Arithmetic Gevrey series and transcendence. A survey

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

Y

VES

A

NDRÉ

Arithmetic Gevrey series and transcendence. A survey

Journal de Théorie des Nombres de Bordeaux, tome

15, n

o

_{1 (2003),}

p. 1-10

<http://www.numdam.org/item?id=JTNB_2003__15_1_1_0>

© Université Bordeaux 1, 2003, tous droits réservés.

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

Arithmetic

_Gevrey

series and transcendence.

A

_survey

par YVES

ANDRE

RÉSUMÉ. Nous _passons en revue les

principaux

résultats de la

théorie des séries

_Gevrey

de _type

_{arithmétique}

introduite dans

[3] [4],

leurs

_applications

à la

_{transcendance,}

_{ainsi que}

_quelques

conjectures

diophantiennes

sur la sommation de séries

divergentes.

ABSTRACT. We review the main results of the

_theory

of

arith-metic

_Gevrey

series introduced in

_{[3] [4],}

their

_applications

to

transcendence,

and a few

_{diophantine conjectures}

on the

sum-mation of

_divergent

series.

1. Introduction:

_ubiquity

of

_Gevrey

series in

_complex

_analysis

1.1. Let us

_begin

with a classical definition.

~.1.1. I3efiniti®n. Let s be a rational number. A formal _power series

with

_complex

coefHcients is

_Gevrey

_of

order s if and

_only

if the associated

series

has a non-zero radius of _{convergence, i. e.} if and

_only

if

It is

_{Gevrey of precise}

order s if and

_only

if

_fhl

has a

_finite

non-zero

radius of _convergence.

1.1.2. Remarks.

_i)

for s =

0,

a

Gevrey

series is

_just

a

_convergent

series.

ii)

Gevrey

series of order s &#x3E; 0 are in

_{general divergent.}

iii)

For s

_0,

_Gevrey

series define entire functions. More

_precisely,

it is not diff.cult to see that

_f

is

_Gevrey

of order s if and

_only

if it is entire.

of

exponential

ord er

_1/lsl.

1.2. At the

_begining

of the 20th

_century,

it was discovered that

_Gevrey

series are

_remarkably

_ubiquitous

in

_{complex analysis.}

This was

_emphasized

by

M.

_Gevrey

_[8].

For

_instance,

it turns out that any formal power series

arising

in the

asympotic expansion

of _any solution of a linear or non-lineax

analytic

dif-ferential

_equation

is

_Gevreyl

of

_precise

order s for some rational number s.

Gevrey

series also occur in the context of

_singular

_{perturbations,}

differ-ence

_{equations ...}

Besides their

_{ubiquity, divergent}

_Gevrey

series have the remarkable

prop-erty,

in _many natural

_analytic

_contexts,

of

_being

summable in an

essentially

canonical _way, in suitable sectors. This is the

_{starting point}

_(cf.

_[14])

of

the extensive

_theory

of

_{summability/ multisummability/}

_{resurgence. We}

refer to

_[11]

for an

_inspiring

overview.

1.2.1.

_Examples.

_i)

The

_Airy

function is

Its

_{Taylor expansion}

at the

_origin

is

Here both formal _power series in x are

_Gevrey

_{of precise}

order

_-2/3.

Its

_{asymptotic expansion}

at _c~ca, in a suitable sector bissected

_by

the

_positive

real

_axis,

is

_{given by}

Here both formal power series in

1/~

are

Gevrey of precise

order

~-~/~.

(The

Pochhammer

_symbol

_{(~z) ~}

stands for

_{a (a}

+

_{1 ) ... (rz}

+ n -

_1)).

ii)

The Barnes

_{generalized hypergeometric}

series

_{pFq-1 (xT)}

are

Gevrey

of

_precise

order s =

~-~.

ithe fact that any formal power solution of a non-linear _analytic differential _equation is always Gevrey was _already known in 1903 _(9; and, for the precise order - in the algebraic linear case

2. Arithmetic

_Gevrey

series

2.1. The idea of arithmetic

_Gevrey

series came out from an

empirical

obser-vation :

_{leafing through}

the classical treatises on

_{special functions,}

one

re-marks that the

_Gevrey

series

_occurring

in

_Taylor

or

asymptotic expansions

of classical

_special

functions have

_algebraic

or even rational

_{coefficients,}

and that the

_growth

of the denominators of these coefficients is

_inversely

proportional

to the archimedean

_growth.

