The Behavior of Rigid Analytic Functions Around Orbits of Elements of <span class="mathjax-formula">$\mathbb {C}_p$</span>

The Behavior of Rigid Analytic Functions Around Orbits of Elements of C

S. ACHIMESCU(*) - V. ALEXANDRU(**) - N. POPESCU(*) M. VAÃJAÃITU(*) - A. ZAHARESCU(*)(***)

ABSTRACT- Given a prime numberpand the Galois orbitO(x) of an elementxofCp, the topological completion of the algebraic closure of the field ofp-adic numbers, we study the behavior of rigid analytic functions around orbits of elements ofCp.

Introduction

Let pbe a prime number,Qpthe field of p-adic numbers,Qp a fixed algebraic closure ofQp, andCpthe completion ofQpwith respect to thep- adic valuation. LetO(x) denote the orbit of an elementx2Cp, with respect to the Galois groupGˆGal_cont(C_p=Q_p). We are interested in the behavior of rigid analytic functions defined onE(x)ˆ(C_p[ f1g)nO(x), the com- plement ofO(x). The paper consists of five sections. The first one contains notations and some basic results. In the second section we study the zeros of rigid analytic functions, which are not rational, around finite sets ofCp

and, in particular, around orbits of algebraic elements ofC_p, see Theorem 1 and Corollary 1. The next section is concerned with the transcendental case. One has a similar result of a theorem of Barsky, see Theorem 2, and then we prove that if all the points ofO(x) are singular points for a rigid

(*) Indirizzo degli A.: Simion Stoilov Institute of Mathematics of the Romanian Academy Research Unit 5, P.O. Box 1-764, RO-014700 Bucharest, Romania.

E-mail: sachimescu@yahoo.com E-mail: Nicolae.Popescu@imar.ro E-mail: Marian.Vajaitu@imar.ro

(**) Indirizzo dell'A.: Department of Mathematics, University of Bucharest, Romania.

E-mail: vralexandru@yahoo.com

(***) Indirizzo dell'A.: Department of Mathematics, University of Illinois at Urbana-Champaign, Altgeld Hall, 1409 W. Green Street, Urbana, IL, 61801, USA.

E-mail: zaharesc@math.uiuc.edu

analytic functionF then one has lim infz!xjF(z)j ˆ0, see Theorem 3. In section four we give a few examples of rigid analytic functions onE(x) with and without zeros. The last section is concerned with some remarks related to zeros on a corona.

1. Background material

Letpbe a prime number andQ_pthe field ofp-adic numbers endowed with thep-adic absolute valuej j, normalized such thatjpj ˆ1=p. LetQ_p be a fixed algebraic closure ofQ_pand denote by the same symbolj jthe unique extension ofj jtoQp. Further, denote by (Cp;j j) the completion of (Qp;j j) (see [Am], [Ar]). LetGˆGal(Qp=Qp) endowed with the Krull topology. The group G is canonically isomorphic with the group Gal_cont(C_p=Q_p), of all continuous automorphisms of C_p over Q_p, see [APZ2]. We shall identify these two groups. For any x2C_p denote O(x)ˆ fs(x):s2Ggthe orbit ofx, and letQgp[x] be the closure of the ring Q_p[x] inC_p.

For any closed subgroup Hof Gdenote Fix(H)ˆ fx2C_p:s(x)ˆx for all s2Hg. Then Fix(H) is a closed subfield of C_p. Denote H(x)ˆ fs2G:s(x)ˆxg. Then H(x) is a closed subgroup of G, and Fix(H(x))ˆQgp[x].

The map s?s(x) from G to O(x) is continuous, and it defines a homeomorphism from G=H(x) (endowed with the quotient topology) to O(x) (endowed with the induced topology fromCp) (see [APZ1]). In such a wayO(x) is a closed compact and totally disconnected subspace ofCp, and the group G acts continuously on O(x): if s2G and t(x)2O(x) then s?t(x):ˆ(st)(x).

Now, ifXis a compact subset ofC_pthen by an open ball inXwe mean a subset of the form B(x;e)\X, x2Cp, e>0 where B(x;e)ˆ fy2Cp:jy xj5eg. Let us denote byV(X) the set of subsets ofXwhich are open and compact. It is easy to see that anyD2V(X) can be written as a finite union of open balls inX, any two disjoint.

