J
OURNAL DE
T
HÉORIE DES
N
OMBRES DE
B
ORDEAUX
V
ICTOR
A
LEXANDRU
N
ICOLAE
P
OPESCU
A
LEXANDRU
Z
AHARESCU
A representation theorem for a class of rigid
analytic functions
Journal de Théorie des Nombres de Bordeaux, tome 15, n
o3 (2003),
p. 639-650.
<http://www.numdam.org/item?id=JTNB_2003__15_3_639_0>
© Université Bordeaux 1, 2003, tous droits réservés.
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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
A
representation
theorem for
aclass
of
rigid
analytic
functions
par VICTOR
ALEXANDRU,
NICOLAE POPESCU et ALEXANDRUZAHARESCU
RÉSUMÉ. Soit p un nombre
premier,
Qp
le corps des nombresp-adiques
etCp
lacomplétion
d’une clôturealgébrique
deQp.
Dans cetarticle,
nous obtenons un théorème dereprésentation
pour les fonctions
analytiques rigides
surP1 (Cp)BC(t,03B5)
qui
sontéquivariantes
par le groupe de Galois G =Galcont (Cp/Qp),
où tdésigne
un élémentLipschitzien
deCp
etC(t, e)
un03B52014voisinage
de la G-orbite de t.
ABSTRACT.
Let p
be aprime number,
Qp
the field ofp-adic
num-bers and
Cp
thecompletion
of thealgebraic
closure ofQp.
In this paper we obtain arepresentation
theorem forrigid analytic
functions onP1 (Cp)BC(t,03B5)
which areequivariant
with respectto the Galois group G =
Galcont(Cp/Qp),
where t is aLips-chitzian element of
Cp
andC(t, 03B5)
denotes thee-neighborhood
of the G-orbit of t.1. Introduction
"
Let p
be aprime number,
~,
the field ofp-adic numbers,
Qp
a fixedalgebraic
closure of~
andCp
thecompletion
ofQp
withrespect
to thep-adic
absolute value. Let t ECp
and setE(t)
= =Cp
U{oo}BC(t)
whereC(t)
denotes the orbit of t withrespect
to the group G ofall continuous
automorphisms
ofCp
overQ§ .
In this paper we are interestedin the
G-equivariant
rigid
analytic
functions onE(t)
and their restrictionsto affinoids of the form
E(t, e) _ Cp
Uloo}BC(t,
c)
whereC(t, c)
stands for thee-neighborhood
ofC(t).
These functions are
easily
described in case t isalgebraic
overQ§ .
Forinstance,
if t E Qp
then one can use theequivariant
transformation zto send t to the
point
atinfinity.
Then theequivariant
rigid
analytic
func-tions onE(t)
willcorrespond
to the entire functions which areequivariant
and these are
simply
power seriesf (x) _
anzn
with cn EQp
for any1
=
-n and such that
lani n
= 0.If t is transcendental over
Q,
it is not obvious that there are anynoncon-stant
equivariant
rigid
analytic
functions onE(t).
For certain elements t(called Lipschitzian)
such a function z HF(t, z)
is constructed in[APZ2].
In this paper we define for any
Lipschitzian
element t of~
and anynatu-ral numbers m, n an
equivariant
rigid analytic
functionFm,n(t,
z)
onE(t),
which is related to our basic trace series
F(t, z).
Then in Theorem 4.2 belowwe express any
equivariant
rigid analytic
function on an a,ffinoidE(t, e)
interms of the above functions
F m,n (t,
z).
2.
Background
material2.1. Let p be a
prime
number and~
the fieldof p-adic
numbers endowedwith the
p-adic
absolute valueII,
normalized such thatlpi
=1/p.
LetQp
bea fixed
algebraic
closure ofQp
and denoteby
the samesymbol ~ ~ the
unique
extension
of [
toQp.
Further,
denoteby
(Cp,
II)
thecompletion
ofII)
(see [Am],
[Ar]).
Let G = endowed with the Krulltopology.
The group G is
canonically
isomorphic
with the group ofall continuous
automorphisms
ofCp
over For any x ECp
denote
-C (x) =
EG}
the orbit of x and let be the closure of thering
Q§ [x]
inCp.
For any x EQp
denotedeg(x) =
~~ (x) : Qp].
2.2. Let x E
Cp.
