We first provide a detailed proof of Kato’s classification theorem of log p-divisible groups over a Noetherian Henselian local ring. Exploring Kato’s idea further, we then define the notion of a standard extension of a classical finite étale group scheme (resp. classical étale p-divisible group) by a classical finite flat group scheme (resp. classical p-divisible group) in the category of finite Kummer flat group log schemes (resp. log p-divisible groups), with respect to a given chart on the base. These results are then used to prove that log p-divisible groups are formally log smooth. We then study the finite Kummer flat group log schemes $T_n(\mathbf {M}):=H^{-1}(\mathbf {M}\otimes _{{\mathbb Z}}^L{\mathbb Z}/n{\mathbb Z})$ (resp. the log p-divisible group $\mathbf {M}[p^{\infty }]$) of a log 1-motive $\mathbf {M}$ over an fs log scheme and show that they are étale locally standard extensions. Lastly, we give a proof of the Serre–Tate theorem for log abelian varieties with constant degeneration.

Let A=A$_g, 1, n$ denote the moduli scheme over Z[1/N] of p.p. g-dimensional abelian varieties with a level n structure; its generic fibre can be described as a Shimura variety. We study its ’Shimura subvarieties‘. If x ∈ A is an ordinary moduli point in characteristic p, then we formulate a local ’linearity property‘ in terms of the Serre–Tate group structure on the formal deformation space (= formal completion of A at x). We prove that an irreducible algebraic subvariety of A is a ’Shimura subvariety‘ if, locally at an ordinary point x, it is ’formally linear‘. We show that there is a close connection to a differential-geometrical linearity property in characteristic 0.

We apply our results to the study of Oort‘s conjecture on subvarieties Z [rarrhk] A with a dense collection of CM-points. We give a reformulation of this conjecture, and we prove it in a special case.