DEFINITION1. By a distribution on X with values inC_pwe mean a map m:V(X)!C_p which is finitely additive, that is, if Dˆ Sⁿ

iˆ1D_i with D_i2V(X) for 1in and D_i\D_jˆ ; for 1i6ˆjn, then m(D)ˆPⁿ

iˆ1m(D_i). (See also [MS].)

The norm ofmis defined byk km :ˆsupfjm(D)j:D2V(X)g. Ifk k51m we say thatmis a measure onX.

DEFINITION 2. Let D be a infinite subset of P¹(C_p). A function f :D!C_pis said to be rigid analytic (or Krasner analytic) on D provided that f is a uniform limit with respect to the topology of uniform con- vergence on D of a sequence of rational functions on D. (See also [FP], [Am].) We denote byA(D;Cp)the set of all rigid analytic functions defined on D.

The set XC_p is said G-equivariant provided that s(x)2X for any x2Xand anys2G. (XˆO(x) is such an example.)

DEFINITION3. Let X be a G-equivariant compact subset ofC_pandma distribution on X with values inC_p. We say thatm is G-equivariant if m(s(B))ˆs(m(B))for any ball B in X and anys2G.

REMARK. On a Galois orbit O(x) there exists a uniqueG-equivariant probability distribution with values in Q_p, namely the Haar distribution px, see [APVZ1] and [APZ2].

According to [APZ2], a rigid analytic function defined on a subsetDof P¹(Cp) is said to be equivariantif for any z2D one has O(z)Dand f(s(z))ˆs(f(z)) for alls2G.

Let A^G(D;C_p) be the set of equivariant rigid analytic functions on D with values inCpandA^G₀(D;Cp) the subset of those that vanish at1when 1 2D.

2. The algebraic case

LetSbe a finite set inCp and letF:P¹(Cp)nS!Cpbe a rigid ana- lytic function which is not rational.

We study the zeros of such functions around S. In particular, one considers Sof the formO(a)ˆ fs(a):s2GˆGal_cont(C_p=Q_p)gthe orbit ofa2Q_p. Let us recall the analogue of a classical result of Picard.

PROPOSITION 1 (see [AR]). Let FˆP¹

1a_nXⁿ2C_p[[X;X ¹]] be a Laurent series that has infinitely many coefficients an 6ˆ0for n50and such that R_F:ˆlim sup_n!1ja _nj¹⁼ⁿ ˆ0 i.e. 0 is an essential isolated

singularity for F. Then there are infinitely many critical radii ristrictly decreasing to 0 and for each e>0, y2Cp, the equation F(x)ˆy has infinitely many solutions05jxj5e.

For F as above, we define R^‡_F:ˆsupfr0:ja_njrⁿ!0; n! 1g ˆ 1=lim sup_n!1ja_nj¹⁼ⁿ, by the Hadamard formula, see [AR]. One has the following result.

THEOREM1. Let S be a finite set inC_pand let F:P¹(C_p)nS!C_pbe a rigid analytic function. If F is not rational then F has infinitely many zeros in any neighborhood of at least one point of S.

PROOF. For the sake of simplicity we supposeF(1)ˆ0. Let us con- sider the Mittag-Leffler decomposition

F(z)ˆX

a2S

F_a(z)ˆX

a2S

X

n1

b^(a)_n (z a)ⁿ; …1†

where Fa(z)ˆ P

n1

b^(a)_n

(z a)ⁿ and jb^(a)_n j

eⁿ !0, n! 1, for any positive real number e and any a2S. It is easy to see that this condition is further equivalent tojb^(a)_n j¹⁼ⁿ!0, whenn! 1. BecauseF is not rational there existse2Ssuch thatF_eis not rational. In such a way from (1) one has

F(z)ˆF_e(z)‡H(z);

…2†

where H(z)ˆP

a2Sa6ˆe

P

n1

b^(a)_n

(z a)ⁿ. We have y:ˆH(e) is well defined and denote Aˆ jyj ˆ jH(e)j. It is clear that R_Fe ˆlim supjb^(e)_n j¹⁼ⁿˆ0 and R^‡_Feˆsupfr0:jb^(e)_njrⁿ !0; n! 1g ˆ 1. (We note here that b^(e)_nˆ0, for anyn0, by (1).) BecauseF_eis not rational we have thate is an essential isolated singularity forFe. By Proposition 1 we have that Fe has infinitely many critical radii ri strictly decreasing to 0 and for each y2Cp the equation Fe(x)ˆy has infinitely many solutions 05jx ej5e. One knows that lim sup_z!ejF_e(z)j ˆ 1. Then there exists e_A>0 such that if 05e5e_A and jz ej e5e_A we have jH(z)j5A‡1.

