Strong approximation for the total space of certain quadric fibrations

TL;DR

This paper investigates strong approximation for solutions to certain quadratic polynomial equations in four variables over integers and number fields, establishing conditions under which solutions are dense and identifying Brauer-Manin obstructions.

Contribution

It proves strong approximation holds for these equations when P(t) is not a perfect square times a constant, and shows Brauer-Manin obstructions are the only barriers in the general case.

Findings

Strong approximation holds when P(t) is not a constant times a square.

Brauer-Manin obstructions are the only obstructions in the general case.

Results extend to arbitrary number fields.

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 \'etudions les \'equations \`a quatre variables (x,y,z,t) \`a coefficients entiers du type q(x,y,z)=P(t), o\`u q(x,y,z) est une forme quadratique enti\`ere ternaire ind\'efinie sur les r\'eels, et P(t) un polyn\^ome \`a coefficients entiers en une variable. Lorsque le polyn\^ome n'est pas le produit d'une constante et d'un carr\'e de polyn\^ome, nous \'etablissons l'approximation forte pour les solutions de ces \'equations en entiers (x,y,z,t). Dans le cas g\'en\'eral, nous montrons que l'obstruction de Brauer-Manin enti\`ere est la seule obstruction \`a l'approximation forte. Nous \'etudions la situation sur un corps de nombres quelconque.