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
= /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
is a cube in
. Note that p0 ≡ 1 (mod 3) implies that only every third element
is a cube.
Observe that the reduction of X modulo p0 is given by
x3(a1x0 + d1x3)(a2x0 + d2x3) = x0
This means, there are three planes intersecting in a triple line. No
point may reduce to the triple line. Thus, there are three different planes to which
-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
x3(x0 + x3)(12x0 + x3) =
Then, in 19, the cubic equation
x(1 + x)(12 + x) − 1 = 0
has the three solutions 12, 15, and 17. However, in
, 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
= /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]