Let z02B(e;e) such thatjFe(z0)j>A‡1. In fact one has

i!1lim sup

z2S(e;ri)jF_e(z)j ˆ 1 ˆlim sup_z!ejF_e(z)j;

where S(e;r_i) is the sphere centered at eand of radius r_i. There exists

z12S(e;jz0 ej),jz0 ej ˆria critical radius, such thatjFe(z1)j 6ˆ jFe(z0)j becausejFejis not constant on the spheres of radiiri, which are the critical radii, andFehas zeros on these spheres. It is clear thatjF(z0)j ˆ jFe(z0)j.

Two cases may appear:

(i) jF_e(z₁)j5jF_e(z₀)jso one hasjF(z₁)j maxfjF_e(z₁)j;A‡1g5jF_e(z₀)j ˆ jF(z₀)j.

(ii) jF(z₁)j ˆ jF_e(z₁)j>jF_e(z₀)j ˆ jF(z₀)j.

This implies thatjFjis not constant onS(e;r_i) soF has zeros on this

sphere. The proof is now complete. p

COROLLARY 1. Let a2QpnQp be an algebraic element and let F:P¹(Cp)nO(a)!Cp be a rigid analytic function. If F is not rational then F has infinitely many zeros in any neighborhood of at least one point of O(a).

REMARK. IfF:P¹(C_p)nO(a)!C_pis a rigid analytic function that is equivariant and is not rational, then it takes any values in any neighbor- hood ofO(a), so all the conjugates ofaare essential singular points ofF. In what follows we give another proof of Theorem 1 without the ana- logue of the classical result of Picard, by exploiting the fact thatF_egiven above has a Weierstrass product and it is uniquely determined by its fa- mily of zeros. So, letF be as in Theorem 1. Let us consider the Mittag- Leffler decomposition

F(z)ˆX

a2S

fa(z)ˆX

a2S

X

n0

b^(a)_n (z a)ⁿ; …3†

where f_a(z)ˆ P

n0

b^(a)_n

(z a)ⁿ and jb^(a)_n j

eⁿ !0, n! 1, for any positive real numbereand anya2S. This condition is equivalent to havingjb^(a)_n j¹⁼ⁿ !0, whenn! 1. SupposeF(1)6ˆ0 andF(0)6ˆ0. It is easy to see that there existse2Ssuch thatfeis not rational. In such a wayfehas the following decomposition (see [L])

f_e(z)ˆcY

n1

1 a_n z e

…4†

where c is a nonzero constant, an2Cp such that ja1j ja2j janj . . ., limn!1janj ˆ0 and eachja_ijappears in the decomposition offe

only a finite number of times. Let us consider Fˆfe‡g; gˆ X

a2Snfeg

fa: …5†

One has lim sup_z!ejf_e(z)j ˆ 1 and denote Aˆ jg(e)j51. There exists e_A>0 such that ifjz ej5e_A then jg(z)j5A‡1. We have the following result that implies Theorem 1.

PROPOSITION 2. Let f_e be as above and let e>0. Then there exists z₀2C_psuch that:

1) jz₀ ej5eandjf_e(z₀)j>A‡1.

2) there exists z₁2C_p,jz₀ z₁j5jz₀ ejandjf_e(z₁)j5jf_e(z₀)j.

Let us see that Proposition 2 implies Theorem 1. We choose a positive real numberesuch thate5e_Aande5inffje aj:a6ˆe; a2Sg. Because jf_e(z₀)j>A‡1 andjg(z₀)j5A‡1 one hasjF(z₀)j ˆ jf_e(z₀)j>A‡1. From jz₀ z₁j5jz₀ ej5ewe havejz₁ ej5esojg(z₁)j5A‡15jf_e(z₀)jand by jf_e(z₁)j5jf_e(z₀)jone obtainsjF(z₁)j5jF(z₀)j. On the other hand there are no elements ofSn fegin the closed ballB[z0;jz1 z0j] and even in the open ballB(z0;e). But the functionFis defined onB(z0;e) andjFjis not constant on this open ball soFhas zeros inB(e;e), see also [AR].

