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A conjecture of Yves André’s. (English) Zbl 1097.11032

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.

MSC:

11G18 Arithmetic aspects of modular and Shimura varieties
14G35 Modular and Shimura varieties
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