Bad reduction of the Brauer-Manin obstruction

Abstract

We relate the Brauer group of a smooth variety over a p-adic field to the geometry of the special fibre of a regular model, using the purity theorem inétale cohomology. As an illustration, we describe how the Brauer group of a smooth del Pezzo surface is determined by the singularity type of its reduction. We then relate the evaluation of an element of the Brauer group to the existence of points on certain torsors over the special fibre; we use this to describe situations when the evaluation is constant, and situations when the evaluation is surjective. In the latter case, we describe how this surjectivity can be used to prove vanishing of the Brauer-Manin obstruction on varieties over number fields.

Failures of weak approximation in families

“…Under the assumption that the number field k contains a primitive cube root of unity, it follows from their work that H 1 (K, Pic Xη) ∼ = Z/3Z. In Lemma 2.1 we shall show that this continues to hold over any number field k. This takes care of Condition (7) and so all of the hypotheses of Theorem 1.6 are met. Thus, when ordered by height, 100% of diagonal cubic surfaces over k which are everywhere locally soluble fail weak approximation.…”

“…(3) the fibre of π at each codimension 2 point of P n has a geometrically reduced component; (4) H 1 (k, PicX) = 0; (5) BrX = 0; (6) H 2 (k, Pic P n k ) → H 2 (k, PicX) is injective; (7) either H 1 (K, Pic Xη) = 0 or Br X η / Br K = 0.…”

Obstructions to the Hasse principle in families

“…The definition is easily seen to be independent of the choice of the P v . A straightforward extension of [2,Proposition 7.3] shows that, if A ⊂ Br Y / Br k is prolific at any set S of places, then A gives no Brauer-Manin obstruction to the existence of rational points on Y .…”

“…To prove Theorem 1, then, it suffices to show that 100% of the fibres X P admit a set S of primes at which the relevant subgroup of Br X P / Br k is prolific. It was shown in [2] that, at least for v sufficiently large, the evaluation map at v corresponding to an element A of Br 1 X P only depends on the residue of A at v. To control these residues, we use the philosophy of [3]. Specifically, we generalise the following proposition.…”

“…The first introduces a method of associating a torsor to a finite subgroup of H 1 (T, Q/Z), for a scheme T . In [2,Section 5.3], this was approached by choosing a basis for a subgroup and then taking the fibre product of the corresponding torsors. The following lemma gives a coordinate-free way of achieving the same result.…”