On sums of three squares

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

J

AMES

W. C

OGDELL

On sums of three squares

Journal de Théorie des Nombres de Bordeaux, tome

15, n

o

_{1 (2003),}

p. 33-44

<http://www.numdam.org/item?id=JTNB_2003__15_1_33_0>

© Université Bordeaux 1, 2003, tous droits réservés.

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

On

sums

of three

_squares

par JAMES W. COGDELL

RÉSUMÉ. Nous nous intéressons à la

_question

de savoir

_quand

un entier d’un _corps de nombres totalement réel est somme de trois carrés d’entiers du corps, et

_plus

_{généralement,}

s’il _peut être

représenté

par une forme

_quadratique

ternaire définie

_positive

et

entière sur le _corps. Dans un travail récent avec

Piatetski-Shapiro

et

_Sarnak,

nous avons montré que tout entier totalement

positif

et

sans facteur carré assez

grand

possède

une

_{représentation}

_intégrale

globale

si et seulement s’il en est de même localement _partout, résolvant ainsi pour l’essentiel le dernier cas ouvert du onzième

problème

de Hilbert. Dans cet

_article,

nous _exposons les idées de

la démonstration de ce résultat.

ABSTRACT. We address the

_question

of when an

_integer

in a

to-tally

real number field can be written as the sum of three

_squared

integers

from the field and more

generally

whether it can be

repre-sented

_by

a

_positive

definite

_integral

_ternary

_quadratic

form over

the field. In recent work with

_{Piatetski-Shapiro}

and Sarnak we

have shown that _every

_{sufficiently large}

_{totally positive}

_square free

_integer

is

_{globally integrally}

_represented

if and

_only

if it is

so

_locally

at all

_places,

thus

_{essentially resolving}

the

_remaining

open case of Hilbert’s eleventh

problem.

In this paper we

_give

an

exposition

of the ideas in the

_proof

of this result.

The

_question

of when an

_integer

is

_{representable}

as a sum of _squares has a

_long

venerable

history.

More

_generally,

Hilbert’s eleventh

problem

asks

(among

other

_things)

which

_integers

are

integrally represented by

a

_given

quadratic

form over a number field. The case of

_{binary quadratic}

forms

is

_equivalent

to the

_theory

of relative

_quadratic

extensions and their class

groups and class fields as

_{developed by}

Hilbert. For forms in four or more

variables the situation is

_quite

different and has been understood for some

time. The case of three variables has remained _open.

Manuscrit requ le 19 novembre 2001.

The author was _supported in part by the NSA. While at the Institute for Advanced Study,

where the _{original stages} of the work was _performed, the author was _supported in _{part by} a

grant from the Ellentuck Fund and in part the research was conducted for the Clay Mathematics Institute.

The

_problem

of which

_integers

in a number field k are

represented by

the genus of a

quadratic

form is

completely

answered

by

Siegel’s

mass

formula,

which

_gives

the number of solutions in terms of

_products

of local densities

_[23].

If there is

_only

one _{class in the genus} of

_q(x),

then this answers the

_question

of

_{representability}

for _q. If n &#x3E; 4

_{and q}

is indefinite

at some archimedean

place

of k then

Siegel

showed

by

analytic

methods

that a number is

represented by

one form in a _{genus iff it is}

represented by

all forms in the genus, thus

reducing

the

_{global representability question}

to local ones

[24].

These results were recovered and extended to indefinite

ternary

forms via

_algebraic

_{techniques by}

Kneser

_[12]

and Hsia

_[7]

_utilizing

spinor

genera. So we will restrict ourselves to the case of

_positive

definite

integral

forms over a

totally

real field k. If the number of variables is

at least

_five,

one can

proceed

either

by analytic

methods,

using

bounds

towards

_Ramanujan

for Hilbert modular

_forms,

or

_algebraic

methods

[8]

to _prove that there is an effective constant

_Cq

such that if a the

ring

of

_integers

of

_k,

is

_{totally positive}

and its norm

N(a)

&#x3E;

_Cq

then a is

represented by

q iff it is

represented

by

q for every

completion kv

of k. When the number of variables is four one must add a

_primitivity

condition on the

representations,

both

_locally

and

_globally

_[8].

_Recently,

in collaboration with I. I.

_{Piatetski-Shapiro}

and P.