Given a real number e > 0 letB(x, c) = {y
ECp,
y|
el
the open ball of radius e centered at x. If M is acompact
subsetof
Cp
and c > 0 is a realnumber,
denoteby
N(M, c)
the number of alldisjoint
balls of radius e which have anon-empty
intersection with M. Wesay that M is
Lipschitzian
if limc
= 0. We call an element x ECp
’
Lipschitzian.
ifC(x)
isLipschitzian.
According
to[APZ2]
if x isLipschitzian
then one canintegrate
Lips-chitzian functions
(see
definition in 2.3below)
withrespect
to thep-adic
Haar measure ~t induced
by
G on the setC(x).
Let
Gx
=la
E G :u(x)
=x}
and P a closedsubgroup
of G whichcontains
Gx.
ThenCp(x) = la(x) : a
EP},
the orbit of x withrespect
toP,
is acompact
subset ofC(x)
=CG(x).
If x isLipschitzian
thenCp(x)
is aLipschitzian
compact
set for any P withGx
C P. This follows from the fact that for any 6- >0,
N(Cp(x),e)
dividesN(C(x), e).
Let x E
Cp
and P a closedsubgroup
of G which containsGx.
For any c > 0 letHp(x, c) =
{7
E P :Ix-q(x)1
[
c}
andNp(X,ê)
=N(Cp(x),e).
Then
Hp (x, c)
is an opensubgroup
of P andNp(X,ê) =
[P :
Inparticular
N(x, 6) =
[G : H(X,ê)],
whereH(X,ê)
=Ha(X,ê).
2.3. The notion of
rigid
analytic
function is defined in[FP]
(see
also[Am]).
According
to[APZ2],
arigid
analytic
function defined on a subsetD of
Cp
is said to beequivariant
if forany z E D
one hasC(z)
C D andf (a(z)) = a(f (z))
for all a E G. A functionf :
C(t)
-Cp,
t ECp
isLipschitzian
if there exists a real number c > 0 such thatf (y)~
yl
for all x, y EC(t).
Let t be a
Lipschitzian
element ofCp
andf :
C(t)
-Cp
aLipschitzian
function.
Then theintegral
is well defined
(see
[APZ2]).
Inparticular
for anypolynomial
P(X)
ECp[X],
any z U{00}
B C(t)
and any natural number n the functionz 1-7 = is
Lipschitzian
on and we consider theintegral
Let us denote
According
to[APZ2]
for any m > 0 one hasThis shows that
F,.,z,o (t,
z)
= E~,
Fo,o (t,
z)
= 1 and1 +Fl,l
t, 1)
=F(t, z),
the trace function associated to t. Alsoby
theequality
valid for any
positive integer
m and any u withJul
1,
it follows that forIzi
>lxl
one hasThen one may write:
This formula
represents
theexpansion
ofFm,n(t,
z)
in a suitableneigh-borhood of
infinity.
As in Theorem 6.1 of[APZ2]
one shows that for all m >0, n
>0,
is anequivariant
rigid analytic
function defined onRemark 2.1. = for any m >
0,’n ~~’
1. As aconsequence one has where the derivative
is taken with
respect
to z.n! m,
2.4. The above considerations can be
generalized
as follows: Let c > 0 be a real number and S asystem
ofright
representatives
of G withrespect
to the
subgroup
H(t, e).
Assume that theidentity
element e of Gbelongs
to S and that t is
Lipschitzian.
For any cr ~ S the subsetCu(t, c)
={r(t) :
T ~ is acompact
subset ofC(t).
For m >0, ~ >
0 denoteIt is clear that
In fact this formula
represents
theMittag-Leffier
decomposition
ofFm,n(t,
z)
viewed as a
rigid
analytic
function in the connected affinoidCp
Ufool
B
Uo-EsB(u(t), c)
=E(t, ê).
In what follows wetry
to obtain a similardecom-position
for any element of the setA(E(t, e))
ofequivariant
rigid analytic
functions onE(t, c).
3. A combinatorial Lemma
Let a, x, y, be variables. For any m >
1,
let us denote, . 6 ,
where as
usually
1)
..~I(m -
k +
1) ,
For anyinteger
1 we set
..
I
~ . %--l
where
h~,n~(a)
denotes the formal k-th derivative of thepolynomial
hm(a)
withrespect
to a.Lemma 3.1. For any x, y and any m > 1 one has:
Proof.
