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Roth’s theorem, integral points and certain ramified covers of $$\mathbb P_1$$. (English) Zbl 1231.11069
Chen, W. W. L. (ed.) et al., Analytic number theory. Essays in honour of Klaus Roth on the occasion of his 80th birthday. Cambridge: Cambridge University Press (ISBN 978-0-521-51538-2/hbk). 471-491 (2009).
Let $${\mathcal C}$$ be a curve in affine space $${\mathbb A}^n$$ and assume that its projective closure $$\widetilde{{\mathcal C}}$$ in $${\mathbb P}_n$$ is non-singular. Then $$\{A_1,\ldots , A_r\}=\widetilde{{\mathcal C}}\setminus{\mathcal C}$$ is the set of points at infinity of $${\mathcal C}$$. Siegel’s Theorem on integral points on curves (in an extended form proved by Lang) may be stated as follows: let $${\mathcal C}$$ be an affine curve over $$\overline{{\mathbb Q}}$$ and suppose that either $$\widetilde{{\mathcal C}}$$ has genus $$\geq 1$$ or that $$r\geq 3$$. Then for any number field $$k$$ and finite set of places $$S$$ of $$k$$, the set of $$S$$-integral points $${\mathcal C}(O_S)$$ is finite. The author discusses various methods to prove this theorem. The first method, followed by Lang, uses the Thue-Siegel-Roth-theorem, together with the Mordell-Weil theorem on the structure of the group of $$k$$-rational points of the Jacobian of $$\widetilde{{\mathcal C}}$$. The second method, discovered by Corvaja and the author, uses Schmidt’s Subspace Theorem and avoids Jacobians completely.
In his paper, the author focuses on a third method to prove Siegel’s Theorem, which is based on the Thue-Siegel-Roth-theorem. In fact, using the latter, the author shows that if Siegel’s Theorem is false for some $$k,S$$, then for any rational function $$\varphi\in \overline{{\mathbb Q}}({\mathcal C})\setminus\overline{{\mathbb Q}}$$ one has
$\sum_{j=1}^r \text{ord}_{A_j}(\varphi -\varphi (A_j))\leq 2\deg\varphi\tag{$$*$$}$ (with the convention $$\varphi -\varphi (A_j)=\varphi^{-1}$$ if $$A_j$$ is a pole of $$\varphi$$). So to prove Siegel’s Theorem for all $$k,S$$, it suffices to find $$\varphi$$ for which (*) does not hold. In fact, it is already sufficient to find $$\varphi$$ such that $$2\deg\varphi$$ is smaller than the number $$r$$ of points at infinity of $${\mathcal C}$$. The author shows that there is a covering $${\mathcal C}'$$ of $${\mathcal C}$$ admitting a rational function $$\varphi '$$ such that $$2\deg\varphi '$$ is smaller than the number of points at infinity of $${\mathcal C}'$$. In fact, if we embed $${\mathcal C}$$ in the Jacobian $${\mathcal J}$$ of $$\widetilde{{\mathcal C}}$$ then for $${\mathcal C}'$$ one may take the inverse image of $${\mathcal C}$$ under the multiplication by $$n$$-map on $${\mathcal J}$$ for sufficiently large $$n$$. As a consequence, Siegel’s Theorem holds true for $${\mathcal C}'$$ and then by the Chevalley-Weil Theorem also for $${\mathcal C}$$. This method has some similarities with the proof of Robinson and Roquette of Siegel’s Theorem which is based on non-standard analysis, but it is not a ‘standardization’ of their proof.
The author discusses in more detail the problem whether for a given non-singular projective curve $$\widetilde{{\mathcal C}}$$ over $$\overline{{\mathbb Q}}$$ and given $$A_1,\ldots A_r\in \widetilde{{\mathcal C}}(\overline{{\mathbb Q}})$$ there is a rational function $$\varphi\in \overline{{\mathbb Q}}({\mathcal C})\setminus\overline{{\mathbb Q}}$$ which violates (*). For $$r=2$$ such a function does not exist. For $$r\geq 3$$ this general problem is still open, but the author constructs such $$\varphi$$ in a few special cases.
The paper finishes with an appendix containing a letter of Coates to Cassels, in which Coates discusses yet another method to prove Siegel’s Theorem, for curves $${\mathcal C}$$ of genus $$1$$.
For the entire collection see [Zbl 1155.11004].

MSC:
 11G30 Curves of arbitrary genus or genus $$\ne 1$$ over global fields 11G32 Arithmetic aspects of dessins d’enfants, Belyĭ theory 11J68 Approximation to algebraic numbers 11J87 Schmidt Subspace Theorem and applications