_Sarnak,

we have established an

_analogue

of this result in the case of

_positive

definite

_integral

_ternary

_quadratic

forms

via

_analytic

methods

_[3].

Theorem. Let k be cc

_totally

real number

_field

and let

_q(x)

be a

positive

definite integrals

ternary

quadratic

form

over k. Then there is an

ineffective

constant

_Cq

such that

_{if a}

is a

totally positive

_square

free integer of

k with

N(a)

&#x3E;

_Cq

then a is

represented integrally by q

locally integrally

represented

over _every

completions kv of

k.

The result was

previously

known for k

= ~ by

Duke and Schulze-Pillot

[6, 18].

Of

_special

interest is the case of the

_ternary

form

+ ~2 +

x3

giving

the result on sums of _squares in a number field alluded to in the title.

Corollary.

Let k be a

_totally

real number

field.

Then there is an

ineffec-tive constant

_Ck

such that _every

_{totally positive}

_square

_{free integer}

a with

N(a)

&#x3E;

_Ck

is the sum

_of

three

_integrals

_squares

_iff

it is the sum

_of

three

integral

squares

locally for

each

place

v

_of

k.

Of course

over Q Legendre

has

_given

us the

_precise

answer: a E Z is the

sum of three _squares iff a is not of the form

_4’~(8m

+

_7).

Over a number

field, partial

results have been obtained

_{by algebraic}

methods

_by

Donkar

such a can be

represented

but do not

_give

the local to

_global

result we

present

here.

In this note I would like to describe our

proof

of this theorem in the

simplified

case of k

_totally

real of class number one.

This _paper is an

_expanded

version of the lecture I

_presented

at the XXII Journées

_{Arithmétiques}

2001 in Lille. I would like to thank the

_organizers

of JA 2001 for the

_opportunity

to

_speak

on this

_topic.

I would also like

to thank _my

_{collaborators,}

I. I.

_{Piatetski-Shapiro}

and P.

_Sarnak,

for

allow-ing

me to

_present

_{this summary of} our work here. In

particular,

I thank

P. Sarnak for

_reading

and

_providing

critical comments on an earlier version

of this note.

1. Theta series

We take k to be a

_totally

real number field

_having

class number one. Let

d =

(k : Q)

be the

degree

of 1~ _over

Q.

We let 0 be the

ring

of

integers

of k. Let V be a vector _space of dimension three over k

_equipped

with a

positive

definite

_integral

_quadratic

form

_{q (x) .}

We will let L =

C73

denote

the

_integral

lattice in V. So

_q(x)

E 0 for every x E L.

The

_proof

we will

_give

is

analytic

in nature. Hence we

_begin

with the

theta series associated to

_q(x)

and

_{L, Siegel’s analytic}

class invariant

_[23],

where T . This is a Hilbert modular form of

weight

3/2

for an

appro-priate

congruence

subgroup

r c

_8L2(O).

Its Fourier

_expansion

is

_given

by

where E

_Llq(x)

=

all

[

is the

_{representation}

number of a

_by

L.

There are two related theta series. Let

Spn(L)

denote the

spinor

_genus

of L and

_Gen(L)

_{the genus of L}

_{[18, 19]. (Whether}

we fix the lattice and

work with the _genus of the form or fix the form and work with the genus

of the lattice is all the same. Here we fix the

_{form q}

and _vary the lattice in

to be the

_weighted

average over _{the genus} of

_L,

_Siegel’s

_analytic

_genus

invariant

_[23],

where

_o(M)

_{is the order of the group} of units of

_M,

and

similarly

for

_Spn(L)).

By

the work of

_Siegel

_[23]

we know that the coefficients

_rq(T,

Gen(L))

are

_{given by}

a

_product

of local densities and is non-zero iff a is

_locally

represented

integrally

by

(q, Lv)

for all

_completions.

So we need to be able

to relate

_{rq(a, L)}

and

_rq(a,

_{Gen (L)).}

By

classical results of

_Siegel,

an

_{algebraic proof}

of which can be found

in

_Walling

_[27],

we know that both

and

are _cusp forms of

_weight

3/2.

Now there are two

_types

of _cusp forms of

weight

3/2.

Recall that we have the Shimura

correspondence

between _cusp

forms of

_weight

_3/2

for F and modular forms of

_weight

2 for an

appropriate

subgroup

r’ c

_PGL2(O).

