Subvarieties of Shimura varieties (Q1425320)

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 \textit{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.