Yafaev, Andrei A conjecture of Yves André’s. (English) Zbl 1097.11032 Duke Math. J. 132, No. 3, 393-407 (2006). Let \((G,X)\) be a Shimura datum, and let \(K\) be an open compact subgroup of \(G (\mathbb A_f)\). If \(C\) is an irreducible closed algebraic curve contained in the Shimura variety \(\text{Sh}_K (G,X)\) such that \(C\) contains an infinite set of special points, then Yves Andre conjectured in 1989 that \(C\) is of Hodge type. In this paper the author proves this conjecture assuming the generalized Riemann hypothesis. The main ingredient in the proof is a theorem on lower bounds for Galois orbits of special points of Shimura varieties. Reviewer: Min Ho Lee (Cedar Falls) Cited in 1 ReviewCited in 16 Documents MSC: 11G18 Arithmetic aspects of modular and Shimura varieties 14G35 Modular and Shimura varieties Keywords:Shimura varieties; Hodge structures; Mumford-Tate groups PDF BibTeX XML Cite \textit{A. Yafaev}, Duke Math. J. 132, No. 3, 393--407 (2006; Zbl 1097.11032) Full Text: DOI arXiv OpenURL References: [1] Y. André, \(G\)-Functions and Geometry , Aspects of Math. E13 , Vieweg, Braunschweig, Germany, 1989. · Zbl 0688.10032 [2] D. Bertrand, “Galois orbits on abelian varieties and zero estimates” in Diophantine Analysis (Kensington, Australia, 1985) , London Math. Soc. Lecture Note Ser. 109 , Cambridge Univ. Press, Cambridge, 1986, 21–35. · Zbl 0597.10032 [3] B. Edixhoven, “On the André-Oort conjecture for Hilbert modular surfaces” in Moduli of Abelian Varieties (Texel Island, Netherlands, 1999) , Progr. Math. 195 , Birkhäuser, Basel, 2001, 133–155. · Zbl 1029.14007 [4] B. Edixhoven and A. Yafaev, Subvarieties of Shimura varieties , Ann. of Math. (2) 157 (2003), 621–645. JSTOR: · Zbl 1053.14023 [5] S. J. Edixhoven, B. J. J. Moonen, and F. Oort, Open problems in algebraic geometry , Bull. Sci. Math. 125 (2001), 1–22. · Zbl 1009.11002 [6] J. S. Milne, “Canonical models of (mixed) Shimura varieties and automorphic vector bundles” in Automorphic Forms, Shimura Varieties, and \(L\)-Functions, Vol. I (Ann Arbor, Mich., 1988) , Perspect. Math. 10 , Academic Press, Boston, 1990, 283–414. · Zbl 0704.14016 [7] B. Moonen, Linearity properties of Shimura varieties, II , Compositio Math. 114 (1998), 3–35. · Zbl 0960.14012 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.