A finiteness theorem for canonical heights attached to rational maps over function fields.

*(English)*Zbl 1187.37133Let \(K\) 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.

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.

Reviewer: Paul Vojta (Berkeley)

##### 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 |

##### Keywords:

Northcott finiteness theorem; canonical height; Green function; Berkovich space; good reduction; Mordell-Weil theorem##### References:

[1] | Baker M., Math. Res. Lett. 13 (2) pp 245– (2006) |

[2] | Baker M., Math. 585 pp 61– (2005) |

[3] | DOI: 10.1155/IMRN.2005.3791 · Zbl 1120.11025 · doi:10.1155/IMRN.2005.3791 |

[4] | Baker M., Ann. Inst. Fourier (Grenoble) 56 (3) pp 625– (2006) |

[5] | DOI: 10.1006/jnth.2000.2577 · Zbl 0978.37039 · doi:10.1006/jnth.2000.2577 |

[6] | DOI: 10.1155/IMRN.2005.3855 · Zbl 1114.14018 · doi:10.1155/IMRN.2005.3855 |

[7] | DOI: 10.1007/s00208-002-0404-7 · Zbl 1032.37029 · doi:10.1007/s00208-002-0404-7 |

[8] | DOI: 10.1007/s002220050358 · doi:10.1007/s002220050358 |

[9] | Compos. Math. 138 (2) pp 199– (2003) |

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