introduction 3

It has no solution in

6

except for (0, 0, 0, 0, 0, 0). This may be seen by a repeated

application of an argument modulo 7. A more formal reason is the fact that the

projective algebraic variety defined by (§) has no point defined over 7.

One might ask whether or to what extent solvability over

p

for every prime num-

ber p together with solvability in real numbers implies the existence of a ratio-

nal solution. This question has been very inspiring for research over many decades.

As early as 1785, A.-M. Legendre gave an aﬃrmative answer for equations of

the type

q(x, y, z) = 0,

where q is a ternary quadratic form. Legendre’s result was generalized to quadratic

forms in arbitrarily many variables by H. Hasse and H. Minkowski. The term “Hasse

principle” was coined to describe the phenomenon.

A totally different sort of examples where the Hasse principle is valid is provided

by the circle method originally developed by G. H. Hardy and J. E. Littlewood.

The circle method uses tools from complex analysis to study the asymptotics of

the number of points of bounded height on complete intersections in a very high-

dimensional projective space. It provides an asymptotic formula and an error term.

The main term is of the form

τBn+1−d1− ... −dr

for a complete intersection of multidegree (d1, . . . , dr) in

Pn.

The reader might

want to consult [Va] for a description of the method and references to the origi-

nal literature.

The exponent of the main term allows a beautiful algebro-geometric interpretation.

The anticanonical sheaf on a complete intersection of multidegree (d1, . . . , dr)

in

Pn

is precisely O(n + 1 − d1 − . . . − dr)|X . This means, when working with

an anticanonical height instead of the naive height, the circle method proves linear

growth for the -rational points.

The coeﬃcient τ of the main term is a product of p-adic densities together with a

factor corresponding to the Archimedean valuation.

Unfortunately, it is necessary to make very restrictive assumptions on the number

of variables in comparison with the degrees of the equations. These assumptions on

the dimension of the ambient projective space are needed in order to ensure that the

provable error term is smaller than the main term. One might, nevertheless, hope

that there is a similar asymptotic under much less restrictive conditions. This is

the origin of Manin’s conjecture.

However, as was observed by J. Franke, Yu. I. Manin, and Y. Tschinkel [F/M/T],

the main term as described above is not compatible with the formation of di-

rect products. Already on a variety as simple as

P1 ×P1,

the growth of the number

of the -rational points is actually asymptotically equal to τB log B. This may be

seen by a calculation, which is completely elementary.

Thus, in general, the asymptotic formula has to be modified by a log-factor.

Franke, Manin, and Tschinkel suggest the factor

logrk Pic(X)−1

B and prove that

this factor makes the asymptotic formula compatible with direct products.