It has no solution in
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
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
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
The reader might
want to consult [Va] for a description of the method and references to the origi-
The exponent of the main term allows a beautiful algebro-geometric interpretation.
The anticanonical sheaf on a complete intersection of multidegree (d1, . . . , dr)
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
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
B and prove that
this factor makes the asymptotic formula compatible with direct products.