Failures of weak approximation in families

Abstract

Given a family of varieties $X\to {\mathbb{P}}^n$ over a number field $k$, we determine conditions under which there is a Brauer-Manin obstruction to weak approximation for $100\%$ of the fibres which are everywhere locally soluble.

44 pages; there is a new application to integral points (Theorem 1.4) and Cor 1.5 recovers a well-known result concerning the Hasse principle

References in corpus (1)

• On the Brauer group of diagonal cubic surfaces

Cited by in corpus (11)

• Integral Hasse principle and strong approximation for Markoff surfaces
• Hasse principle for generalised Kummer varieties
• Obstructions to the Hasse principle in families
• Many cubic surfaces contain rational points
• Failures of the Integral Hasse Principle for Affine Quadric Surfaces
• Schinzel Hypothesis on average and rational points
• Second descent and rational points on Kummer varieties
• Rational points on del Pezzo surfaces of degree four
• On the proportion of locally soluble superelliptic curves
• Points of bounded height in images of morphisms of weighted projective stacks with applications to counting elliptic curves
• Local-global principles for semi-integral points on Markoff orbifold pairs