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Points entiers et théorèmes de Bertini arithmétiques. (Integral points and arithmetic Bertini theorems). (French) Zbl 1020.11044

Ann. Inst. Fourier 51, No. 6, 1507-1523 (2001); corrigendum ibid. 52, No. 1, 303-304 (2002).
This paper is concerned with the problem of solutions in \(\overline\mathbb{Z}\) (the ring of algebraic integers) of systems of Diophantine equations and in particular with its translation into the language of schemes.
Let \(K\) be an algebraic number field and \(O_K=K\cap \overline\mathbb{Z}\) its ring of integers. Let \(B=\text{Spec}(O_K)\) and denote by \(Y\) a closed sub-scheme of \(P^n_{O_K}\), defined by homogeneous polynomials \(G_1,\dots,G_r\) in \(n+1\) variables, \(X_0,X_1, \dots,X_n\), and with coefficients in \(O_K\). Finally \(U\) denotes the complement of the closure of \(Y\) defined by \(X_0=0\).
The author is interested in the structure of the set \(U(\overline\mathbb{Z})\) of points of \(U\) defined over \(O_K\) and having values in \(\overline\mathbb{Z}\). That set is a bijection on the set of solutions \((x_1,x_2, \dots,x_n) \in\overline \mathbb{Z}^n\) of the system of equations defined by \[ \forall i\in\{1,\dots,r\}, G_i(1,x_1,x_2, \dots,x_n) =0. \] Given an integral, projective scheme \(Y\) over \(O_K\) and \(U\) an open subset of \(Y\), an integral point over \(U\) is defined in terms of closed subsets of \(Y\) of dimension 1 and contained in \(U\). There is a natural bijection between the integer points so defined and the quotient of \(U(\overline\mathbb{Z})\) by the action of the Galois group of \(\overline\mathbb{Q}\) over \(K\). A theorem of R. Rumely [J. Reine Angew. Math. 368, 127-133 (1986; Zbl 0581.14014)] affords a criterion for \(U\) to possess integral points, as follows. Suppose that the generic fibre \(Y_K\) is geometrically irreducible over \(K\). Then \(U\) admits an integer point if and only if the morphism \(U\to B\) is surjective.
The author then formulates an effective version of Rumely’s theorem, using Arakelov’s theory of heights, in which he obtains a condition for the existence of infinitely many integer points. The theorem so obtained is then used to prove the following result. Let \(A\) be an Abelian variety over \(K\) of dimension \(g\geq 1\) and let \(X\) be a model of \(A\) over \(O_K\). Let \({\mathfrak L}\) be an invertible, ample sheaf over \(X\) such that \(L={\mathfrak L}_K\) is symmetric on \(A\), let \(\widetilde h_L\) denote the Néron-Tate height relative to \(L\) and \({\mathfrak A}_\varepsilon\) the set of closed points, \(x\), of \(A\) such that \(\widetilde h_L(x)\leq \varepsilon\) and \(\{\overline x\}\in X\) is an integer point of \(U\). Then \({\mathfrak A}_\varepsilon\) is Zariski-dense in \(A\).
The paper concludes with an arithmetic analogue of a theorem of Bertini.

MSC:

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G50 Heights
11G35 Varieties over global fields
11G10 Abelian varieties of dimension \(> 1\)

Citations:

Zbl 0581.14014
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