Study of rational cubic forms via the circle method [after D.R. Heath-Brown, C. Hooley, and R.C. Vaughan]

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

J

EAN

-M

ARC

D

ESHOUILLERS

Study of rational cubic forms via the circle method [after

D.R. Heath-Brown, C. Hooley, and R.C. Vaughan]

Journal de Théorie des Nombres de Bordeaux, tome

2, n

o

_{2 (1990),}

p. 431-450

<http://www.numdam.org/item?id=JTNB_1990__2_2_431_0>

© Université Bordeaux 1, 1990, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Study

of

rational cubic forms

via

the

circle

method.

[after

D.R.

_Heath-Brown,

C.

_Hooley,

and R.C.

_Vaughan]*

par JEAN-MARC DESHOUILLERS

The circle method due to

_Hardy,

Littlewood and

_Ramanujan,

is a

pow-erful tool in the

_{study of Diophantine problems}

of the

_type

_{P(x, ..., x,q =}

_0,

when the number of variables is

_{large compared}

with the

_degree

of the

polynomial

P. The last incarnations of this method in

_particular

enabled Heath-Brown to show that _every

_non-singular

rational cubic form in 10 variables

_represents

_0,

_Hooley

to shew that 9 variables suffice when there is no local obstruction and

_Vaughan

to

_give

an

asymptotic expression

for

the number of

_{representations}

of an

integer

as a sum of 8

cubes,

as well as a

good

lower bound for the number of

_{representations}

as a sum of 7 cubes.

CONTENTS

Introduction

31

Heuristic

_{approach :}

the

_{singular integral}

1.1 Some

_computed

data

1.2

_Study

of the number of

_{representations}

1.3

_{Integral expression}

for the number of

_{representations}

1.4 Heuristic order of

_magnitude

of

§2

A

_p-adic

_{approach :}

the

_singular

series

2.1 _{Number of solutions of the congruence +} ... +

m~

N(mod q)

2.2

_{Interpretation}

of the

_singular

series

§3

Classical

_Farey

dissection :

_major

_arcs,

_singular

series and

integral

*

L’auteur remercie le Séminaire de Théorie des Nombres de Bordeaux et le Séminaire Bourbaki pour leur permission de _publier simultanément la version _française de ce texte

dans les actes du Séminaire Bourbaki

_(exp.

_720).

Manuscrit reçu le 4 mai 1990

3.1 Contribution of the first

_major

arc

(a

close to

0)

3.2 Contribution of the

_major

arcs : distribution

of powers

in arithmetic

progressions.

§4

Classical

_Farey

dissection : minor arcs

4.1

_Weyl

_upper bound : 13 cubes 4.2 Parseval

_{identity :}

9 cubes

4.3 Number of

_{representations}

as a sum of 8

cubes,

after

Vaughan

4.4 Number of

_{representations}

as a sum of 7 cubes.

§5

Non

_diagonal

cubic forms

5.1 A non

typical

situation : the case of norm forms

5.2 Extension of the

_major

arcs, after

Vaughan

5.3 Intermezzo :

_{Fa,rey-Kloosterman}

dissection 5.4 Cubic forms in 10

variables,

after Heath-Brown

5.5 Cubic forms in 9

variables,

after

_Hooley

Introduction

The

_problem

whether _every

_{positive integer}

is a sum of four _squares,

which _may

_already

have been considered

_by

the ancient

_Greeks,

has been studied

_by

_many a 111athematician in the seventeenth and

eighteenth

cen-turies,

and was

finally

solved

by Lagrange

in 1770. The interest in and the

difficulty

of this

_question

come from the

interplay

between the additive and

multiplicative

structures of the

_integers.

Similar

_conjectures

can be

easily formulated ;

most of the time

they

are

intractable,

but raise new mathematical

developements. Historically

important

are Goldbach’s

problem

(1742),

_every even

integer

is a sum

of two

_primes

_(at

that

_time,

1 was considered a

prime),

and

_Waring’s

problem

(1770),

every

integer

is a sum of at most 4 squares, 9

cubes or 19

biquadrates,

extended to

_higher

_powers in 1782.

