brauer groups, tamagawa measures, and rational points

xviii
INTRODUCTION
extensionÉ(ζ p0 )/É. Fix the explicit generator θ ∈ K given by
θ := trÉ(ζ p0 )/K(ζ p0 − 1) = −2n + ∑
i∈(∗ p0 ) 3 ζ i p 0
for n := p 0−1
. Then, consider the cubic surface X ⊂ P 3É, given by
6
3 ( )
T 3 (a 1 T 0 + d 1 T 3 )(a 2 T 0 + d 2 T 3 ) = T0 + θ (i) T 1 + (θ (i) ) 2 T 2 .
Here, a 1 , a 2 , d 1 , d 2 ∈. The θ (i) denote the three images of θ under Gal(K/É).
Proposition III.5.3 provides criteria to verify that such a surface is smooth
and has p-adic points for every prime p. More importantly, the Brauer-Manin
obstruction can be understood completely explicitly. At least for a generic
choice of a 1 , a 2 , d 1 , and d 2 , one has that H 1( Gal(k/k), Pic(X k
) ) =/3.
Further, there is a class α ∈ Br(X ) with the following property. For an adelic
point x = (x ν ) ν , the value of ev(α, x) depends only on the component x νp0 .
Write x νp0 =: (t 0 : t 1 : t 2 : t 3 ). Then, one has ev(α, x) = 0 if and only if
a 1 t 0 + d 1 t 3
t 3
is a cube in∗ p0
. Note that p 0 ≡ 1 (mod 3) implies that only every third element
of∗ p0
is a cube.
∏
i=1
Observe that the reduction of X modulo p 0 is given by
T 3 (a 1 T 0 + d 1 T 3 )(a 2 T 0 + d 2 T 3 ) = T 3
0 .
This means, there are three planes intersecting in a triple line. NoÉp 0
-rational
point may reduce to the triple line. Thus, there are three different planes, a
Ép 0
-rational point may reduce to. The value of ev(α, x) depends only the plane
to which its component v νp0 is mapped under reduction.
For instance (cf. Example III.5.24), for p 0 = 19, consider the cubic surface X
given by
3 ( )
T 3 (T 0 + T 3 )(12T 0 + T 3 ) = T0 + θ (i) T 1 + (θ (i) ) 2 T 2 .
Then, in19, the cubic equation
∏
i=1
T (1 + T )(12 + T ) − 1 = 0
has the three solutions 12, 15, and 17. However, in19, 13/12 = 9, 16/15 = 15,
and 18/17 = 10 which are three non-cubes. This shows that X (É) = ∅. It is
easy to check that X (É) ≠ ∅. Therefore, X is an example of a cubic surface
violating the Hasse principle.
We construct a number of similar examples. For instance, Example III.5.24
describes a cubic surface X such that H 1( Gal(k/k), Pic(X k
) ) =/3but the