## 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

### Keywords:

Shimura varieties; Hodge structures; Mumford-Tate groups
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### References:

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