Additive number

_theory

deals with those

_{questions ;}

we aim here to

present

the so-called circle method which has been introduced

_{by Hardy}

and

_Ramanujan

in

_1917,

and

_improved

upon

by Hardy

and Littlewood a

still used with

_benefit,

as well for

_getting

_{asymptotic evaluations,}

as we

shall see

here,

as for effective and efficient numerical estimates.

_Through

the -limited-

_study

of

_diagonal

forms

_(and

_essentially

of

_diagonal

cubic

forms)

we

perform

this

exhibition, suggesting

some

_steps

in the

proofs

of

the

_following

results

THEOREM 1.-

_{(D.R.Heath-Brown, 1982).}

Let

_F(x)

:=

F(xi, ...x,g)

be a non

singular

rational cubic form. For s

_larger

than or

_equal

to

_10,

there exists

x in

Q’ B {0}

such that

F(x)

= 0.

THEOREM 2.-

_(C.

_Hooley,

_{19$7).Let F(x)}

be a non

singular

rational cubic

form in 9 variables. The

_equation

_F(x)

= 0 has a rational non trivial

solution if and

_only

if it has a non

_singular

solution in each

p-adic

field

Theorem 3.-

_(R.C.

_Vaughan,

_{1985).Let rs(N)}

denote the number of

representations

of the

_integer

N has the sum of the cubes of

eight

_positive

rational

_integers.

As N tends to

_infinity,

one has

where

One

_conjectures

that the

_{non-singularity}

condition _may be raised in the

statement of Theorem ~.

_Davenport

showed in ~963 that _every rational cubic form in at least ~6 variables

_represents

_{0 ;}

even in the case of non

singular

forms this was the best known result before Heath-.Brown’s. In the case

_{of nonary forms,}

the Iocal condition can’t be

_{dispensed with,}

as Mordell

showed in ~936. Theorem 3 was one of

_{Davenport’s}

favorite

_conjectures

and

I was told that he

suggested

its

study

to aII of his research students. To end this

_{introduction,}

let us mention the work of

CoIIiot-Thélène,

Sansuc and Sir

_{Swinnerton-Dyer,}

1986

_(a

survey of their work is

presented

in

_{CoIIiot-Thélène,}

_1986).

_Attacking

the

_problem

_through

the

_algebraic

geometry,

they get

among other the

presently

best known results

concerning

systems

of

_quadratic

forms. As far as cubic forms are

concerned,

this

approach

leads to the

_{proof of}

the

_following,

which can

usefully

be

compared

with Theorem 2

_(less

_variables,

but more subtle

_geometric

conditions).

THEOREM 4.-

_{(J.-L.Colliot-Thélène}

et P.

_Salberger,

_~988).

be a

If X conta,ins a set of 3

conjugate singular

points,

and if X admits rational

points

over aII the

completions

then X admits a rational

point

over

K.

1 Heuristic

_{approach :}

the

_{singular integral}

1.1 Some

_computed

data

Let us look at a table which

gives,

for _every

integer

N _{up to}

40,000

the minimal number of summands needed to writer N as a sum of

positive

integral

cubes : the numbers 23 and 239 are the

only

ones for which 9 cubes are

needed ;

fifteen other

integers,

the

largest

of which is

8,402, require

8

cubes,

and for all the other

_integers

7 cubes suffices. Would this table be extended to

_1015,

, we would

get

the

feeling that,

from some

point onward,

4

cubes suoEce...

_Thus,

it seems easier to

_represent

large integers

than small ones.

1.2

_Study

of the number of

_{representations}

This

_aspect

was not taken into account in the studies

performed

during

the second half of the nineteeth

_century,

from Liouville who showed in 1859

that _every

_integer

is a sum of at most 53

_biquadrates,

to Hilbert who

_proved

in 1909 that _every

_integer

is a sum of a finite number of k-th _powers, and

hempner

who concluded a work

_{by Wieferich,}

_establishing

in 1912 that

every

integer

is a sum of at most 9 cubes.

