Strong approximation for certain quadric fibrations with compact fibers

TL;DR

This paper proves strong approximation with Brauer-Manin obstruction for specific quadratic fibrations and extends classical quadratic form representation results over number fields.

Contribution

It establishes strong approximation for certain quadratic fibrations and develops the representation theory of quadratic Diophantine equations over number fields.

Findings

Strong approximation with Brauer-Manin obstruction holds for specific quadratic fibrations.

Representation theory of quadratic Diophantine equations is extended over number fields.

Equivalence between representability of quadratic polynomials and classical quadratic forms is demonstrated.

Abstract

In this paper, we will show that strong approximation with Brauer-Manin obstruction holds for certain quadratic fibration such that none of fibers satisfies strong approximation with Brauer-Manin obstruction. Moreover, we develop the representation theory of quadratic Diophantine equations and explain that the representability of quadratic polynomials is equivalent to the classical result of representability of quadratic forms with congruent conditions and extend the classical result over number fields.