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A finiteness theorem for canonical heights attached to rational maps over function fields. (English) Zbl 1187.37133
Let $$K$$ be a function field, and let $$\varphi\in K(T)$$ be a rational map of degree $$n\geq2$$. We say that $$\varphi$$ is isotrivial if there is a finite extension $$K'$$ of $$K$$ and a Möbius transformation $$M\in\text{PGL}_2(K')$$ such that $$M^{-1}\circ\varphi\circ M$$ lies in $$k_0'(T)$$, where $$k_0'$$ is the field of constants of $$K'$$.
The main result of this paper is that, if $$\varphi$$ is not isotrivial, then a point $$P\in\mathbb P^1(\overline K)$$ has $$\varphi$$-canonical height zero if and only if $$P$$ is preperiodic for $$\varphi$$. This answers affirmatively a question of L. Szpiro and T. Tucker, and generalizes a result of R. Benedetto [Int. Math. Res. Not. 2005, 3855–3866 (2005; Zbl 1114.14018)] (in which $$\varphi$$ is a polynomial).
The above result is derived as a corollary of the following result, which is a variant of the Northcott finiteness principle. If $$\varphi$$ is not isotrivial, then there is $$\varepsilon>0$$ such that the set of points $$P\in\mathbb P^1(K)$$ with $$\varphi$$-canonical height at most $$\varepsilon$$ is finite.
If $$L$$ is a valued field and $$\mathcal O_L$$ is its valuation ring, then we say that $$\varphi\in L(T)$$ has good reduction over $$L$$ if there is a homogeneous lifting $$(F_1,F_2)$$ of $$\varphi$$ such that $$F_1,F_2\in\mathcal O_L[x,y]$$ and $$\text{Res}(F_1,F_2)\in\mathcal O_L^{*}$$. We say that $$\varphi$$ has potentially good reduction over $$L$$ if there is a finite extension $$L'$$ of $$L$$ and a Möbius transformation $$M\in\text{PGL}_2(L')$$ such that $$M^{-1}\circ\varphi\circ M$$ has good reduction over $$L'$$. Finally, $$\varphi\in K(T)$$ has potentially good reduction at a place $$v$$ of $$K$$ if it has potentially good reduction over the completion $$K_v$$. Then it is shown that $$\varphi$$ is isotrivial if and only if it has potentially good reduction at all places $$v$$ of $$K$$.
The heart of the proof involves potential theory on Berkovich spaces. Specifically, the paper recalls the dynamical Green function attached to $$\varphi$$, as defined by M. Baker and R. Rumely [Ann. Inst. Fourier 56, 625–688 (2006; Zbl 1234.11082)]. It is then shown that $$\varphi$$ does not have potentially good reduction over a valued field $$L$$, if and only if the Green function $$g_\varphi$$ satisfies $$g_\varphi(x,x)>0$$ for all $$x\in\mathbb P^1_{\text{Berk}}\setminus\mathbb P^1(L)$$.
In an appendix, the paper shows how the above Northcott-like result leads to a new proof of the Mordell-Weil theorem over function fields, more closely modeled on the classical proof in the number field case.

##### MSC:
 37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps 37P15 Dynamical systems over global ground fields 37P30 Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems 14H52 Elliptic curves 14H05 Algebraic functions and function fields in algebraic geometry 14G40 Arithmetic varieties and schemes; Arakelov theory; heights
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##### References:
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