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arXiv · 1112.2991

Strong approximation for the total space of certain quadric fibrations

Jean-Louis Colliot-Thélène·Fei Xu
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Abstract

We study equations in four variables (x,y,z,t) of the shape q(x,y,z)=P(t), where q(x,y,z) is an indefinite ternary quadratic form over the integers and P(t) is a polynomial in one variable with integral coefficients. If P(t) is not the product of a constant and the square of a polynomial, strong approximation holds for integral solutions (x,y,z,t). In the general case, we show that the integral Brauer-Manin conditions are the only obstructions to strong approximation. We actually study the analogous situation over an arbitrary number field. --- Nous étudions les équations à quatre variables (x,y,z,t) à coefficients entiers du type q(x,y,z)=P(t), où q(x,y,z) est une forme quadratique entière ternaire indéfinie sur les réels, et P(t) un polynôme à coefficients entiers en une variable. Lorsque le polynôme n'est pas le produit d'une constante et d'un carré de polynôme, nous établissons l'approximation forte pour les solutions de ces équations en entiers (x,y,z,t). Dans le cas général, nous montrons que l'obstruction de Brauer-Manin entière est la seule obstruction à l'approximation forte. Nous étudions la situation sur un corps de nombres quelconque.

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BibTeXRIS

Jean-Louis Colliot-Thélène, Fei Xu. 2011-12-13. Strong approximation for the total space of certain quadric fibrations. https://arxiv.org/abs/1112.2991

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