The circle

_method,

introduced in the context of

_Waring’s

_{problem by}

Hardy

and

_Littlewood,

takes into account this

_ability

of

_{large integers}

to be

more

easily represented ;

to know whether the

integer

N can be

represented

as a sum of s

_integral

k-th _powers,

_they

tried to determine the

_asymptotic

behaviour of the number of

_{representations}

of N as a sum of k-th

powers.Thus, they

concentrated on

large integers,

hence the success of their method.

1.3

_{Integral expression}

for the number of

_{representations}

Let

us first fix some notation. k

_represents

an

integer

which is at least

2. We can restrict ourselves to the value 3.

N denotes an

integer,

which is assumed to tend to

_infinity.

For a

_given

_positive

_integer

_s, we denote

by

the number of

repre-sentations of N as a sum of s k-th _powers

The reader will

_easily

_give

a

proof

of relation

by appealing,

either to the

_{orthogonality}

of characters

_e(h.),

or to

integral

formula. We

_give

here a

probabilistic proof,

leaving

aside its inter-est for a

_given

explicit

computation

(Deshouillers, 1985).

we consider s

_independent

random variables

_{~~ , ... , X,q}

wi th common

law

_6(n),

where

_b(a~

denotes the Dirac measure at the

point

a, and the

normalizing

factor A is

equal

to +

IJ -1.

Each random

variable

_~~

follows the law A The law of the variable X :=

xf

+ ... +

X q

is the s-th convolution _power of that of each it is thus

_equal

to ~’9

_~~

_r,:(m)8(m),

where

_{r~ (m~}

is the number of _ways to

write m as a sum of s

integral

k-th _powers, each of which is at most

_{N ;}

the coefficient is

_readily

obtained from the characteristic function

(Fourier

series)

of X,

which is the s-th _power of that Formula 1.3.a.

follows from th at fac t an d th e trivial,

_{eq u aIi ty}

and

1.4 Heuristic order of

_magnitude

of

We

_give

a continuous

_{approximation}

to the

_problem

under consideration : let

_...Z,q

be s

_independent

random variables each of which is

_uniformly

distributed on

[0,

N~ ~k~,

and let us denote

by

Z the sum

Z~ + ... + ZR .

The

characteristic function

_(alias

Fourier

_transform)

of Z is the s-th power of

that

_by

inverse Fourier

_transform,

we

_get

the

density

ps (t)

of Z at

t:

The

_right-hand

side is called the

_{singular integral.}

When t =

N,

we

get

Thus,

the mean-value of

_r,,(N)

is _up to an Eulerian factor. It

is worth

_noticing

that the

_singular

_integral

does occur in the

asymptotic

formula for the number of

_{representations}

as a sum of 8 cubes

(cf

Theorem

3).

2 A

_{p-adic approach :}

the

_singular

series

Having explained

the

_occuring

of the

_{singular integral}

in Theorem

3,

we

turn to the

_{interpretation}

of the

_singular

series

_C~(N),

_according

to

_Hardy

and Littlewood’s

_terminology.

2.1 Number of solutions of the _congruence

_{m~ +...~mq -}

_{N(mod q)}

Let us denote

_by

M(q)

the number of solutions of the _congruence under

consideration. The

_equality

follows

_easily

from the

_{orthogonality}

relations

and is the

_counterpart,

in

_ZlqZ,

to the

_integral

formula 1.3.a.

By

rear-ranging

the

_right

hand side of 2.1.a

_according

to the value of

_{(r, q,}

we

get

-

-or

where

2.2

_{Interpretation}

of the

_singular

series

As it can be seen either

directly

from the Chinese remainder

theorem,

or

indirectly

through

2.~.b,

the function G is

_{multiplicative,}

and this enables

one to write the

singular

series

as an Euler

produet

By 2.1.b,

we thus

_get

Noticing

that

_{p~~ R-~ ~}

is the

_cardinality

of a

_hyperplane

in

_(l/pIZ)",

we _may

interpret

each factor in

_6(N)

as a

density

for the solutions of

To the reader who

_might

be

_tempted

to _{go further in this direction} we

recommand the _paper of

_{Lachaud, 1982,}

for an adelic formulation of the

circle method.

