 Villeneuve d'Ascq cedex, France |
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Abstract: | Using geometric tools introduced by P. Cohen, H. Shiga, J. Wolfart and G. Wüstholz, we show in Theorem 1 that when a certain Gauss hypergeometric function takes an algebraic value at an algebraic point, then another Gauss hypergeometric function takes a transcendental value at a related algebraic point. Using Appell hypergeometric functions, which generalize to two variables the Gauss functions, we study values at algebraic points of a new transcendental function defined in terms of these two functions. By Theorem 2, these values correspond to abelian varieties in the same isogeny class. Using a result of Edixhoven-Yafaev [B. Edixhoven, A. Yafaev, Subvarieties of Shimura varieties, Ann. of Math. 157 (2003) 621-645], this last result is in turn related to the distribution of the moduli of such abelian varieties in certain Shimura varieties. |
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Keywords: | Varié té abé lienne Multiplication complexe Pé riodes Fonctions hypergé omé triques de Gauss et d'Appell |
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