If we denote

by

53~2(r)

the _cusp forms associated

to theta series attached to one dimensional

_quadratic

forms and its

orthogonal

complement

with

_respect

to the Petersson inner

_product

then

53~2(T)

consists of

_precisely

those _cusp forms that lift _{to cusp} forms in

52 (F’)

[25].

Following

the work of Schulze-Pillot

_(see

_{[19, 20]}

for related

_results)

we can conclude that

and

Fortunately

the Fourier coefficients of the forms in

_(r)

are

_quite

_sparse.

In

_fact,

it is known

_(see

_{[7, 12])}

that outside of an

explicitly

computable

finite number of

_{"genus exceptional"}

_square classes we have

So for square free a there are

only

a finite number of such _genus

exceptions

which we can avoid

by

taking

~x

sufficiently

large.

Hence for all but

finitely

many square free cx we have

From the

computation

of the local densities

_(see

_[19])

one can conclude

that for a

locally represented

a one has an ineffective lower bound

with the

_implied

constant

_independent

of a. The

_{ineffectivity}

comes from

an

_application

of the

_{Brauer-Siegel}

theorem and is the source of the

inef-fectivity

of our theorem. Thus our result will follow if we can

_produce

an

estimate for the Fourier coefficients

_a(a)

of

_{half-integral weight}

forms in

Sg/2

(r)

of the form

for some fixed 6 &#x3E; 0. This _program was carried out in the case of k

_{= Q}

by

Duke and Schulze-Pillot

_[6]

with a

_{key ingredient being}

estimates on

the Fourier coefficients of

_{half-integral weight}

forms due to Iwaniec

_[9]

and Duke

_[5].

2. Fourier coefficients and L-functions

By

now, a common _{way to} estimate Fourier coefficients of modular forms

of half

_{integral weight}

is to

_appeal

to

_{Waldspurger’s}

formula

_[26]

_relating

these coefficients to central values of the L-functions of the Shimura lift.

Waldspurger

established this relation

_only

for k

_{= Q}

but

_recently

his

re-sult has been

_generalized

to

_totally

real fields

_by

Shimura

_[22]

_using

the Shimura

_{correspondence}

and

_by

Baruch and Mao

_{using Jacquet’s}

relative

trace formula

_[I].

We will use Baruch and Mao’s version of this relation.

Note that it suffices _{to prove} our estimate for Hecke

eigenforms

since

there is

_always

a basis

_coming

from such. Let if denote the

cuspidal

repre-sentation of

_{SL2 (A)}

_{generated by}

our

_f

E Fix an additive

charac-ter 1f;

of

_kBA

such that 1f =

8(if,1/J)

is the Shimura lift of 7r to a

_cuspidal

representation

of

_PGL2(A).

If a is a _square free

_integer

of O and we let

then we know

[25]

where Xa is the

quadratic

character associated to the extension

Classically

xa is a _ray class character mod

(a).

Let cp

be the new form on

S5d

associated to the new vector in the space of 7r and let

a(a)

denote its

a-Fourier coefficient.

Let S denote the finite set of

_places

of k where 7r,

7i-,

or 1/J

is ramified.

Let

Soo

denote the archimedean

_places

of k and let

Sa

denote the set of finite

_{places v}

where 0. Let

be the finite

_{(or classical)}

L-function of 7r. Then the formula of Baruch and

Mao

_[1]

can be stated as:

where the are certain local constants

_given

as ratios

involving

local

norms, local Fourier

coefficient,

and local L-values. Similar formulas had

been

_given

earlier

_by

Shimura

_[22].

_{For v 0}

S Baruch and Mao can

_explicitly

compute

the

_{c, (a)}

and for v E

_S,

a finite set of

_{places, they}

can estimate

them as a varies. As a _consequence one

_gets

an estimate

with the

_implied

constant

_independent

of a.

Note that the

_convexity

bound on

L(l,

7r (9 as cx varies

(see

[10])

is

which would

_give

This is better than the Hecke

_bound,

but not sufficient for our _purposes. We

must beat the

_convexity

bound to obtain a first non-trivial estimate towards

Ramanujan

for the Fourier coefficients of

_{half-integral}

_weight

forms over a

totally

real number field.

3.

_Subconvexity

Let 1r be a

_{cuspidal automorphic representation}

of

PGL2 (A)

correspond-ing

to a

_holomorphic

Hilbert modular form of even

_weight

k =

(1~, ... , k) .