3. Classical

_Farey

dissection :

_major

_arcs,

_singular

series and

integral

We are now concerned with the

_computation

of

r,(N)

from the

_integral

formula 1.3.a. The main

_expression

in the

_integrand

is the

_exponential

sum

Its modulus is

_clearly

maximal when a is an

integer ;

we start

_by

_studying

the contribution to

_r,,(N)

of those a’s close to 0 in

_R/Z,

_or, what is the

same under the identification

of RIZ

to a

_{neighbourhood}

of 0 in

_R,

of those a’s close to 0.

3.1 Contribution of the first

_major

arc

(a

close to

₀₎

Since

_S(a)

is a continuous function

_{of a,}

and

_S(0)

= P

(this

equality

justifies

the

_clumsy

definition

_{of P),}

_analysis

allows us to evaluate

_S(a)

for

a close to zero.

Let us _suppose that

fN-1,

wheree is _any function of N

tending

to

zero at

_{infinity. By Taylor’s formula,}

we

_get

One should notice that the contribution to

_r.,(N)

of such an interval

around 0 has the same order of

magnitude

as the heuristic value.

Indeed,

we used

_{Taylor’s formula,}

which is somewhat weak

(don’t

we

teach it to our first _year students

?).

The Poisson’s summation formula

(when

do we teach it to our students

?)

is a much more efhcient tool. Let

and let

_f

denote its Fourier transform

_{( f (t)}

:=

f(z)e(zt)dz) ;

one has

hence

When lai

]

is small

_enough

_1/2),

the function

_e(axk

+

_vx~

does oscillate on

[0, P]

for

_0,

and the

_integral

f P

e(axk -~-

_vx)dx

is

small,

as can be seen

by integrating by

_{parts ;}

it is not difficult to show

and it is then a routine

_computation

to check

_that,

+

1,

the

con tri b u tion of th e in terval

is

_equivalent

to

_{p, (N) ,}

the mean number of

_{representations}

(cf

1.4.a and

1. 4. b) !

3.2 Contribution of the

_major

arcs : distribution of _powers in arithmetic

_progressions

In section

_2.2,

we rewrote the

_singular

series in terms of the number of solutions of the _congruences

_~~

+ ... +

N(

mod

q).

We wish to

explain

here how the mean value

G(q)

of Gauss sums occurs when

_studying

the contribution to

_r.,.(N)

of small intervals centered at rational

_points

with

q as their denominator.

Let us start

by estimating

S(a/q) ;

we have

There is no reason

why

the Gauss sum

~~_~

should

vanish,

and indeed it

_usually

does

_{not ;}

for

_example,

in the case when k = 3

and

_a/q

=

1/4,

it is

equal

_to 2. This

corresponds

_to

irregularities

in the

distribution of the k-th _powers in arithmetic

_{progressions. Hence,}

may have an order of

_magnitude

_comparable

with that of _We

_study

therefore the local behaviour of S around

_a/q

as we did around 0 : we

where 0 a _q,

(a, q) =

1 and

_q

Q.

We should mention that the

choice is at that

_stage

somewhat

_{irrelevant ;}

_generally

_Q

is to be taken

as a _power

_{of P, e.g. P}

itself.

important

to notice that

_{choosing Q}

greater

than P is

_possible,

but _very delicate : it is

_equivalent

to

_studying

the distribution of a set of P

_integers

in classes modulo an

integer larger

than P.

