8 introduction

Proposition IV.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 a1,a2,d1,

and d2, one has that

H1

(

Gal(k/k), Pic(Xk)

)

= /3 .

Further, there is a class α ∈ Br(X) with the following property. For an adelic

point x = (xν)ν , the value of ev(α, x) depends only on the component xνp0 .

Write xνp0 =: (t0 : t1 : t2 : t3). Then one has ev(α, x) = 0 if and only if

a1t0 + d1t3

t3

is a cube in

∗

p0

. Note that p0 ≡ 1 (mod 3) implies that only every third element

of

∗

p0

is a cube.

Observe that the reduction of X modulo p0 is given by

x3(a1x0 + d1x3)(a2x0 + d2x3) = x0

3

.

This means, there are three planes intersecting in a triple line. No

p0

-rational

point may reduce to the triple line. Thus, there are three different planes to which

a

p0

-rational point x may reduce. The value of ev(α, x) depends only the plane,

to which its component xνp0 is mapped under reduction.

For instance (cf. Example IV.5.24), for p0 = 19, consider the cubic surface X

given by

x3(x0 + x3)(12x0 + x3) =

3

i=1

(

x0 +

θ(i)x1

+

(θ(i))2x2

)

.

Then, in 19, the cubic equation

x(1 + x)(12 + x) − 1 = 0

has the three solutions 12, 15, and 17. However, in

19

, 13/12 = 9, 16/15 = 15,

and 18/17 = 10, which are three non-cubes. This shows that X( ) = ∅. It is easy

to check that X( ) = ∅. Therefore, X is an example of a cubic surface violating

the Hasse principle.

We construct a number of similar examples. For instance, Example IV.5.24 de-

scribes a cubic surface X such that

H1

(

Gal(k/k), Pic(Xk)

)

= /3 , but the gen-

erating Brauer class does not exclude a single adelic point. One would expect

that X satisfies weak approximation. Recall that, in similar examples, E. Peyre

and Y. Tschinkel [Pe/T] showed experimentally that there are three times more

-rational points than expected.

The historically first cubic surface that could be proven to be a counterexample to

the Hasse principle was provided by Sir Peter Swinnerton-Dyer [SD62]. We recover

Swinnerton-Dyer’s example (cf. Example IV.5.27) for p0 = 7, d1 = d2 = 1, a1 = 1,

and a2 = 2. L. J. Mordell [Mord] generalized Swinnerton-Dyer’s work by giving a

series of examples for p0 = 7 and a series of examples for p0 = 13. Yu. I. Manin men-

tions Mordell’s examples explicitly in his book [Man]. He explains these counterex-

amples to the Hasse principle by a Brauer class. We generalize Mordell’s examples

further to the case that p0 is an arbitrary prime such that p0 ≡ 1 (mod 3).

We conclude Chapter IV by a section on diagonal cubic surfaces. For these, the

Brauer–Manin obstruction was investigated in the monumental work [CT/K/S]