Let

_W(-r)

be the associated new form. Let _Xi be any _ray class character

modulo an ideal a. We will let _Xl also denote its associated idele class

char-acter. The

_key

to our

_proof

of the stated

_theorem,

and a result of interest

in its own

_right,

is the

following breaking

of

_convexity

for 0

_{Xl) in}

the conductor

_aspect

as _X, varies.

Theorem. We have

2uith the

_implied

constant

_depending

on E but

independent of

a.

Here

_{again by}

_{L(s, 7r}

(8)

_Xl)

we mean the classical

(or finite)

L-function

If we

_apply

this result to our

_previous

situation with _{X, = xa} then this will

_complete

the

_proof

of our first Theorem.

We would now like to describe the

_proof

of this

_Theorem,

still in the case

of class number one. We will work with the

L-function

in its additive form. To this

_end,

we write

where

_tl+

is the _group of

_{totally positive}

units in U = This is related

to the Fourier

_expansion

_{of cp}

where

_dK

is the different of

_K,

_by

=

We first use the

_approximate

functional

_equation

for

_L(s,

(see

for

example

[13, 17]).

This

_gives

an

_expression

of the form

as two sums of

_length

_{essentially X,}

where we have taken X =

N(a).

Here

Vi

and

_V2

are functions

_having

Vi(O)

=

1,

smooth,

and

_rapidly

_decaying

at

infinity. By using

a smooth

dyadic

subdivision it suffices to estimate sums

of the

_shape

where now W is smooth of

_compact

_{support say}

in the interval

(.1, 2),

so

concentrated near 1. There are

_{approximately}

log(N(a))

such sums _up

to

_N(a).

The crucial contribution for us will be

again

when X is of size

N(a).

Note that the _presence of the cutoff function W in

_J(xi)

forces

Q -

so that if we set

_{WV(x) = VX W (x)}

we have W is still

smooth with

_{compact support}

in

_(~,2)

and

where

Our estimate will

_{proceed by placing}

_S(Xl)

into a

family

and then

_using

arithmetic

_{amplification.}

For a

_general

_exposition

of these

_techniques

one can refer to the talk of P. Michel at the XXII Journees

_{Arithm6tiques}

2001

[13].

To

_get

an

_appropriate

_family

we need to work with a sum over the

full set of

_{totally positive integers.}

To this end we let F be a smoothed

characteristic function of a fundamental domain for the action of the

_totally

positive

units

_U+

on the

_hyperboloid

defined

_by

= 1 in

k~

=

Rd+

which satisfies

for _{every x} E

_k+

with

_N(x)

= 1. We extend this to all

totally positive u

by

setting

F(u)

= F

u

Then we can write

Note that both W and F act as cutoff functions - W

_cutting

off in

and F

_cutting

off in the

_"argument"

of _p.

We now

place

S(Xl)

in a

family.

We do not use the

family

of all

S(x)

where X

runs over all _{ray class characters} mod _a, which

_might

seem more

natural but could be _{too sparse.}

_Indeed,

the

_image

of

_U+

in

_(O/a)"

can be

large

[15].

Instead we use the

family

of all

S(X)

_{as X} runs over all characters

of the _group

_(0/a)x.

Let

_C(a)

_{denote this group} of characters. Then we

will _{consider the average value} of in this

_family

_E

_IS(x)12.

Note C(a)

that the

_length

of the sum is

_IC(a)l

which is

_{given by}

the

_generalized

Euler totient function

_~(a)

and is

_trivially

bounded

_by

_N(a)

and is at least of size _{unlike the group} of ray class characters.

In addition to

_averaging

over this

_family,

we will utilize the

_technique

of

arithmetic

_{amplification.}

To this end we will take an

auxiliary

_parameter

M which will be of size

X6

with 5 small. Its

_precise

value will be determined in the course of the

_argument.

We take a set

_fvl

of

_{totally positive integers}

which should be

_relatively

_prime

to a and all have norm bounded

_by

M.

There should be

_roughly

M of them and

_they

should be

_balanced,

in that for the archimedean

_{embeddings v each vv}

should be

_roughly

of the same

size. For each v we take a coefficient

c(v)

such that

I

= 1. We then

To obtain an estimate on our

_original

S(Xl)

we will take

arithmetically

defined coefficients

_c(v)

=

Xi 1(v)

thus

amplifying

the _term

IS(XI)12

by

_a

factor

of M2.