The Poisson’s summation formula allows us to

_approximate

Collecting

the contributions of the different

_major

_arcs, the union of which

is denoted

_by

_9R,

we

_get

PROPOSITION. For s &#x3E; 4k

_{(and s &#x3E;}

4 in the case of

cubes),

there exists an

A &#x3E; 0 such

_that,

when N tends to

_infinity,

one has

We recall that

4 Classical

_Farey

dissection : minor arcs

Although

it is the "core" in a traditional use of the circle

method,

we

treat this

_part

_briefly

which

_depends

in a fundamental _way on the

diagonal

nature of the forms

_implied

in

_{iVaring’s problem.}

The union of the

_major

arcs does not cover

R/Z ;

a

simple

_way to see

this is to

_compute

the

_{total length}

of the

_major

arcs. In the case of

cubes,

we have

a -

-(J _

and so the total

_length

of the

_major

arcs tends to 0 when ~1ï tends to

infinity

i We call minor arcs the

_{complementary}

set of the

_major

arcs,

the minor arcs and denote it

_by

m.

4.1

_Weyl

_upper bound : 13 cubes

The sum

S(a~,

for a

irrational,

has been considered under a "dual" look

S(a~/P

tends to 0. From this

_{point of view,}

that we do not

_{develop here,}

the result

_corresponds

to the uniform distribution of the _sequence

_(ank~

in

R/Z.

Weyl’s

method for

_majorizing

_leads,

in the case of

cubes,

to

the _upper bound valid for every E &#x3E; 0.

(For

k-th _powers, the

_exponent

is 1 -

_{2~ ~).}

From the _upper bound

_4.1.a,

one

easily

deduces

Combining

this result with 3.2.b

_{(contribution}

of

_major

_arcs),

we

_get

an

asymptotic

formula for the number of

_{representations}

as sums of s cubes as soon as

i.e. as soon as s is

_strictly

_larger

than 12. In the same _way, we

_get

an

asymptotic

formula for the number or

_{representations}

as sums of k

2k-l

+ 1

integral

k-th _powers.

4.2 Parseval relation : 9 cubes

Though

relation 4.1.a

_presents

some weakness in that its

_exponent

is

probably

not the best

_possible

one

(considering

the random walk with

_step

one would

_expect

_1/2

to be

_right

_exponent),

the real

_strength

of that formula is its

_uniformity

over m. In

_applying

4.1.a to 4.1.b we waste this

uniformity

since we look

_only

for an É°’ norm. Parseval’s relation

_provides

us

with an

_exponent

sl2

in 4.1.b for s

_{= 2,}

which is _common, and also for s =

4,

which is a little rnore subtle : the

identity

x‘~

+ y~ _

(x+ y~(x2 - xy+y2)

permits

one to

_majorize

_r2(n)

_by

twice the number of divisors of n, hence

the _upper bound

wich leads to an

_asymptotic

formula for the number of

reprentations

of an

integer

as a sum of s cubes as soon as

that is _{to say} as soon as s is

strictly larger

than

8, and,

since s is an

_integer,

it means

_only

that s should be as least 9.

In the ca.se of k-th _powers, Hua could

_{suitably generalize}

Parseval’s

re-lation and obtained the _upper bound

which leads to an

asymptotic

formula for

r,(N)

when s &#x3E;

2k

_{+ 1.}

We just briefly

mention here the works

_{by Vinogradov}

_(improvement

on

M7eyl’s

lemma for k

_12),

Heath-Brown

_(improvement

on Hua’s lemma

for k &#x3E;

_6),

_Vinogradov,

_Davenport,

_Vaughan

and others

_(solubility

of

_~~

+ ... +

zf

= N when s has the

order k log k,

without

estimating

4.3 Number of

_{representations}

as a sum of 8

cubes,

after

Vaughan

Let us _go back to Hua’s

inequality

4.2.b for cubes. It

implies

whereas the contribution of the

_major

arcs has order

P~’~.

As was noticed

by Hooley,

using

mean value results for divisor functions

permits

one to

replace

Pf

_by

a _power of

_log

_{P ;}

a result

_by

Hall and Tenenbaum even leads

to

a,nd therefore to

We can see how narrow is the

_margin

between _4.3.a and 4.3.b. To cover

_it,

Vaughan

introduces several dissections : on _m,

_according

to the

diophan-tine

_properties

of _a, and on the

integers

between 0 and P

(on

which the

summation for

_S(a)

is

_performed),

_according

to their aritmetical

proper-ties. He can

reinterpret

the new

diophantine equations

and save a

logarithm

through

the estimation of the number of their solutions.