This observation led to the

_following

definition:

2.1.1. Definition. Let s be a rational number. A formal _power series

is an arithmetic

_Gevrey

series

_of

order s if and

_only

if

3C &#x3E;

_{0, dn}

&#x3E;

_0,

_{Maxleonjugates}

I

2.1.2. Remarks.

_i)

Since s E

_Q,

one has n!s E

_Q,

so that the

conjugates

and common denominators of

finitely

many 1’n

’s are well-defined.

ii)

Infinite2

arithmetic

_Gevrey

series of order s

_give

rise to

_Gevrey

series of

precise

order s for _any

_embedding

_Q

~ C.

iii)

The arithmetic

_Gevrey

series of order s form a

_subalgebra

stable under derivation and

_integration.

2.2. The above observation

_(about

_growth

of

_{denominators,}

_2.1)

becomes less

_surprising

if one remarks that the coefficients of the

Gevrey

series in

question

are

generally expressed

in terms of Pochhammer

_symbols

with rational

_parameters.

One can then

_apply

the

following

lemma which _goes

back to

_Mayer

and

_Siegel

_[13]:

2.2.1. Lemma. Let _{al, ... , ap,}

_{b1 ... ,}

_bq

be rational numbers

_(dnd

not neg-ative

_integer.).

Then

2.2.2.

_Examples.

_i)

_By

the

_lemma,

the series

_occurring

in the

_expansion

of

_Ai(x)

at

_0,

_namely

are arithmetic

Gevrey of

order

-2/3.

The series

occurring

in its

asympotic

expansion

at _+oo,

_namely

are arithmetic

_{Gevrey of}

order

_+2/3.

ii)

The Barnes

_generalized

_{hypergeometric}

series with rational

parametes

are arithmetic

_Gevrey

of order s =

7.

iii)

Any

series which is

_algebraic

over is arithmetic

_Gevrey

of order 0

_{(Eisenstein).}

More

_{generally, Siegel’s}

G-functions

_[13]

are

nothing

but

holonomic arithmetic

_Gevrey

series of order 0.

Here and in the

_sequel,

holonomic means: solution of a linear differential

equation

with coefficients in

_C(x).

3. Structure of differential

_operators

_annihilating

arithmetic

Gevrey

series 3.1. Let us consider a linear differential

_operator

At _every

_{point ~}

which is not a

singularity

i. e. not a

pole

of _{any cp} admits a basis of

_analytic

solutions at

_~.

The converse is not

_always

_true,

but when it

_holds,

the

_singularity

is called

trivial3.

On the other

_{hand, if ~}

E C is _any

_singularity

of _cp, it is known that there exists a basis of "formal solutions" which are finite linear combinations

where ai E

C, ki

is a

non-negative

integer,

Qi

is a

polynomial,

and ui is a

Gevrey

series of

_(some)

_precise

order s E

_Q.

When no non-zero

polynomial

~72

occurs4,

one _says

_{that ~}

is a

regular

singularity.

In that _case, the

_ui’s

are

Gevrey

series of order

0,

i. e.

convergent

(Frobenius’ theorem).

3traditionally,

the _expression "apparent singularity" is reserves for a _{point ~} where there is a

basis of _meromorphic, not _{necessarily homolomorphic,} solutions.

3.2. The _presence of

_Gevrey

series of certain orders _among the solutions of _cp

_gives

some information about the "formal structure" of _cp at the

_given

point

~,

but

_{nothing beyond.}

We shall _see, in

_strong

_contrast,

that the _presence of arithmetic

_Gevrey

series in the solutions of _cp at some

_point

determines somehow the

global

behaviour of _cp. In

_{particular, together}

with the remark at the

_beginning

of

_§2,

this

_provides

an "arithmetic

_explanation"

for another

_elementary

observation which can be made when

_{leafing through}

the treatises on

_special

functions:

_namely,

that the classical linear differential

_operators

_occurring

there are of two

_types:

either fuchsian

_(the

"non-confluent

_type"),

or with

two

_{singularities,}

0 and _oo, one of them

being regular

(the

"confluent

type"

or

_"Hamburger

type" ) .

