Edixhoven, Bas; Yafaev, Andrei Subvarieties of Shimura varieties. (English) Zbl 1053.14023 Ann. Math. (2) 157, No. 2, 621-645 (2003). Let \(\text{Sh}_K (G,X)\) be the Shimura variety associated to a Shimura datum \((G,X)\) and a compact open subgroup \(K\) of \(G(\mathbb A_f)\). Given a set \(S\) of special points in \(\text{Sh}_K (G,X) (\mathbb C)\), according to a conjecture of André and Oort, every irreducible component of the Zariski closure of \(S\) in \(\text{Sh}_K (G,X)_{\mathbb C}\) should be a subvariety of Hodge type. In this paper the authors prove this conjecture for a special case. The choice of the special case was motivated by the work of J. Wolfart [Invent. Math. 92, No.1, 187–216 (1988; Zbl 0649.10022)] on algebraicity of values of hypergeometric functions at algebraic numbers. More specifically, let \(V\) be a finite-dimensional faithful representation of \(G\), and let \(V_h\) for each \(h \in X\) denote the corresponding \(\mathbb Q\)-Hodge structure. If \(x = \overline{(h,g)}\) is an element of \(\text{Sh}_K (G,X) (\mathbb C)\), let \([V_x]\) be the isomorphism class of \(V_h\). The authors prove that an irreducible closed algebraic curve \(Z\) contained in \(\text{Sh}_K (G,X)_{\mathbb C}\) is of Hodge type if \(Z(\mathbb C)\) contains an infinite set of special points \(x\) such that all \([V_x]\) are equal. This result implies in particular that \(Z\) is of Hodge type if \(Z(\mathbb C)\) contains an infinite set of special points that lie in one Hecke orbit. Reviewer: Min Ho Lee (Cedar Falls) Cited in 7 ReviewsCited in 49 Documents MSC: 14G35 Modular and Shimura varieties 11G18 Arithmetic aspects of modular and Shimura varieties 14D07 Variation of Hodge structures (algebro-geometric aspects) 11J81 Transcendence (general theory) Keywords:Hodge structures; Hecke operators; Hecke orbit Citations:Zbl 0649.10022 × Cite Format Result Cite Review PDF Full Text: DOI arXiv