_Upper

bounds of

type

4.3.b are then

suffisant,

and one can thus take as a suitable C in 4.3.c any

integer strictly

less than 1.

4.4 Number of

_{representations}

as a sum of 7 cubes This iterative method is the root of the lower bound

obtained

_by

_Vaughan

in 1987 for some c &#x3E; 0 and any

sufficiently large

N.

R. C. Baker and J. Brüdern

_recently

entended this result to

_diagonal

forms in 7 variables.

In

_1942,

Linnik could _prove the

_positivity

for

_{sufficiently large}

_N,

by

different considerations

_(ternary

_{quadratic forms,}

distribution of

_prime

numbers in arithmetical

_{progressions).}

An

_{asymptotic expression}

for

_{r7 (N)}

has been

_{conditionally given by}

Hoo-ley,

1984,

under the

_validity

of the Riemann

_hypothesis

for some

global

Hasse-Weil L-functions.

5 - Non

_diagonal

cubic forms

5.1 A non

typical

situation : the case of norm forms

As we have

just

_seen,

proving through

the circle method that _every

suf-ficiently

large

integer

is a sum of seven

_cubes,

is

_fairly

recent. The above-mentionned result

_{by Davenport concerning}

cubic forms in 16

_variables,

as

well as Theorem 1 and 2

_suggest

that

_general

forms are harder to handle. Theorem 4 illustrate a

possible

use of certain

geometrical

considerations. It

is also

_possible

to rriake use of some

algebraic features,

as in the

following

results,

the first of which deals with cubic forms in 7 variables

_(and

more

generally

with forms of

_{degree k}

in 2k + 1

_variables).

THEOREM 5. -

_(B.J.

_Birch,

H.

_Davenport,

D.J.

_Lewis,

_1962).

Let

_(resp.

K-2)

be a cubic number field and

(resp.

N2(X2,Y2,Z2))

the norm form

_expressed

with

_respect

to a

given

integral

bases. If the

equation

has a

non-singular

solution modulo _p for _every

prime

_p which divides

(the

product

of the discriminants of

_!1

and

_K2),

then it has

infiiitely

many solutions with rational

integral

arguments.

By

sieve

_methods,

H. Iwaniec has even obtained results

concerning

the

number of

_{representations by .some}

cubic forms in 6 variables. For the sake of

_simplicity,

we

_quote

_only

a

corollary

of this result :

THEOREM 6. -

_(H.

Iwaniec,

1977)

Let

_lilo

be a cubic Galois

extension ;

we

denote

_by

the norm form with

_respect

to a

_given

integral

basis.

(There

exists an

integer

D and a

family

H of classes mod D such that a

prime

number _p

_totaly

_splits

in K if and

_only

_{if p}

is

_congruent

to an element of

H module

_D).

Every sufficiently

large

even

integer

which is

_congruent

modulo D to

the sum of two elements of H can be

represented

as _Yl, +

y2 ~ z2 ~

wi th

_{in tegral}

_{argumen ts.}

5.2 Extension of the

_major

_arcs, after

_Vaughan

In the first four

_sections,

we consider

only

diagonal

forms. All that concerns

majors

arcs can indeed be extended to non

diagonal

forms,

thanks

to Fourier transform on 7L.4J. The first

really

curcial use of the

diagonal

aspect

occured in

_{Weyl’s inequality}

_{(formula 4.1.a).}

At the end of the

_70’s,

_Vaughan

noticed that the choice of the

_major

arc

_931o,l

we made in 3.1.c was

_only

_governed

_by

a will to

neglect

as _many

terms as

possible

in Poisson summation formula 3.1.a in order to

_get

3.1.b.

if a

larger

major

arc is

chosen,

the function +

_vx

does not

necessar-ily

oscillate any

longer,

and the number of terms that can’t be

neglected

increases with the size of the

_major

arc. A decent extension still

permits

to

control the in which each term is taken

care of

_by

the

_{stationary phase}

method

(expansion

around the

_point

where

kaxk-1

+ v

vanishes, leading

to a Fresnel

integral).