Now, let us give the proof of Proposition 2. As we know, the sequence (ja_nj)_n1 is decreasing and its limit is zero. We choose z₀ of the form z0ˆe‡as(1‡u) withs sufficiently large andjuj51,u6ˆ0 that will be chosen later. One has

fe(z)ˆc Y¹

iˆ1

1 ai

z e

!

ˆcY¹

iˆ1

z e ai

z e ; …6†

so

f_e(z₀)ˆcY¹

iˆ1

a_s(1‡u) a_i as(1‡u) : …7†

It is clear thatja_s(1‡u)j ˆ ja_sj. Now, ifja_ij5ja_sjthen ja_s(1‡u) a_ij ˆ jas(1‡u)j ˆ jasj, and using (7) we have

jfe(z0)j ˆ jcj Y

jaii1jjasj

as(1‡u) ai

a_s(1‡u) : …8†

Let us give an explanation about the form ofz0as above,z0ˆe‡as(1‡u),

withjuj51. If there existsz1that satisfiesjz0 z1j5jz0 ejandjfe(z1)j5 jfe(z0)j then je z1j ˆ je z0j so on the sphereS(e;je z0j) the function jfe(z)jis not constant and this implies thatje z0jis the same with one of the moduli ja_sj, s1. In such a way one has z₀ˆe‡a_st, jtj ˆ1 and z₁ˆz₀‡a_st⁰,jt⁰j51, i.e.z₁ˆe‡a_s(t‡t⁰). Denoteja_sj ˆr_s. Let us com- putejf_e(z₀)jandjf_e(z₁)j. We factorize

fe(z)ˆcY¹

iˆ1

z e a_i

z e ˆhs(z)Hs(z);

…9†

where

hs(z)ˆ Y

je‡ajjˆrs

z e a_j …10† z e

and

H_s(z)ˆ Y

je‡ajj6ˆrs

z e aj

z e : …11†

In (10) the product has a finite number of factors. Let us see thatjHsjis constant on the sphereS(e;rs), because ifjHsjis not constant thenHshas zeros on this sphere that is not the case. The inequalityjf_e(z₁)j5jf_e(z₀)jis equivalent to jh_s(z₁)j5jh_s(z₀)j that is equivalent to hs(z1)

h_s(z₀)51. Because jz₀ ej ˆ jz₁ ejit results that

hs(z1)

h_s(z₀)ˆ Y

je‡ajjˆrs

z1 e aj

z₀ e a_jˆ Y

je‡ajjˆrs

1 z0 z1

z₀ e a_j51:

There existsjanda_jwithje‡a_jj ˆr_ssuch that1 z₀ z₁ z0 e aj

51, which meansjz₀ z₁j ˆ jz₀ e a_jj5je z₀j, soz0 e aj

z₀ a ˆ1 aj

z₀ e51.

Again, one obtains aj

z₀ eˆ1 soz₀ eˆa_j(1‡w), withjwj51, for some a_j with ja_jj ˆrs ˆ jasj i.e. z0ˆe‡a_j‡a_jw, with jwj51. Denote rˆminfjz₀ e a_jj:je‡a_jj ˆr_sg. We takez₁such thatjz₀ z₁j rand for at least a j such that jz₀ e a_jj ˆr one has z0 z1

z₀ e a_j 1ˆ z₁‡e‡a_j

z₀ e a_j 51. This implies jfe(z1)j5jfe(z0)j and also jz0 z1j rˆ jz0 e a_jj5jz0 ej. So z1ˆz0‡ast⁰ will be determined such that for

some a_j with jz₀ e a_jj ˆr, z₀ z₁

z₀ e a_j 151. For this z₀ z₁ z₀ e a_j must have the residual image equal to 1. This implies

z₁ z₀ z0 e a_j

ˆ a_s

z0 e a_j

t⁰ˆ1;

…12†

so the condition that determinez₁is t⁰ˆ z₀ e a_j

a_s

: …13†

In such a way z₁ is determined by z₀. It remains to find z₀, z0ˆe‡as‡asu, withjuj51, such thatjfe(z0)jis big enough (>A‡1) and je z0j ˆ jasjwithjasjsmall enough.