Equality
(3)
states that =x),
which followsdirectly
from the
Taylor
expansion
(2).
As for the secondequality, by
applying
(3)
andusing
theidentity
1
one contains
This
equals
by
(3)
and so the lemma isproved.
04.
Equivariant
rigid
analytic
functions onE(t, e)
4.1. Let t be an element of
Cp,
let e > 0 be a real number and denoteby
B(C(t),
c)
the union of alldisjoint
open ballsB(x, c)
which have anonempty
intersection withC(t).
Choose a EQp
suchthat it
-al
e.Then one has
H(t, c)
=H(a,E).
Let S be asystem
ofright
representatives
of G withrespect
toH(t, E)
and assume e E S. One hasB(C(t),E) =
UUES
B(u(a),E).
Consider the affinoidE(t,ê) = ~
UloolBB(C(t), 6)
and letA(E(t, E))
be the set ofequivariant
rigid
analytic
functions onE(t, e).
If t is
Lipschitzian
then the functionsFm,n(t,
z)
defined in Section 2 areelements of
A(E(t, e)).
In this section we shall prove that all the elementsof
A(E(t, E))
can beexpressed
in terms of the functionsFm,n(t,
z),
m, n > 0. 4.2. We have thefollowing
proposition.
Proposition
4.1. Let t be an elemento/Cp.
DenoteKt
=n Qp
and let e > 0 and a EKt
such that c. There exists a sequenceof
elementsof
Kt
and a sequencesof
positive
real numbers suchthat:
(i)
W = ~~ ai = a,(ii)
Forany n >
1 one hasIt
-anll,
(iii) It
-The
proof
easily
followsby
induction on n since any ballB (t, c)
containselements of
Kt
(see
[APZ1]).
In what follows we work with sequences and
fcnln
as inPropo-sition 4.1. It is clear that
limen
=0,
and t =liman.
Note also thatn n
the ball is contained in
B (an, cn)
for all n > 1. Let usconsider the
subgroup
H(t, en)
=H(an, en)
defined in Section 2. Denotedn
=[G :
H(T, en)]
=N(t, en),
and letSn
be a fixedsystem
ofrepresen-tatives of
right
cosets of G withrespect
toH(t, en).
We shall assume thatthe
identity
element e of Gbelongs
to eachSn.
We remark thatSl
= Sand
dn
divides for all n > 1.4.3. Let
f
EA(E(t,ê)).
Then(see
[FP],
ChI)
f
admits aMittag-Leffier
decomposition:
f(z) =
¿uES
fa(z)
+f (oo)
wheref (oo)
is the value off
at
infinity
andSince
f
isequivariant
then for any z EE(t, e)
and any T E G one hasand for E S one can write:
Next we remark that for any T E
H(t,
e)
and any a E S the elementu(r(a))
belongs
toB (a (a), ê),
and so the functionfu(z) =
E,.~, 1aa,,m ).
can alsobe written as
where
In what follows we shall assume that
f (oo)
= 0.4.4. At this
point
we derive another convenientexpression
forf (z),
using
the above elements an. Fix n > 1. Then d =
d¡
dividesdn
= [G :
H(t, en)~.
Denote qn =
dn/d
and leten),
1 j
qn be all the balls of radius en centered at suitableconjugates
of an and such that these balls coverwhere
According
to(5)
for all a E S one hasAs a consequence of
(4)
there exists apositive
realnumber M such that for any
m 2:
1 one has:It follows from
(7)
thatfor 2 and
1 j
4.5. At this
point
we assume that t is aLipschitzian
element ofCp,
e > 0 and a EB(t, e),
a EKt.
Letf
EA(E(t, e)),
f
=EqES
fu(z)
withf,(z)
given by
(4).
For any m > 1 denoteAlso,
for a E S consider the function170, , (t, Z)
defined in Section 2. Theorem 4.2. Let t be aLipschitzian
elementof Cp,
e >0,
a Eand
f
EA(E(t,e)).
Thenfor
any z EE(t,e)
one hasProof.
For any ~n > 1 let andStep
1. Fix zo EE(t,
ê).
We assert that for any z EB(zo, ê),
the functionz e
A(x, z)
is defined and isLipschitzian
onB(t, c).
Firstly
we remarkand
Since
c
1,
by
(4)
and the above considerations it follows that I - . Iwhen m 2013~ oo. Then the function
A(x, z)
is defined onB(t, ê),
as
claimed. Now let x, y EB (t, ~) .