Global out

_{put :}

in

the case of

_cubes,

one

_gets

exactly

the

3/4

_exponant

of

Weyl

_upper

bound,

and for

_higher

_powers an

_exponant

which is at least

1,

and this has little value.

_However,

one should notice that

Vaughan’s

idea has been used for

biquadrates,

in a context where a certain numerical

subtlety

was looked

for.

_{(Deshouillers, 1985).}

_Vaughan’s

method

_permits

to write

_S(a~

as a

trigonometrical

sum

which, trivially treated,

leads to

_Weyl

_{upper ;} but as a further

_advantage,

we have the

possibility

_{of keeping}

that sum

_explicit,

_for

subsequent

cancellation. The

_reader,

whom I thank for still

_being

_aboard,

in 13 variables

_depends

on a

_{good knowledge}

of

_generalized

Gauss sums

(Deligne’s Theorem)

in the same _way as "13 cubes"

depends

on the mere

Weyl’s

upper bound.

Before

_explaining

how one

gains

3 extra

_variables,

1 mention the use of

Vaughan’s

method

_{by Cherly,}

_1989,

who obtained a

Weyl

_type

extimate for

cubic

_{trigonometrical}

sums on

F2 ~X~,

which is not

possible

to

_get

by

the

classical tools

_since,

in

_F2,

one has

3!

= 0.

5.3 Intermezzo :

_{Farey -}

Kloosterman dissection

Since we wish to

_{keep explicit}

the above-mentionned

_{trigonometrical}

sums for _any a E

_R/Z,

we need a

partition

of

R/Z

consisting

of

major

arcs. When

studying diagonal

quadratic

forms in four

variables,

Kloost-erman met the same

difficulty

over

sixty

_{years ago. The solution he gave}

turns out to be one of the

keys

to Theorems 1 and 2. it is worth

men-tionning

that Kloosterman himself re-used it to obtain the first non trivial

upper bound for Fourier coefficients of

holomorphic

modular cusp

forms ;

polished by

Peterson,

this idea was to be further

_{developped by}

_Selberg,

Kuznecov,

Iwaniec et al. for

_studying

Kloosterman sums on _average

(via

non

holomorphic

_cusp

forms),

introducing

what

Hooley

rediscover in a

dif-ferent

_context,

and calls "double Kloosterman reffinement" .

To the

_Farey

sequence of order

Q,

i.e. the set of rationals ~ in

R/Z

with

(a, q)

= 1

and q Q,

Kloosterman connects the

partition

5.3.a

where

5.3.b

the

_neighbours

_{of §}

in the

_Farey

_sequence of order

_Q

have been denoted

.

The

_challenge

is to determine the

_endpoints

of

_~~~,~~

from the mere

knowl-edge

of a

and q !

In that

aim,

we use the fundamental

_property

of

_Farey

This

_implies

where nor a’ nor a" are

explicity

refered to.

It remains to _express

_q’

_(and

_similarly

_q")

in terms of a

and q only.

We

first notice that

_q’

is well localised in an interval

of lenght

_{q ;} we have indeed

(the

lower bound comes from the fact that £ and

2013

are

neighbours

in the 17 q

Farey

sequence of order

Q,

so that

° °’,

can’t

belong

to

it).

q q

Kloosterman then notices that relation 5.3.c

_permits

to localise

_q’

modulo

q ;

denoting by 7i

the inverse of a

modulo q,

we have

Both relations 5.3.d and 5.3.e

_perfectly

determine

_q’,

thus the left

_endpoint

of

_~,~~~~

in terms of a

and q only.

The

right

endpoint

is treated in a similar

way.