If in the decompositon of jfej as in (8) one has ja_ij>jasj, then jas(1‡u) a_ij ˆ ja_ijsoas(1‡u) ai

a_s(1‡u) ˆjaij

ja_sj>1. This implies jf_e(z₀)j ˆM_sM⁰_s

…14†

whereMsˆ jcj Q

jaij>jasj

ai

as

andM⁰_sˆ Q

jaijˆjasj

as(1‡u) ai

as(1‡u)

1) We see thatMs is large enough as soon asas is small enough that meanssis large enough.

2) One has as(1‡u) a_i

a_s(1‡u) ˆ1 a_i

a_s(1‡u)ˆ(1‡u) a_i

a_s where wˆ1‡uis also a unit inU1(Cp).

Now, letd2(0;1) and denote

Ud(Cp)ˆ fz:zˆ1‡y; jyj5dg:

…15†

ThenU_d(C_p)U₁(C_p) and the factor groupU₁(C_p)=U_d(C_p) is an infinite group (because of the residual field ofC_p that isF_p i.e. infinite). Denote msˆ#fa_i:ja_ij ˆ jasjg. BecauseU1(Cp)=U_d=ms(Cp) is infinite we can find wˆ1‡u2U1(Cp) such that (1‡u) ai

a_s d

m_s, for each 1ims. From this one obtains

Y

jaijˆjasj

as(1‡u) a_i a_s(1‡u)

d;

so by (14) we have

jfe(z0)j jcj Y

jaij>jasj

a_i as

d:

…16†

Finally it is clear that forslarge enough, by (16) one hasjf_e(z₀)j>A‡1, and the proof is done.

3. The transcendental case

Let x2C_p. For any e>0 denote B(x;e)ˆ fy2C_p:jx yj5eg and B[x;e]ˆ fy2C_p:jx yj eg. For anyd>e>0 also denoteE(x;e;d)ˆ fy2B(x;d):jy tj e, for all t2O(x)\B(x;d)g and E[x;e;d]ˆ fy2 B[x;d]:jy tj e; for allt2O(x)\B(x;d)g. The complement ofE(x;e;d) inB(x;d) is denoted byV(x;e;d). Both setsE(x;e;d) andV(x;e;d) are open and closed, and one has: \_eV(x;e;d)ˆO(x)\B(x;d). Denote E(x;d)ˆ [eE(x;e;d)ˆB(x;d)nO(x) andE[x;d]ˆ [eE[x;e;d]ˆB[x;d]nO(x). For anyx2Cpande>0 denoteH(x;e)ˆ fs2G:js(x) xj5egandH[x;e]ˆ fs2G:js(x) xj eg. LetSe (respectivelySe) be a complete system of representatives for the right cosets of Gwith respect toH(x;e) (respec- tivelyH[x;e]). ThenE(x;e;d)ˆ [_s2SeB(s(x);e)\B(x;d).

(Whendˆ 1we dropdin the above notations and obtain the standard notations and results as in [APVZ1].)

Letdbe a fixed positive real number and letF:E[x;d]!Cpbe a rigid analytic function such that the set of real numbers fekFk_E[x;e;d]gd>e>0 is bounded. (HerekFk_E[x;e;d]ˆsup_z2E[x;e;d]jF(z)j.)

By the Mittag-Leffler Theorem (see [FP]) one can write:

F(z)ˆF₁^(e)(z)‡F^(e)₂ (z) where

F₁^(e)(z)ˆ X

s(x)2B(x;d)s2Se

X

n1

a^(e)_n;s

(z s(x))ⁿ; a^(e)_n;s2Cp; ja^(e)_n;sj eⁿ !0;

and

F^(e)₂ (z)ˆ X

s(x)2B(x;d)s2Se

X

n0

b^(e)_n;s(z s(x))ⁿ; b^(e)_n;s2C_p; jb^(e)_n;sjdⁿ!0;

for any z2E(x;e;d). It is easy to see that F^(e)₂ (z)ˆF^(e₂⁰⁾(z) for any 05e⁰5e5d, soF₂^(e)(z) has analytic continuation on the entireB[x;d], which we denote it byF₂(z). DenoteF₁(z)ˆF(z) F₂(z).

