Strong approximation and descent

Abstract

Abstract We introduce descent methods to the study of strong approximation on algebraic varieties. We apply them to two classes of varieties defined by P ⁢ ( 𝐭 ) = N K / k ⁢ ( 𝐳 ) {P(\mathbf{t})=N_{K/k}(\mathbf{z})} : firstly for quartic extensions of number fields K / k {K/k} and quadratic polynomials P ⁢ ( 𝐭 ) {P(\mathbf{t})} in one variable, and secondly for k = ℚ {k=\mathbb{Q}} , an arbitrary number field K and P ⁢ ( 𝐭 ) {P(\mathbf{t})} a product of linear polynomials over ℚ {\mathbb{Q}} in at least two variables. Finally, we illustrate that a certain unboundedness condition at archimedean places is necessary for strong approximation.