A result on formal linearity (Q1972431)

Let \(\Lambda\) be a complete discrete valuation ring of mixed characteristic \((0, p)\) with an automorphism \(\tau\). Let \(X\) be the formal \(n\)-dimensional torus \(\widehat\mathbb{G}_m^n\) over \(\Lambda\) with coordinates \(q_1,\dots,q_n\), and let \(\Phi: X \rightarrow X\) be the \(\tau\)-linear morphism such that \(\Phi^*(q_i) = q_i^q\), where \(q=p^t\) for some integer \(t > 0\). Given a closed formal subscheme \(Z\) of \(X\) define \(\Phi(Z)\) to be the minimal closed subscheme of \(X\) such that the morphism \(Z \rightarrow X \rightarrow_\Phi X\) factors through \(\Phi(Z)\). Define inductively \(\Phi^a(Z)\) for a positive integer \(a\). The paper is concerned with the characterization of formal subtori of \(X\), and generalizations thereof, in terms of the map \(\Phi\). For example, the author proves the following theorem: Let \(Z\) be a closed formal subscheme of \(X\) that is equi-dimensional, reduced and flat over \(\text{Spf}(\Lambda)\). Then \(Z\) is a formal subtorus of \(X\) iff \(Z\) is irreducible and \(\Phi^a(Z) \subseteq Z\) for some \(a \geq 1\). This result is useful in the study of reduction of Shimura varieties because Serre-Tate theory identifies the deformations of an ordinary abelian variety to \(\text{Spf}(\Lambda)\) with a formal torus over \(\Lambda\) with canonical coordinates. The study of the closed subscheme along which certain endomorphisms of the special fibre deform leads to formal subtori [see \textit{B. Moonen}, Compos. Math. 114, No. 1, 3-35 (1998; Zbl 0960.14012)].