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On algebraic points on curves. (English) Zbl 0731.14015

Let C be a curve of genus \(g(C)>1\) over a function field K of characteristic 0. For any point \(P\in C(K')\) where \(K'\) is a finite extension of K, its height \(h_ K(P)\) is defined by the formula \(h_ K(P)=\deg (s^*_ P\omega_{X/B})/[K':K]\). Here \(f:X\to B\) is a minimal model of \(C/K\) and \(s:B'\to X\) is a morphism associated to the normalization of the closure of P in X.
The author proves the following bound for the height. For any \(\epsilon >0\), \(h_ K(P)\leq (2+\epsilon)d(P)+O(1)\), where \(d(P)=(2g(B')- 2)/[K':K]\), and O(1) depends on C and \(\epsilon\). This is a geometric analog of a conjectural bound of the height of an algebraic point in the arithmetic case which implies the \(abc\quad conjecture\) and the asymptotic Fermat conjecture. The proof follows the method of Grauert’s proof of the Mordell conjecture.
Note that there exist the following effective bounds for \(h_ K(P):\) \(h_ K(P)\leq 8\cdot 3^{3g+1}(g-1)^ 2(s+1+3^{-g}d(P)+3^{-3g})\) [L. Szpiro, Astérisque 86, 44-78 (1981; Zbl 0517.14006)] and \(h_ K(P)\leq 2(2g-1)^ 2(d(P)+s)\) [H. Esnault and E. Viehweg, Compos. Math. 76, No.1/2, 69-85 (1990)]. Here s denotes the number of degenerate fibres of \(f:X\to B\).

MSC:

14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14H05 Algebraic functions and function fields in algebraic geometry
14H10 Families, moduli of curves (algebraic)
14G05 Rational points
11G05 Elliptic curves over global fields

Citations:

Zbl 0517.14006
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References:

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