3.3. To _express our main result in the most economic _way, it is

conve-nient to

_consider,

besides arithmetic

_{Gevrey series,}

finite

_{ramifications,}

logarithms,

and transcendental constants. This leads to the

_following

3.3.1. Definition. An arithmetic

_{Nilsson-Gevrey}

series of order s E

_Q

is

a finite linear combination

where

_Xi

E

_C,

ei E

Q, ki

E

_N,

and ui is an arithmetic

_Gevrey

series of

order s.

They

form a

C(x)-algebra (stable

by

derivation and

integration)

denoted

by

NGAIXI,

in

_[3].

3.3.2.

_Example.

The

_{Taylor expansion}

at 0 of the

_Airy

function

_belongs

to The factor of

_{exp(-2/3x3/2)}

in the

_asymptotic

_expansion

at +oo

_belongs

to

3.4. Let _y be a holonomic arithmetic

_{Nilsson-Gevrey}

series of order s

(we

assume for

_{simplicity that y}

is not a

polynomial).

Let

Dy

be a

non-zero-element of

_C(x)[lxJ

of minimal

_degree

in

_d

such that

Dy (y) =

0

(such

a differential

_operator

is

_unique

_up to left

_{multiplication}

_by

an element

of

3.4.1. Theorem.

₁₎

Assume s = 0. Then

Dy

is

_fuchsian,

i.e. has

_only

regular

singularities

(even

at

_oo).

Moreover, for any ~’

E

(a,

_Dy

has a basis

_of

solutions in

_{NGAIX - fl)o}

and a basis

_of

solutions in

2)

Assume s 0. Then

_Dy

has

only

two non-trivial

_{singularities:}

0 and 00

(in

other words:

_Dy

has a basis

_{of analytic}

solutions at _any

_{point 0}

_0,

_(0).

i)

0 is a

_{regular singularzty}

with rational

_exponents,

and

_Dy

has a basis

of

solutions in

_NGAfxl,,

ii)

oo is an

irregular singularity,

and

_Dy

has a basis

of

solutions

of

the

form

fi.exp«(ix-1/S)

with

_fi

E

_~’i

E

Q.

3)

Assume s &#x3E; 0. Then

_Dy

has

_only

two non-trivial

_{singularities:}

0 and

00 . Moreover:

i)

0 is an

_{irregular singularity,}

and

_Dy

has a basis

_of

solutions

_of

the

form fi.

exp«(ix-l/S)

with

_fi

E

_{NGAIXI, ,}

_(i

E

Q,

ii)

oo is a

_{regular szngularity}

with rational

_exponents,

and

_Dy

has a basis

of

solutions in

The theorem is

_proved

in

_[3].

The

_proof

involves an arithmetic

_study

of

the

_{Fourier-Laplace}

transform

_(which

reduces

₂₎

and

₃₎

to

_{1) ).}

The main feature of this theorem is that an arithmetic

_property

of one

solution at one

point

accounts for the

_global

behaviour the

_operator.

In

particular,

whenever arithmetic

_Gevrey

series

_of

order s occur at

0,

then

arithmetic

_Gevrey

series

_of

order -s occur at oo, and

conversely.

The

Airy

differential

operator

=

d2 / dx2 - x

offers _a

typical

illustration of this

phenomenon,

yvith s =

-2/3.

4. On

_special

values of arithmetic

_Gevrey

series

4.1. Let K be a number

field,

and let

f

E

_xJ

be a holonomic arithmetic

Gevrey

series

_of

order s E

_Q.

We may assume that

_{D f}

E

_d/dx].

We shall

_study

_separately

the cases s

0, s

&#x3E; 0 and s = 0.

4.2. The case s 0. Recall that for _any

embedding v : K

y.

_{C, f ~,}

defines an entire function of

exponential

order -1/s.

In

particular,

it can

be evaluated at _any

_{point ~}

E I~* .

4.2.1.

_Corollary.

Assume that

_for

_every

_complex

_embedding

_v,

_{f~, (~)}

= 0.

Then ~

is a

_singularity

_of

_{D f.}

Proof.

Let us write := and show

that g

is

also a holonomic arithmetic

Gevrey

series of order s.