For any m > 1 we havewhere
Di
are suitable natural numbers. Then one can writeBut
(see
(8))
for any i > 1 and z EB(zo,
e)
one has..
Also
by
an easycomputation
one sees that:Finally,
one has isLipschitzian
on
J3(t,c).
The above considerationsalso
show that for any J > 0 we haveStep
2. Let us denote D =Ce (t, e)
=B (t, c) nC(t).
Then D is acompact
Lipschitzian
subset ofCp
and we consider theintegral
Here we use the definition of the
integral
withrespect
thep-adic
measure~t as in
[APZ2].
We assert thatwhere e is the
identity
element of G.To see
this,
consider the sequences and~an }n
fromProposition
4.1. Let
H(t, en),
dn,
Sn
be as above. Inparticular
el = e, al =a,
d,
= d.For
any n >
1 letCn),
1 i qn be the open balls of radius enwhich cover D. Then one has:
where
is the Riemann sum associated to
(see[APZ2]).
We haveFrom
(6)
it now follows thatSince this
equality
is valid for any n we conclude thatStep
3. We nowapply
formula(2)
to obtain anotherexpression
forTherefore
We claim that
In order to prove this formula we need the
following
result:Lemma 4.3. Let t be a
Lipschitzian
elementof
e > 0 a realnumber,
g :
B(C(t), e)
-->Cp
aLipschitzian function,
and let c be a real numbersuch that
lg(x) - g(y)~ clx -
all x, y EC(t).
Then there exists areal number k
independent
of
g such that:Proof.
Letbe
adecreasing
sequence ofpositive
real numbers suchthat
limén
= 2 andC(t)
g
Then one has:-
n
where
(see
Section2)
dn
=[G :
Sn
is asystem
ofright
cosets of G withrespect
H (t, En) and ~ (g, T (t),
1 .
is the Riemann sum associated to
Sn
and g(see
[APZ2]).
Inparticular
E1)
=g(t).
Let n > 1. Then
dn
dividesdn+1
and for any T E there existsexactly
one element a ESn
such thatT(t)
E Then we haveThen
by
the above considerations one has:Now let us take since
being
Lipschitzian by
hypothesis.
Let 9 > 0 be a real number. Then
by
(4), (11)
and(12)
it follows that for mlarge
enough
one has:E D. Lemma 4.3
implies
that Thereforeand
using
(13)
one obtains(14).
Step
4. Let a E S and denote DO’=B(u(a),e) fl C(t)
=Cu(t,e).
Working
asabove,
onegets:
Finally by
adding
theseequalities
for Q E S one obtains theexpression
off (z)
stated in Theorem 4.2 0Corollary
4.4. The notations andhypothesis
are as in Theorem4.2
References
[APZ1] V. ALEXANDRU, N. POPESCU, A. ZAHARESCU, On closed subfields of Cp. J. Number
Theory 68 (1998), 131-150.
[APZ2] V. ALEXANDRU, N. POPESCU, A. ZAHARESCU, Trace on Cp. J. Number Theory 88
(2001), 13-48.
[Am] Y. AMICE, Les nombres p-adiques, Presse Univ. de France, Collection Sup. 1975.
[Ax] J. Ax, Zeros of Polynomials over Local Fields - The Galois Action, J. Algebra 15 (1970),
417-428.
[PZ] N. POPESCU, A. ZAHARESCU, On the main invariant of an element over a local field,
Portugaliae Mathematica 54 (1997), 73201483.
[Ar] E. ARTIN, Algebraic Numbers and Algebraic Functions, Gordon and Breach, New
York/London/Paris, 1967.
[FP] J. FRESNEL, M. VAN DER PUT, Géometrie Analytique Rigide et Applications, Birkhäuser.
Boston. Basel. Stuttgart, 1981.
Victor ALEXANDR,U
Department of Mathematics
University of Bucharest
Romania
Nicolae POPESCU
Institute of Mathematics of the Romanian Academy P.O. Box 1-764
70700 Bucharest, Romania
E-mail : nipopescGstoilov.imar.ro Alexandru ZAHARESCU
Department of Mathematics
University of Illinois at Urbana-Champaign
Altgeld Hall, 1409 W. Green Street
Urbana, IL, 61801, USA