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Uniform bounds for periods of endomorphisms of varieties. (English) Zbl 1472.11197

Let \(K\) be a finite extension of \({\mathbb Q}_{p}\) and \(R\) be its ring of integers. Assume that \(X\) is an algebraic variety, defined as the common zeroes of polynomials with coefficients in \(R\), and that \(f:X \to X\) is an endomorphim, also defined over \(R\).
A point \(P \in X\) is a periodic point of \(f\) if there is a positive integer \(n\) with \(f^{n}(P)=P\). The minimal of these integers \(n\) is called the primitive period of \(P\).
Let \(P \in X(R)\) (i.e, a \(R\)-rational point of \(X\)) be a periodic point with primitive period \(n\). The main result of this paper is an explicit upper bound for \(n\) in terms of the residue field of \(K\) and the valuation of \(K\). The proof is short (just two pages).
This result is related to Morton-Silverman’s Conjecture on the existence of an upper bound \(C(D,N,d)\) for the cardinality of preperiodic \(K\)-rational points of \(f\) in the case that \(K\) is a finite extesion of degree \(D\) of \({\mathbb Q}\), \(X={\mathbb P}^{N}\) and \(f\) has degree \(d\).

MSC:

11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
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