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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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