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
Previous Page Next Page