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A generalization of theorems of Faltings and Thue-Siegel-Roth-Wirsing. (English) Zbl 0778.11037

The author proves the following general theorem, which is a common generalisation of Roth’s theorem and Mordell’s conjecture. Let \(C\) be a curve defined over a number field \(k\), let \(K\) be a canonical divisor on \(C\), and let \(\pi: X\to B\) be a regular model, where \(B=\text{Spec}(O_ k)\). Define local Arakelov intersection numbers \((D\cdot E)_ v\) for arithmetic divisors \(D\), \(E\) on \(X\) and places \(v\) on \(k\) and the global intersection number \((D\cdot E)= \sum_ v(D\cdot E)_ v\) as in [S. Lang, Introduction to Arakelov theory (1988; Zbl 0667.14001), Chapter 3]. For algebraic points \(P\in C(\overline{k})\), let \(H_ P\) be the corresponding horizontal divisor on \(X\). Fix a finite set of places \(S\) of \(k\). For arithmetic divisors \(D\) on \(X\), let \(h_ D(P)=(H_ P\cdot D)/[k(P):\mathbb{Q}]\) and for \(P\not\in\text{Supp } D\), let \(m_ S(D,P)=\sum_{v\in s} (H_ P\cdot D)_ v/[k(P):\mathbb{Q}]\). Further, let \(\omega=\omega_{X/B}\) be the dualizing sheaf, with admissible metric. Define the arithmetic discriminant of \(P\), \(d_ a(P)=(H_ P\cdot H_ P+\omega)/ [k(P):\mathbb{Q}]\), and let \(h_ K(P)\) be the height relative to \(\omega\) with this metric. One has \(d_ a(P)\geq d(P)=\log| D|/[k(P):\mathbb{Q}]\), where \(D\) is the discriminant of \(k(P)\). The author’s main result is as follows:
Theorem. In addition to above, let \(\nu\) be a positive integer, \(\varepsilon\) a positive real, \(D\) an effective divisor on \(X\) with no multiple components, and \(A\) a divisor on \(X\) which is ample on the generic fibre. Then for all points \(P\in C(\overline{k})\setminus \text{Supp }D\) with \([k(P):k]\leq\nu\), \[ m_ S(D,P)+h_ K(P)\leq d_ a(P)+\varepsilon h_ A(P)+O(1), \] where the constant in \(O(1)\) depends on \(X\), \(D\), \(\nu\), \(A\), \(\varepsilon\).
In his proof, the author uses diophantine approximation techniques of Roth and Wirsing and combines these with ideas of Faltings and his own. In particular, instead of constructing a suitable auxiliary polynomial, the author constructs a suitable section of a certain line bundle which is shown to be ample. The author mentions several corollaries of this result, of which we mention two. First, by taking \(C=\mathbb{P}^ 1\), one gets Wirsing’s generalisation of Roth’s theorem, that for given algebraic \(\alpha\) there are only finitely many \(x\in\overline {\mathbb{Q}}\) such that \(| x-\alpha| <H(x)^{-2\nu-\varepsilon}\). Second, by taking \(D=0\), one gets the following generalisation of Mordell’s conjecture: Let \(C\) be a curve of genus \(g\) defined over a number field \(k\), let \(\nu\) be a positive integer, and let \(f:C\to \mathbb{P}^ 1\) be a dominant morphism. Assume that \(g-1>(\nu-1)\deg f\). Then there are only finitely many \(P\in C(\overline{k})\) with \([k(P):k]\leq\nu\) and \(k(f(P))=k(P)\).

MSC:

11J25 Diophantine inequalities
14G40 Arithmetic varieties and schemes; Arakelov theory; heights

Citations:

Zbl 0667.14001
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References:

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