We have vve ave

_{- - n -’ ’t rs}

_nis which shows

_{that g satisfies}

like

e

0 ( ki ) k!s W IC sows a 9 satls es, e

f ,

the denominator condition in the definition of arithmetic

_Gevrey

series of order s . To _prove

_{that g}

is arithmetic

_Gevrey

of order _s, it remains to

prove that for any

complex embedding

v, gv is

Gevrey

of order s in the

ordinary

sense. Since s

0,

this amounts to

_being

entire of

_exponential

(like f). Since gv

=

~2013

has no

_{pole at ~ by assumption,}

this is clear. The

_{holonomicity of g}

is also clear:

_D9

=

D f

o

(x - ~).

We can therefore

_apply

the theorem to _g. We

_get

that

_Dg

has a basis of

that

_{D j}

has a basis of solutions in

_{(x - ç)K[[x - 6]],}

hence

_{that 6}

is a

(trivial)

singularity

of 0

4.3. This

_corollary

is in fact a transcendence theorem in

_disguise.

Let us first indicate how it

_{easily implies}

the Lindemann- Weierstrass the-orem. We embed K in C. Let ~i,..., am be distinct elements of K and

,(31, ... ,

be non-zero elements of K. Let us show that

Let L be the Galois closure of in C. We set

It is

_enough

to show that

_{f (1) #}

0. Note that

_f

is an

_exponential

polyno-mial with constant

_coefficient,

hence an arithmetic

_Gevrey

series of order

-l,

with

_{D f}

E

_{(constant coefficients).}

In

_particular,

1 is not a

singularity

of

_{D f,}

and the

_corollary

shows that

_{f (1) ~}

0 as wanted. 0

4.4. A modification of this

_{argument -}

_replacing

the fact that

_{D f}

has constant coefficients

_by

a zero lemma for solutions of linear differential

operators -

leads to a new and

qualitative

proof

of the

Siegel-Shidlovsky

theorem,

which can be

expressed

as follows:

4.4.1. Theorem. Let

_f

be a holonomic arithmetic

_Gevrey

series

_of

order s 0.

Let p

be the order

of

Df.

Then

_for

E

Q*

distinct

_from

the

_{singularities}

_{of D f,}

the transcendence

degree of

f (~), f’(ç), ...

f ~~-1~

(ç)

over

Q

equals

the transcendence

degree

~f f, f ~, ... , f ~~‘-1~

over

cf.

[4] (and

also

_[5]

for details about the zero

_lemma).

The theorem

_applies

to confluent

_{hypergeometric}

series

_{PFq_1 (x)}

with rational

parameters

and _{p q.}

4.5.

_{Actually, although}

theorem 3.4.1 deals with arithmetic

_{Nilsson-Gevrey}

series and not

_only

with arithmetic

_{Gevrey series,}

we have not been able

to

_generalize

the above

_corollary

to that more

_general

_case, hence to derive transcendence results for arithmetic

_{Nilsson-Gevrey}

series. We nevertheless

propose the

following

4.5.1.

_{Conjecture. Let y}

be a holonorraic arithmetic

_{Nilsson-Gevrey}

series

of

order s 0. _{Let ~c} be the order

of

Dy.

Assume that _y,

_y’,

... , y~~‘-l

are

algebraically independent

over

Then

_{for any ~}

E

Q*

distinct

_from

the

_{singularities of}

_Dy,

with

_finitely

many

exceptions,

y(~~, y~(~),

... ,

y~~-1~ (~~

are

algebraically independent

The

_necessity

to allow some

_exceptions

is

_already

_patent

on the

_example

y = exp x - e. The

conjecture

would

_imply

for instance that the

_Airy

function takes transcendental values at almost _every

_algebraic

_point.

4.6. The case s &#x3E; 0. A

_prototype

is the Euler series Eval-uations at

_{points ç i=}

0

_diverge

but one can turn the

_difficulty

in two _ways:

either

_by

_evaluating

such series

_p-adically

for almost all _p, or

_{by using}

some

canonical _process of

_{(archimedean)}

resummation in a sector bissected

_by

~ 5

.

We concentrate on the second

_viewpoint,

which is

_actually

much older than the first one: L. Euler

already proposed

four methods for

_summing

the series _{and gave} some evidence that

they

lead to the same

result

_[7].

4.6.1.

_Conjecture.

The

_{prevzous conjecture}

should also hold

_{if s}

&#x3E;

_0,

pro-vided

_{y(~), y’(~),}

... ,

(ç)

are

interpreted

as canonical resummations.

4.6.2.

