introduction 5

defines restrictions of α to Br(Spec ) and Br(Spec p) for each p. If the sum of all

invariants is different from zero, then, according to the computation of Br(Spec ),

x may not be approximated by -rational points.

As α ∈ Br(X) then “obstructs” x from being approximated by rational points, the

expression Brauer–Manin obstruction became the general standard for this famous

observation of Manin.

In the counterexamples to the Hasse principle, which were known to Manin in

those days, one typically had a Brauer class, the restrictions of which had a totally

degenerate behaviour. For example, on Lind’s curve, there is a Brauer class α such

that its restriction is independent of the choice of the adelic point. α restricts to

zero in Br(Spec ) and Br(Spec p) for p = 17 but non-trivially to Br(Spec 17).

This suﬃces to show that there is no -rational point on that curve.

In general, the Brauer–Manin obstruction defines a subset X(

)Br

⊆ X( )

consisting of the adelic points that are not affected by the obstruction. At least

for cubic surfaces, there is a conjecture of J.-L. Colliot-Thélène stating that

X(

)Br

is equal to the set of all adelic points that may actually be approximated

by -rational points.

Thus, X( )Br = ∅, while X( ) = ∅ means that X is a proven counterexample

to the Hasse principle. If X( )Br X( ), then we have a counterexample

to weak approximation. If Colliot-Thélène’s conjecture were true, then one could

say that all cubic surfaces that are counterexamples to the Hasse principle or to

weak approximation are of this form.

The Brauer group of an algebraic variety X over an algebraically non-closed field k

admits, according to the Hochschild–Serre spectral sequence, a canonical fil-

tration into three terms. The first term is given by the image of Br(Spec k)

in Br(X). Second, Br(X)/)Br(Spec k) has a subgroup canonically isomor-

phic to

H1

(

Gal(k/k), Pic(Xk) . The remaining subquotient is a subgroup of

Br(Xk)Gal(k/k).

It turns out that only the second and third parts are relevant for the

Brauer–Manin obstruction. The third one causes the so-called transcenden-

tal Brauer–Manin obstruction, which is technically diﬃcult. We will not cover

the transcendental Brauer–Manin obstruction in this book. The subquotient

H1

(

Gal(k/k), Pic(X

k

)

)

= 0 is responsible for what might be called the algebraic

Brauer–Manin obstruction.

In the cases where the circle method is applicable, the Noether–Lefschetz The-

orem shows that Pic(Xk) = with trivial Galois operation. Consequently,

H1

(

Gal(k/k), Pic(Xk)

)

= 0, which is clearly suﬃcient for the absence of the alge-

braic Brauer–Manin obstruction. This coincides perfectly well with the observation

that the circle method always proves equidistribution.

By consequence, in a conjectural generalization of the results proven by the

circle method, one can work with X(

)Br

instead of X( ) without mak-

ing any change in the proven cases. However, in the cases where weak ap-

proximation fails, this does not give the correct answer, as was observed

by R. Heath-Brown [H-B92a] in 1992. On a cubic surface such that

H1

(D.

Gal(k/k), Pic(Xk)

)

= /3 and a non-trivial Brauer class excludes two thirds