##
**On integral points on surfaces.**
*(English)*
Zbl 1146.11035

Let \(k\) be a number field and \(S\) a finite set of places including the archimedean places. The main theorem of the paper imposes geometric conditions on an algebraic surface \(X\) that force all integral solutions \(X(\mathcal{O}_S)\) to be located in a curve \(C \subset X\). Similar work was carried out by the authors in dimension one in [P. Corvaja and U. Zannier, C. R., Math., Acad. Sci. Paris 334, No. 4, 267–271 (2002; Zbl 1012.11051)]. The conditions appear natural when using the Riemann-Roch theorem to find zeros of large order along prescribed divisors. The proofs are not effective, but allow, in principle, quantitative estimations for the degree of the curve \(C\). As an application it is shown that an affine curve with at least five points at infinity has at most finitely many quadratic \(S\)-integral points.

Theorem: (Sketch of Main theorem) Let \(\tilde{X}\) be a geometrically irreducible projective surface defined over \(k\), and let \(X\) be an affine open set. Assume that \(\tilde{X} - X=D_1 \cup D_2 \cup \dots \cup D_r\), \(r \geq 2\), where no three of the \(D_i\) shares a common point and also the intersection matrix of the \(D_i\)’s satisfies certain conditions, then there is a curve on \(C \subset X\) containing \(X(\mathcal{O}_S)\).

The first thing to note is that the main theorem will follow if it can be proved that for every infinite sequence of integral points on \(X\), there is a curve defined over \(k\) and containing an infinite subsequence. Taking a sequence of \(S\)-integral points \(P_i\) we may assume that for each valuation \(v \in S\), the points \(P_i^v\) converge to \(P^v \in \tilde{X}(k_v)\).

The conditions on the intersection matrix for the divisors \(D_i\) allow to build a divisor \(D=\sum_i p_i D_i\) and a vector space \(V=V_N=\{\varphi \in k(X): \text{Div}(\varphi) + ND \geq 0 \}\) with a basis \(\varphi_1,\dots,\varphi_d\). For every \(v \in S\) the authors build linearly independent linear forms \(L_{1v},\dots,L_{dv}\) on the \(\varphi_j\), such that for some positive number \(\mu_v\) and for a suitable infinite subsequence of \(P_i\) we have \[ \prod_{j=1}^d | L_{jv}(P_i)| _v \ll \big(\max_j(| \varphi_j(P_i)| _v)\big)^{-\mu_v}, \] where the implied constant does not depend on \(i\). The main theorem is now an application of the subspace theorem [W. M. Schmidt, Diophantine Approximations and Diophantine Equations, Lectures Notes in Math. 1467, Springer-Verlag, New York (1991; Zbl 0754.11020), Thm. 1D’, p. 178].

The subspace theorem states that giving \(v \in S\) and \(d \geq 2\) linearly independent linear forms \(L_{1v},\dots,L_{dv}\) in the variables \(X_1,\dots,X_d\) with coefficients in \(k\), the solutions \((x_1,\dots,x_d) \in \mathcal{O}_S^d\) to the equation \[ \prod_{v \in S} \prod_{j=1}^d | L_{jv}(x_1,\dots,x_d)| _v \leq H^{-\epsilon}(x_1:\dots:x_d) \] lie in the union of finitely many proper subspaces of \(k^d\). The curve \(C\) we are looking for can be obtained as linear combination of the \(\varphi_j\).

Theorem: (Sketch of Main theorem) Let \(\tilde{X}\) be a geometrically irreducible projective surface defined over \(k\), and let \(X\) be an affine open set. Assume that \(\tilde{X} - X=D_1 \cup D_2 \cup \dots \cup D_r\), \(r \geq 2\), where no three of the \(D_i\) shares a common point and also the intersection matrix of the \(D_i\)’s satisfies certain conditions, then there is a curve on \(C \subset X\) containing \(X(\mathcal{O}_S)\).

The first thing to note is that the main theorem will follow if it can be proved that for every infinite sequence of integral points on \(X\), there is a curve defined over \(k\) and containing an infinite subsequence. Taking a sequence of \(S\)-integral points \(P_i\) we may assume that for each valuation \(v \in S\), the points \(P_i^v\) converge to \(P^v \in \tilde{X}(k_v)\).

The conditions on the intersection matrix for the divisors \(D_i\) allow to build a divisor \(D=\sum_i p_i D_i\) and a vector space \(V=V_N=\{\varphi \in k(X): \text{Div}(\varphi) + ND \geq 0 \}\) with a basis \(\varphi_1,\dots,\varphi_d\). For every \(v \in S\) the authors build linearly independent linear forms \(L_{1v},\dots,L_{dv}\) on the \(\varphi_j\), such that for some positive number \(\mu_v\) and for a suitable infinite subsequence of \(P_i\) we have \[ \prod_{j=1}^d | L_{jv}(P_i)| _v \ll \big(\max_j(| \varphi_j(P_i)| _v)\big)^{-\mu_v}, \] where the implied constant does not depend on \(i\). The main theorem is now an application of the subspace theorem [W. M. Schmidt, Diophantine Approximations and Diophantine Equations, Lectures Notes in Math. 1467, Springer-Verlag, New York (1991; Zbl 0754.11020), Thm. 1D’, p. 178].

The subspace theorem states that giving \(v \in S\) and \(d \geq 2\) linearly independent linear forms \(L_{1v},\dots,L_{dv}\) in the variables \(X_1,\dots,X_d\) with coefficients in \(k\), the solutions \((x_1,\dots,x_d) \in \mathcal{O}_S^d\) to the equation \[ \prod_{v \in S} \prod_{j=1}^d | L_{jv}(x_1,\dots,x_d)| _v \leq H^{-\epsilon}(x_1:\dots:x_d) \] lie in the union of finitely many proper subspaces of \(k^d\). The curve \(C\) we are looking for can be obtained as linear combination of the \(\varphi_j\).

Reviewer: Jorge Pineiro (Bronx)