4 introduction

Furthermore, it turns out that the coeﬃcient τ has to be modified when

rk Pic(X) 1. There appears an additional factor, which today is called α(X).

This factor is defined by a beautiful yet somewhat mysterious elementary geomet-

ric construction.

Another problem is that the Hasse principle does not hold universally. Con-

sider the following elementary example, which was given by C.-E. Lind in 1940.

Lind [Lin] dealt with the Diophantine equation

2u2

=

v4

−

17w4

defining an algebraic curve of genus 1. It is obvious that this equation is non-

trivially solvable in reals, and it is easy to check that it is non-trivially solvable

in

p

for every prime number p.

On the other hand, there is no solution in rationals except for (0, 0, 0). Indeed, as-

sume the contrary. Then there is a solution in integers such that gcd(u, v, w) = 1.

For such a solution, one clearly 17 u. Since 2 is a square but not a fourth power

modulo 17, we conclude that

(has

u

17

)

= −1. On the other hand, for every odd prime

divisor p of u, one has(

v4)

−

17w4

≡ 0 (mod p). shows

(

17

p

)

= 1. By the low of

quadratic reciprocity,

p

17

= 1. Altogether,

(

u

17

)This

= 1, which is a contradiction.

One might argue that this example is not too interesting since, on a curve of genus 1,

there are relatively few -rational points to be expected. Thus, it might happen

that there are none of them without any particular reason.

However, several other counterexamples to the Hasse principle had been invented.

Some of them were Fano varieties. For example, Sir Peter Swinnerton-Dyer

[SD62] and L.-J. Mordell [Mord] (cf. our Chapter IV, Section 5) constructed exam-

ples of cubic surfaces violating the Hasse principle. A few years later, J. W. S. Cas-

sels and M. J. T. Guy [Ca/G] as well as A. Bremner [Bre] even found isolated

examples of diagonal cubic surfaces showing that behaviour. Typically, the proofs

were a bit less elementary than Lind’s in that sense that they required not the

quadratic but the cubic or biquadratic reciprocity low.

In the late 1960s, Yu. I. Manin [Man] made the remarkable discovery that all the

known counterexamples to the Hasse principle could be explained in a uniform man-

ner. There was actually a class α ∈ Br(X) in the Brauer group of the underlying

algebraic variety responsible for the lack of -rational points.

This may be explained as follows. The Brauer group of is relatively complicated.

One has, by virtue of global class field theory,

Br(Spec ) = ker

(

s:

p prime

/ ⊕

1

2

/ −→ /

)

.

Here, s is just the summation. The summand / corresponding to the prime

number p is nothing but Br(Spec

p

) while the last summand is Br(Spec ).

Let α ∈ Br(X) be any Brauer class of a variety X over . An adelic point

x = (xν)ν∈Val(

)

∈ X( )