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Variation of periods modulo \(p\) in arithmetic dynamics. (English) Zbl 1153.11028
Let \(V\) be a quasi-projective variety over a number field \(K\), and let \(\varphi: V \to V\) be a \(K\)-morphism. We fix a \(K\)-rational point \(P\) on \(V\), and assume that \(O_{\varphi}(P) := \{ \varphi^n(P)\mid n \in \mathbb{N} \}\) is an infinite set. If \(\varphi\) has good reduction at a finite place \(\mathfrak{p}\) of \(K\), then we set \(m_{\mathfrak{p}}(\varphi, P) := \# \{ \widetilde{Q} \mid Q \in O_{\varphi}(P) \}\), where \(\widetilde{Q}\) is the reduction of \(P\) at \(\mathfrak{p}\). Otherwise, we set \(m_{\mathfrak{p}}(\varphi, P) := \infty\). The author proves that, roughly speaking, \(m_{\mathfrak{p}}\varphi, P)\) is almost as large as \(\log N_{K/\mathbb{Q}} \mathfrak{p}\) for most \(\mathfrak{p}\). To state precise statements, given a set \(\mathcal{P}\) of finite places of \(K\), we define \(\delta(\mathcal{P})\) (resp. \(\underline{\delta}(\mathcal{P})\)) to be the limit (resp. liminf) of \((d \log \zeta_K(\mathcal{P}, s))(d \log \zeta_K(s))^{-1}\) as \(s \to 1+\), where \(\zeta_K(s)\) is the Dedekind zeta function of \(K\) and \(\zeta_K(\mathcal{P}, s) = \prod_{\mathfrak{p} \in \mathcal{P}} (1 - (N_{K/\mathbb{Q}} \mathfrak{p})^{-s})^{-1}\) is the partial zeta function for \(\mathcal{P}\). The main results of the paper under review are the following:
(1) For any \(\gamma < 1\), we have
\[ \delta\{ \mathfrak{p}\mid m_{\mathfrak{p}}(\varphi, P) \geq (\log N_{K/\mathbb{Q}} \mathfrak{p})^{\gamma} \} = 1. \] (2) There exists \(C=C(K, V,\varphi, P)\) so that for all \(\varepsilon >0\)
\[ \underline{\delta}\{ \mathfrak{p}\mid m_{\mathfrak{p}}(\varphi, P) \geq \varepsilon \log N_{K/\mathbb{Q}} \mathfrak{p} \} \geq 1- C \varepsilon. \] The author expects this lower bound is far from strict. Based on experimental and heuristic arguments, he made a conjecture that for any \(\varepsilon >0\), one should have \[ \delta\{ \mathfrak{p} ~| ~ m_{\mathfrak{p}}(\varphi, P) \leq N_{K/\mathbb{Q}} \mathfrak{p}^{\frac{N}{2} - \varepsilon} \} =0 \] where \(\varphi: \mathbb{P}^N \to \mathbb{P}^N\) is a morphism of degree \(\geq 2\) and if \(O_{\varphi}(P)\) is Zariski dense in \(\mathbb{P}^N\).

11G35 Varieties over global fields
11B37 Recurrences
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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