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A lower bound for average values of dynamical Green’s functions. (English) Zbl 1173.11041
The paper provides a lower bound for the average of dynamical Green’s function on the projective line. Suppose that \(K\) is a number field with an absolute value, which can be either archimedean or not archimedean. Let \(\varphi \in K(T)\) be a rational function of degree \(d \geq 2\) and let \(g_{\varphi}\) be the normalized Arakelov-Green’s function associated to \(\varphi\). Consider the \(g_{\varphi}\)-discriminant sum \[ D_{\varphi}(z_1,\dots,z_N)=\sum_{1 \leq i,j \leq N, i \neq j} g_{\varphi}(z_i,z_j). \] The paper presents the following result:
Theorem: There is an effective constant \(C>0\), depending on \(\varphi\) and \(K\), such that if \(N \geq 2\) and \(z_1,\dots,z_N\) are distinct points on \({\mathbb P}^1(K)\) then \(D_{\varphi}(z_1,\dots,z_N) \geq -CN \log(N)\).
Lattès maps can be used to obtain as a corollary the following result of Elkies over elliptic curves. If \(E/{\mathbb C}\) is an elliptic curve and \(g(z,w)\) is the normalized Green function associated to the Haar measure on \(E\), then there is a constant \(C > 0\), such that for \(N \geq 2\) and points \(z_1,\dots,z_N \in E\) we have \(\sum_{i \neq j} g(z_i,z_j) \geq -CN \log(N)\).
Also when we apply the theorem to the map \(t \rightarrow t^2\) on \({\mathbb P}^1(K)\) we get the bound \(\sum_{i \neq j} g(z_i,z_j) \geq -N \log(N)\) obtained by Mahler for the classical Green function.
As an application to the theory of canonical heights \(\hat{h}\) and in relation to Lehmer classical problem the following result is presented:
Theorem: There exist constants \(A,B > 0\), depending only on \(\varphi\) and \(K\), such that for any extension \(K'/K\) with \(D=[K':\mathbb Q]\), we have \(\#\{ P \in {\mathbb P}^1(K') : \hat{h}_{\varphi}(P) \leq A/D \} \leq BD \log(D).\)

MSC:
11G50 Heights
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
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