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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
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