# Heath-Brown: Rational Points and Analytic Number Theory

Analytic number theory is often quite useful in questions on rational points on varieties. For example, using the circle method, weak approximation gives formulas of the type

where

What kind of asymptotic formulae can we expect when weak approximation fails?

Look at

where are linear forms over . This is the intersection of two quadrics in . In this case, the Hasse principle may fail, and weak approximation may fail. We take

We have the following theorem'': There is a modification of the Hardy-Littlewood formula in which

where , vanishes precisely when the Hasse principle fails, is built from information at a finite number of bad' places, and is easily calculable. To do this, use a `descent'' process followed by variation of the circle method.

Theorem. [H-B, Moroz] Let be coprime, with . Then the surface

has a nontrivial rational point.

To prove this, there are two ingredients. First, a result of Satgé: has a rational point if is prime, (proved by Heegner point construction). Second, takes infinitely many prime values .

In the other direction, analytic number theorists are often interested in rational points on varieties. We take the counting function

For instance, take ; there exists an asymptotic formula (Vaughan). In the case , analytic methods establish a local-global principle, but not an asymptotic formula. To handle this case, we would want:

This is known for any , and conjectured to be true for any , so it is reasonable to expect.

What about for ? We can show for . This variety has lines in trivial planes of the type , and no other lines if ; what other quadric or low degree curves can be found?

Proposition. [Green 1975] Any curve of genus 0 or in has constant for some , as soon as .

(Here .) This involves meromorphic functions and Nevanlinna theory.

Proposition. [Davenport 1963] Any cubic form in variables has a nontrivial integer zero.

This applies to an arbitrary cubic form; there is a better result for smooth forms with . Define a matrix with entries

and . Assume that has no nontrivial rational point; then any component of which has a rational point has .

Vinogradov's mean value theorem refers to the counting function of the variety defined by the equations

for , , count . This is a cone with vertex .

One can easily show for all ; if then is a permutation of , . In fact, for (Vaughan, Wooley).

For , we have .

Applications: Exponential sums, zero-free region for and the error term in the prime number theorem. So this question has several far-reaching implications!

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