The circle method is the Hardy-Littlewood method (1920s), ``any method involving Harmonic analysis that counts solutions of Diophantine questions'', including Kloosterman methods.
Example. Consider a homogeneous polynomial of degree . We count
Let as as slowly as you like, . We look at
For , the minor arcs, then should be ``randomly'' behaved, so one tries to show: when . If true, then
In particular, this will not work for varieties which fail the Hasse principle. The basic techniques work for all number fields , or even for or other function fields.
Scope of the Circle Method
The circle method works with ``sufficiently many'' variables.
Proposition. [Birch 1957] Given forms , of respectively odd degrees , and provided that is large enough, then there exists a rational point on .
This method diagonalizes each of the forms, but at a great cost:
Example. For , one has (Davenport 1963); (Schmidt 1984). (Dietmann-W).
Proposition. [Birch 1962] Let be homogeneous of degree . Let . Then whenever , one has asymptotic to a product of local densities as before.
The difficulty of this result depends on the singular locus being reasonably small in dimension. This holds for any number field, and it is probable that this holds for a function field assuming the characteristic is sufficiently large.
Proposition. [Heath-Brown 1983, Hooley 1988] For , we have the Hasse principle for nonsingular cubic forms.
We now turn to some simpler situations.
Proposition. [Brudem-W] If , binary, homogeneous of degree , then is asymptotic to the product of local densities whenever
For a diagonal form , work by Hua, Vaughan, Heath-Brown, the same conclusion holds for
Presumably: should suffice for the method to work.
Simultaneous equations: we expect need variables for form of degree makes it look we need variables for forms of degree . For , ( ), the number of variables required is given by: if the forms are in general position, and , then we have an asymptotic formula. For diagonal cubics, . For diagonal cubics, one has the Hasse principle whenever (Brudeur, W).
Keys to Success
We have the major arcs
For a rational number, we have
For the minor arcs, we want to show . One has Weyl differencing: letting , we have
Finally, there is recent work of Heath-Brown and Skorobogatov: For , a norm form of degree , then the Brauer-Manin obstruction is the only one to weak approximation and the Hasse principle. One uses descent to , where the circle method gives weak approximation and the Hasse principle. One can generalize this to the case
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