## 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 $$> 1$$

Zbl 0581.14014
Full Text:

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