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].

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].

Reviewer: Jan-Hendrik Evertse (Leiden)

##### 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 |