Generalization of Deuring reduction theorem (Q2438364)

The classical Deuring reduction theorem can be formulated in terms of the first truncated Barsotti--Tate group scheme as follows. Let \({\mathcal E}\) be an elliptic curve over a number field. Suppose that \(\text{End}({\mathcal E})\) is an order in an imaginary quadratic field \(K\). Let \({\mathcal P}\) be a place of \(\overline{\mathbb Q}\) over a prime \(p\) where \({\mathcal E}\) has a non-degenerate reduction \(E\). Then \(E[p] \times \overline{{\mathbb F}}_p\) is isomorphic to \({\mathbb Z}/p{\mathbb Z} \times \mu_p\) if \(p{\mathcal O}_K = {\mathcal P}{\mathcal P}^c\) and is isomorphic to \(I_{1,1}\) if \(p{\mathcal O}_K = {\mathcal P}^2\) or \(p {\mathcal O}_K = {\mathcal P}\). Here \({\mathcal P}^c\) is the complex conjugation of \({\mathcal P}\), \(I_{1,1}\) is an indecomposable (but not simple) group scheme of order \(p^2\) fitting into an exact sequence \(0 \to \alpha_p \to I_{1,1} \to \alpha_p \to 0\). First, the author gives a proof of this group scheme version of the classical Deuring reduction theorem. Then the author shows that how a decomposition of \(p{\mathcal O}_K\) into primes gives a decomposition of the first de Rham cohomology \(H_{\text DR}^1(A/\overline{\mathbb F}_p)\) and an action of the Frobenius on it. Because there is an equivalence of categories between the category of the first de Rham cohomologies and the category of Dieudonné modules of \(A[p] \times \overline{\mathbb F}_p\), using Kraft's work on classifications of Barsotti--Tate group of level \(1\), the author is able to generalize the classical Deuring reduction theorem to abelian varieties arising after reduction of an abelian variety with complex multiplication by a CM field over a number field at a place of good reduction. In particular, the author gave a table that describes the relationships between the decomposition of \(p{\mathcal O}_K\) into primes, decomposition of \({\text BT}_1\)-group scheme \(A[p]\), \(p\)-rank and \(a\)-number of dimensions \(1, 2\) and \(3\).