To utilize

_A,

we

expand

the norm _squares,

_interchange

the order of

sum-mation,

and

_perform

the character sum over

_C(a).

This

_yields

We

_split

this sum into two

_terms,

the

_diagonal

D and the off

_diagonal

OD,

where

and

The

_diagonal

term is estimated

_{simply using}

the

_Ramanujan

bounds for the known in this case

by

_Brylinski

and Labesse

[2],

_namely

I

and then

_analysing

the size of the sums determined

_by

the cutoff

functions. These

yield

The off

_diagonal

term is more

_interesting.

Let us write it as

where in

_{B (v1,}

_{v2, h)}

we have resummed over

~~

which

only

has an effect on

the cutoff functions F. The terms _v2,

_h)

are estimated

_using

several

for

_suitably

_large

a where s =

(s1, ... , sd)

_E

cJ

Q

D(s,

vl, v2,

h)

is a

type

of

Dirichlet series

where we have used a multi-index

_notation,

so t-s = where

.~1, - .. , .~d

are the

_images

of under the d

_embeddings

of k into R This Dirichlet series carries the arithmetic information in

_B(vl,

_v2,

_h).

The func-tion

_H(s)

is

_essentially

the Mellin transform of the cutoff functions.

H (s )

is

_relatively

_simple

to handle. It is

entire,

rapidly decreasing

in and can be estimated

_by

where as is common we have written _sj =

aj +

_itj.

The

_interesting

bit of the estimate is in the Dirichlet series

_D(s,

_{vl, v2,}

_h).

It is

_essentially

a Dirichlet series formed with

_products

of shifted Fourier

coefficients.

_Selberg

has shown how to

_approach

such Dirichlet series via Poinear6 series

_[21].

To this

_end,

let us set

_g(T)

= Then

there is a Poincax6 series

Ph(T, s)

such that when we

_compute

the Petersson inner

_product

of _g with

_Ph(s)

we find

One now

_expands

this inner

_{product spectrally}

via Parseval’s formula. If we let

fojl

be a suitable orthonormal basis of Maass _cusp forms then

where

_c(s)

is a similar

_expression

_involving

the continuous

spectrum

and is estimated in a similar manner. Sarnak has

_developed

a

general

method for

estimating

(g,

CPj)

(see

for

_example

_[16])

which in this case

yields

where rj =

(rj,l, - - - , rj,d)

is the

_spectral

_parameter

of

_{Oj -}

The term

(0j, Ph (s))

is

_expressed

in terms of the h-Fourier coefficient

_{pj (h)}

of

_~~

and the associated archimedean r-factors

_involving

the

_spectral

_parameters

_r~. We then estimate these

_using

the bounds towards

_Ramanujan

in both the archimedean and non-archimedean

_aspects

due to Kim and Shahidi

_[11].

The sum is then controlled

using Weyl’s

law. A similar

_type

of estimate can be found

[14]

and in the

appendix

of

[17].

In the final

_analysis,

these estimates

_give

that the Dirichlet series has an

_analytic

continuation to the domain

_Re(sj)

&#x3E; 2 + 1

and in this

_region

satisfies

where

_{the hj}

are the

images

of h under the d

embeddings

of k into IR. Note

that

_{the )}

in the

_boundary

of the domain of continuation comes from the

archimedean estimates towards

_Ramanujan

while

_{the )}

in the

_exponent

of

N (h)

is from the non-archimedean bound towards

_Ramanujan

of Kim and Shahidi.

Returning

now to our

_expression

of

B(V1, V2, h)

in terms of the inverse Mellin

_transform,

we can now shift the lines of

integration

to

_{Re(sj) =}

1/2

which in turn results in

When we combine D and OD and choose M =

X6

to

give

them the

same order of

_growth

in X we find that M =

X7~65.

Now

taking

X =

N(a)

to

_get

the dominant term from our

_partition

we

_get

an estimate of our

amplified

sum

We now take

c(v)

= to

amplify

the term _{we are} interested in.

Then

_estimating

this one term

_by

the entire sum we find

or

which

_{finally gives}

as desired.

Note that to our

knowledge

this estimate is better than the current best bounds even in the case k =

Q.

References °

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Department of Mathematics Oklahoma State _University Stillwater OK 74078 USA