The Oort conjecture for Shimura curves of small unitary rank (Q1757186)

This paper studies the Coleman-Oort conjecture for the open Torelli locus \(\mathcal{T}_g^{\circ}\) of the Siegel modular variety \(\mathcal{A}_g\). This conjecture states that for sufficiently large \(g\), the intersection of \(\mathcal{T}_g^{\circ}\) with any Shimura subvariety \(M \subsetneq \mathcal{A}_g\) of strictly positive dimension is not Zariski open in \(M\). In particular, for Shimura curves \(M\), the conjecture states that for sufficiently large \(g\), the intersections of these curves with the Torelli locus are all finite. In previous work, the authors have shown the Coleman--Oort conjecture for Shimura subvarieties whose canonical Higgs bundle contains a large unitary subbundle. In the current work, they consider Shimura curves \(C\) for which the unitary contribution is smaller. More precisely, they assume that the logarithmic Higgs bundle \((E_{\overline{C}}, \theta_{\overline{C}})\) decomposes as \((A_{\overline{C}}, \theta_{\overline{C}} |_{A_{\overline{C}}}) \oplus (F_{\overline{C}}, 0)\), where \(A_{\overline{C}}^{1,0}\) and where \(F_{\overline{C}}\) is the maximal unitary flat subbundle, and show that \(C\) is not generically contained in \(\mathcal{T}_g^{\circ}\) as long as the rank of \(F_{\overline{C}}^{1,0}\) is at most \((2 g - 22) / 7\). Furthermore, the authors define Shimura curves from an operation called \textit{partial corestriction}, which is a hybrid of the notions of restriction and corestriction. On the level of algebras, constructing a partial corestriction comes down to the following. Let \(A\) be a central simple algebra over \(F\), where \(F\) is in turn an extension of degree \(r\) of a field \(L\). Let \(t\) be a subset of \(\left\{ 1, \dots, r \right\}\). Then the \(t\)-th partial corestriction of of \(A\) along the inclusion \(L \hookrightarrow F\) is the semi-simple \(L\)-algebra \(D (t)\) with \(D (t) \otimes_L \overline{L} = \bigoplus_T \bigotimes_{\sigma \in T} A \otimes_{F, \sigma} \overline{L}\). Here \(T\) runs over the subsets of cardinality \(t\) of the set of \(L\)-embeddings of \(F\) into \(\overline{L}\). Using suitable symplectic representations, the authors obtain Shimura curves in \(\mathcal{A}_g\) from this construction on the level of algebras. By applying the previous result, they show the Coleman--Oort conjecture for certain classes of Shimura curves that are obtained from this construction, among the most notable of which are those obtained from certain quaternion algebras over totally real fields and CM fields.