Strong approximation and descent

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) = NK/k(z): firstly for quartic extensions of number fields K/k and quadratic polynomials P(t) in one variable, and secondly for k = ℚ, an arbitrary number field K and P(t) a product of linear polynomials over ℚ in at least two variables. Finally, we illustrate that a certain unboundedness condition at archimedean places is necessary for strong approximation. © 2015 De Gruyter.