*Introduction*

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

where

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

One can handle the case for small by using the mean value theorem,

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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