_Proposition.

_Conjecture

_{4.5.1 implies 4.6.1.}

This follows from theorem

_3.4.1,

_part

_3ii).

0

By

symmetry,

the

_{Siegel-Shidlovsky}

theorem

_suggests

the

_following

4.fi.3.

_Conjecture.

The statement should also hold

_{if s}

&#x3E;

_0,

pro-vided

_{f (~), f’ (~), ... ,}

_{f ~~‘-1~ (~)}

are

_{interpreted as}

canonical resummations.

A

_proof

of this

_conjecture

would be of considerable interest because

spe-cial values of resummations of arithmetic

_Gevrey

series of

_positive

order

(such

as the Euler

_series)

involve the Euler

_constant,

the values

_of

Riemann

( function

at

_integers,

and more

_generally

the derivatives

_of

Euler’s r

func-tion at rational

_poms

In

_fact,

if we had restricted the definition of

arith-metic

_{Nilsson-Gevrey}

series

_by

_allowing

_only

constants

_Ai

in

Q (r(n)

_{(p/q) ),}

theorem 3.4.1 would still

_hold,

_cf.

_{[3] (remark}

at the end of

_6.2).

This

_might

be a _way of

_{investigating}

the arithmetic nature of _those

mys-terious

_constants,

via arithmetic

_Gevrey

series of

_{order #}

_0,

whereas the usual

_approach

involves arithmetic

_Gevrey

series of order 0

_(variants

of

polylogarithms) .

A lot of work remains to be done in this direction. 4.’T. The case s = 0. This _case is _more delicate than the _{case s}

0,

because there do exist infinite

_"exceptional

sets" of

_{points 6}

E

_Q

in

_general

where such transcendental series take

_algebraic

values 6

5if this is a _{singular (= anti-Stokes)} direction, one has to use the middle summation of Écalle-Ramis-Martinet.

6nevertheless,

under a _{strong assumption} of simultaneous uniformization _imposed on the series under consideration, a statement _analogous to the _{Siegel-Shidlovsky theorem is proven in}

A

_{striking example,}

due to F. Beukers and J. Wolfart involves

2F~ ( 12 , 2 , ~ ; x),

which takes

_algebraic

values at

_infinitely

_many

_algebraic

points,

e.g.

In

_fact,

these "accidental relations" have nice

_{geometric interpretations,}

even in a _very

general setting. Indeed,

it is

expected

that for _any

G-function

_{(i. e.}

holonomic arithmetic

_Gevrey

series of order

_{0), D f}

"comes from

_{geometry" ,}

i. e. is a

_product

of factors of Picard-Fuchs differential

operators

(conjecture

of

_{Bombieri-Dwork).}

This is

_certainly

the case for

any G-function of

generalized

hypergeometric

type.

In such a

_{situation, special}

values of G-fonctions can be

interpreted

as

rational functions in

_periods

of smooth

_projective

_algebraic

varieties de-fined over

Q.

A transcendence

_conjecture

of Grothendieck then

predicts

that _any relation between such

_periods

should come from

algebraic cycles

over suitable _powers of these varieties - or, if one

_prefers,

should have an

interpretation

in terms of

_isomorphisms

of motives. We refer to

_[1]

for a detailed discussion of this

_topic.

This motivic

interpretation

also leads to

the

_{following prediction:}

whenever an "accidental relation" occurs between

special

values of

_G-functions,

one should

_expect

a similar relations between

the

_p-adic

evaluations of the same series at the same

_point

for _every

_prime

p for which the evaluation makes sense

_(this

can be

_{justified, relying}

_upon

some "standard

_conjectures"

in the

_theory

of motives in characteristic zero

and _p, see

[1]

for

det ails ) .

As an

_{illustration,}

one

_gets

in the above

_example

7-adically

(note

that 1323 =

33.72),

_{as was} checked

by

Beukers

[6].

References

[1] Y. _ANDRÉ, Théorie des _motifs et _{interprétation géométrique} des valeurs p-adiques de

G-fonctions (une introduction). Number _{theory (Paris, 1992-1993),} 37-60, London Math. Soc. Lecture Note Ser., 215, Cambridge Univ. Press, Cambridge, 1995.

[2] Y. _{ANDRÉ, G-fonctions} et transcendance. J. Reine _Angew. Math. 476 _(1996), 95-125.

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