## 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 $$\varphi_j$$.

### MSC:

 11G35 Varieties over global fields 11G30 Curves of arbitrary genus or genus $$\ne 1$$ over global fields 14G25 Global ground fields in algebraic geometry

### Citations:

Zbl 1012.11051; Zbl 0754.11020
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