## 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

Zbl 0667.14001
Full Text:

### References:

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