An

_important

and

_annoying

_aspect

of the

_partition

of

_R/Z

considered in

5.3.a is that the arc

~~~,,~

related to the same _q have different

lengths.

In

order to treat at the same time their contributions to the

_integral

_1.3.a,

Kloosterman

_expands

their characteristic functions in Fourier

_series,

thus

introducing

a term where

_~~

is an additif character modulo _q. The

simplest

case leads to the so-called Kloosterman sums

for which he

_gives

a nontrivial _upper bound.

5.4 Cubic forms in 10

_variables,

after Heath-Brown

Actors are now on the

stage !

Let F be a cubic form and P a

_positive

gives

the number of solutions of the

_equation

_{~(x) =}

_0,

under the condition

llxll,,

P. If we show that

rF(P)

&#x3E; 1 for a certa,in P

(for

example

in

getting

an

asymptotic expression,

or an

asymptotic

lower bound for

rF(P)),

we deduce that the

equation

F(x)

= 0 has non trivial solution.

We then use the

Farey-Kloosterman

partition

5.3.a,

with the

order Q =

p3/2

_(one

should

_keep

in mind the remark formulated after definition

_3.2.a):

on each

major

_arc, the sum

p e (,i5F(x» is

transformed

through

the

use of the Poisson summation formula. The contributions of those

major

arcs related to the same denominator are then

gathered

by

Kloosterman’s

method.

_{Output : we just}

ha,ve to evaluate the sums

On the _map, the road is clear

_(Deligne

with us

!),

but on the

ground,

the situation is more

rocky

and

bushy.

Here are two main difficulties that

Heath-Brown ha.s to overcome :

- With the

help

of

_Katz,

he

_gets

the _upper bound

_{0(p~’~+~)l2~}

for the modulus of the sum

Su

(p; b),

which

correspond

to the maximal

expected

cancellation. The

_difhculty

comes from the term 7 in

5.4.b,

which makes

more intricate the

geometrical

nature of the

_variety

on which the sum is

performed

(prism

with

_hyperbolic

_base),

and

_{consequently majorizing}

_{1 Su. 1}

is

_harder,

even in the best case when F is

non-singular.

- In

principle,

it is easier to deal with sums

b),

for É &#x3E;

_{1 ;}

un-fortunately,

the maximal

_expected

ca.ncellation in

_{Su. (pf ;}

_b)

is not

_a.lways

observed. Heath-Brown shows that this cannot occur to often

_(one

can also

consult S.

_Cohen,

₁₉₇₉₎

and _proves

_that,

on _average, is an

admissible upper bound for

1

with

arbitrary

E.

Indeed,

instead ofthe

_integral

_5.4.a,,

Heath-Brown uses the

weighted

one

where the smooth

_weight

w lives in better

harmony

with the Fourier

trans-form than the characteristic function of the box

_llxll

~?.

5.5 Cubic forms in 9

_variables,

after

_Hooley

This section has with the

_previous

a

relationship

similar to that which

correct section 4.3

_(sums

of 8

_cubes)

with section 4.2

_(sums

of s cubes for

A first effort consists in

_pursuing

Heath-Brown method in order to treat

sums

S1,. (k; b)

with the same

efficiency

when k is the _square of a

squarefree

number,

as when k is itself

squarefree ;

the case of

_higher

_powers can be

elementary

treated in most cases. In this _way, a factor of the order

p2

_may

be won.

°

A few _powers of

_log

P are

_gained

as well : some

_by

choosing

a

_compact

supported

weight

w, some others

by

smoothing

the ends of the arcs

~nn,~~

defined in 5.3.b.

The final thrust needs the

_complicity

of Katz. A delicate estimate of four

power moments

of trigonometrical

sums allows to

_get,

for a

family

of

prime

numbers with

_{positive density,}

_upper bounds of the

_shape

with a constant

strictly

less tha,n 1! On _average over the denominators of

Farey points

(cf.

the introduction to section

_5.3),

one saves a _power of

log P, small,

but crucial.

Let us last mention current works

_{by Hooley, concerning}

_nonary cubic forms for which

_singular

loci of dimension zero are

_{only linearly independant}

double

_points.

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33405 Talence _Cedex, FRANCE. dezou@ frbdxll. bitnet