An approximation property for Teichmüller points
Start with a field \(K\) of characteristic zero, complete under a discrete valuation and having an algebraically closed residue field \(k\) of characteristic \(p>0\). Let \(R\) be the valuation ring of \(K\). Assume that we are given a prime element \(\pi\in R\) which is algebraic over \(\mathbb{Q}_p\). Let \(q\) be the cardinality of the residue field of \(\mathbb{Q}_p (\pi)\) and let \(\varphi\) be the unique ring automorphism of \(R\) with \(\varphi (\pi) =\pi\) that lifts the ``Frobenius'' automorphism \(F:k\to k\), \(F(x): =x^q\). Let \(X\) be a scheme of finite type over \(R\) and assume the Frobenius \(x\mapsto x^q\) of the closed fibre \(X_0:=X \otimes k\) lifts to a \(\varphi\)-endomorphism \(\widetilde\varphi\) of \(\widehat X\), the completion of \(X\) with respect to the ideal generated by \(\pi\). For any point \(P\in X(R)= \widehat X(R)\), \(P:\text{Spf} R\to \widehat X\) define the point \(P^{\widetilde\varphi} \in X(R)\) as the composition Spf\( R @>\varphi^{-1}>>\text{Spf} R @>P>> \widehat X @>\widetilde\varphi>> \widehat X\). Call \(P\) a Teichmüller point if \(P^{\widetilde \varphi}=P\) and let \(T=T(X,\widetilde \varphi) \subset X(R)\) denote the set of Teichmüller points. Theorem 1. Let \(X/R\) be a scheme of finite type, \(Y\subset X\) a closed subscheme, \(\widetilde \varphi\) and \(T\) as above. Then there exists a real constant \(c= c(X,Y, \widetilde \varphi)>0\) such that for any \(P\in T\) with \(\text{dist} (P,Y)\leq c\) we have \(P\in Y(R)\), where \(\text{dist} (P,Y)\) means the \(p\)-adic distance. Theorem 2. Assume we are in the situation of theorem 1 and assume moreover that \(K\) is absolutely unramified, \(X/R\) is projective and \(Y/R\) is a smooth curve of genus \(\geq 2\). Then the set of points \(\{P\in T;\text{ dist} (P,Y)<1\}\) is finite.
scientific article; zbMATH DE number 946959