On the order three Brauer classes for cubic surfaces

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On the order three Brauer classes for cubic surfaces

In a final section, which is purely theoretic in nature, we show that U tt actually

fixes four triplets. The corresponding four restriction maps are injections

/3֒→/3×/3. Together, each of the eight non-zero elements of the right

hand side is met exactly once.

2 The Brauer-Manin obstruction – Generalities

2.1. –––– For cubic surfaces, all known counterexamples to the Hasse principle

or weak approximation are explained by the following observation.

2.2. Definition. –––– Let X be a projective variety overÉand Br(X) its

Grothendieck-Brauer group. Then, we will call

ev ν : Br(X) × X(Éν) −→É/, (α, ξ) ↦→ inv ν (α| ξ )

the local evaluation map. Here, inv ν : Br(Éν) →É/(and inv ∞ : Br(Ê) → 1 2/)

denote the canonical isomorphisms.

2.3. Observation (Manin). —– Let π: X → SpecÉbe a projective variety

overÉ. Choose an element α ∈ Br(X). Then, everyÉ-rational point x ∈ X(É)

gives rise to an adelic point (x ν ) ν ∈ X(AÉ) satisfying the condition


ev ν (α, x ν ) = 0 .

ν∈Val(É)

2.4. Remarks. –––– i) It is obvious that altering α ∈ Br(X) by some Brauer class

π ∗ ρ for ρ ∈ Br(É) does not change the obstruction defined by α. Consequently, it

is only the factor group Br(X)/π ∗ Br(É) that is relevant for the Brauer-Manin obstruction.

ii) The local evaluation map ev ν : Br(X) × X(Éν) →É/is continuous in the

second variable.

iii) Further, for every projective variety X overÉand every α ∈ Br(X), there exists

a finite set S ⊂ Val(É) such that ev(α, ξ) = 0 for every ν ∉ S and ξ ∈ X(Éν).

These facts imply that the Brauer-Manin obstruction, if present, is an obstruction

to the principle of weak approximation.

2.5. Lemma. –––– Let π: S → SpecÉbe a non-singular cubic surface. Then,

there is a canonical isomorphism

δ: H 1 (Gal(É/É), Pic(SÉ)) −→ Br(S)/π ∗ Br(É)

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