Special subvarieties of Drinfeld modular varieties
Let \(S\) be a Shimura variety over \(\mathbb{C}\) and \(X \subset S\) an irreducible algebraic subvariety. The classical André-Oort conjecture for complex Shimura varieties claims that \(X(\mathbb{C})\) contains a Zariski-dense set of special points of \(S\) if and only if \(X\) is a subvariety of Hodge type. The paper under review tries to find analogous result in the Drinfeld module case (characteristic \(p\)).NEWLINENEWLINELet \(K\) be a global function field with exact field of constants \(\mathbb{F}_q\) and \(A\) be the Dedekind domain of elements of \(K\), regular away from \(\infty\). For any open subgroup \(\mathcal{K}\) of \(\mathrm{GL}_r(\hat{A})\), let \(M_A^r(\mathcal{K})\) be the moduli scheme over \(\mathrm{Spec}(A)\) (of rank \(r\)) of Drinfeld \(A\)-modules with \(\mathcal{K}\)-level structures. Let \(r\), \(r'\) be integers such that \(r' \mid r\). Let \(K'\) be a purely imaginary extension of degree \(r/r'\) and \(A'\) be the integral closure of \(A\) in \(K'\). An irreducible subvariety \(X \subset M_A^r(\mathcal{K})_{\mathbb{C}_{\infty}}\) is said to be special if its image under the canonical projection \(M_A^r(\mathcal{K})_{\mathbb{C}_{\infty}} \to M_A^r(1)_{\mathbb{C}_{\infty}}\) is an irreducible component of \(T_g(M_{A'}^{r'}(\mathcal{K'})_{\mathbb{C}_{\infty}})\) for some \(g \in \mathrm{GL}_r(\mathbb{A}_f)\), \(r'\) and \(A'\). \(T_g\) is the Hecke correspondence. The author made the following analogous conjecture of the André-Oort conjecture for Drinfeld modular varieties: \vskip 0.1in \noindent { Conjecture.} Let \(X \subset M_A^r(\mathcal{K})_{\mathbb{C}_{\infty}}\) be an irreducible subvariety. Then \(X(\mathbb{C}_{\infty})\) contains a Zariski-dense set of CM points if and only if \(X\) is special. \vskip 0.1in The main results of the paper are the following two special cases of the conjecture \vskip 0.1in \noindent (1) Suppose that \(X(\mathbb{C}_{\infty})\) contains a Zariski-dense subset \(\Sigma\) of CM points satisfying the following conditions:NEWLINENEWLINE(a) there is an imaginary extension \(K'\) over \(K\) such that for all \(x\) in \(\Sigma\), \(\mathrm{End}(\varphi^x)\) is an order in \(K'\) andNEWLINENEWLINE(b) there is a set \(Q\) of primes of \(A\), of Dirichlet density zero, such that for all \(x\) in \(\Sigma\), if \(\mathfrak{p} \notin Q\), then \(\mathfrak{p}\) does not divide the conductor of \(\mathrm{End}(\varphi^x)\).NEWLINENEWLINEThen \(X\) is special.NEWLINENEWLINE(2) Let \(X\) be an irreducible subcurve. Then \(X\) contains infinitely many CM points if and only if \(X\) is special.
scientific article; zbMATH DE number 6063234
scientific article