One also has

F^(e)₁ (z)ˆ X

s(x)2B(x;d)s2Se

F^(e)_1;s(z);

where F_1;s^(e)(z)ˆ P

n1

a^(e)_n;s

(z s(x))ⁿ, ja^(e)_n;sj

eⁿ !0. By Cauchy's inequalities, one obtains

…17† ja^(e)_n;sj eⁿkF₁k_E[x;e;d]ˆ

eⁿkF F₂k_E[x;e;d]eⁿmaxfkFk_E[x;e;d];kF₂k_E[x;e;d]g; n1:

The Mittag-Leffler's decomposition is unique, so, for any 05e⁰5eone has F_1;s^(e)(z)ˆ X

t2Se0;^tˆ^s s(x);t(x)2B(x;d)

F_1;t^(e0)(z); z2E[x;e;d]

(here^tˆ^smeanst2sH(x;e)) and so for anys(x)2B(x;d) X

n1

a^(e)_n;s

(z s(x))ⁿˆ X

t2Se0;^tˆ^s t(x)2B(x;d)

X

m1

a^(e_m;t⁰⁾ (z t(x))^m: …18†

Since jz s(x)j e, and js(x) t(x)j5e, it follows that jz t(x)j ˆ jz s(x)jand so:

a^(e_m;t⁰⁾

(z t(x))^mˆ a^(e_m;s⁰⁾ (z s(x))^m

1 t(x) s(x) z s(x)

_m

ˆ a^(e_m;t⁰⁾ (z s(x))^m

X

k0

m‡k 1 k

ht(x) s(x) z s(x)

i_k : …19†

If we denotem‡kˆn, then by identifying the coefficients of the terms of degreenin (18) and (19) one obtains:

a^(e)_n;sˆ X

t2Se0;^tˆ^s t(x)2B(x;d)

X^{n 1}

kˆ0

n 1 k

a^(e_{n k;t}⁰⁾ (t(x) s(x))^k; …20†

wheren1.

Now for any n1 one defines a sequence fm_n;eg_n;e of measures on O(x)\B[x;d] by the equality

m_n;eˆX

s2Se

a^(e)_n;sd_s(x); …21†

wheredydenotes the Dirac measure concentrated aty2Cp.

By (20) one obtains, fornˆ1:

a^(e)_1;sˆ X

t2Se0;^tˆ^s t(x)2B(x;d)

a^(e_1;t⁰⁾; 05e⁰e:

This equality further implies that for any ballBof radiusd⁰,ed⁰5d, one hasm_1;e(B)ˆm_1;e0(B) whenevere⁰e. Then by (17) and the Banach-Stein- haus Theorem (see [R]) there results that the mapping

B?m₁(B)ˆlim

e m_1;e(B)

(where B runs over all the open balls of O(x)\B[x;d]) defines a p-adic measure onO(x)\B[x;d]. One says thatmˆm₁is themeasure associated to the rigid analytic functionF. Here we remark that ifk kF _E[x;e;d]is uni- formly bounded for any 05e5done hasmˆ0.

Furthermore, fornˆ2, by (20) one obtains a^(e)_2;sˆ X

t2Se0;^tˆ^s t(x)2B(x;d)

[a^(e_2;t⁰⁾‡a^(e_1;t⁰⁾(t(x) s(x))]:

IfBis an open ball ofO(x)\B[x;d] of radiusd⁰, by the previous equality and (21) there results that

m_2;e(B) m_2;e0(B)ˆX

s(x)2Bs2Se

X

t2Se0;^tˆ^s t(x)2B(x;d)

a^(e_1;t⁰⁾(t(x) s(x)) 2

64

3 75

and so by (17) one has:

jm_2;e(B) m_2;e0(B)j ee⁰k kF _E[x;e0;d]eM

whereMˆsupd>e>0ek kF _E[x;e;d]51, by hypothesis. Then by (17) and the Banach-Steinhaus Theorem, there exists a measure m₂ onO(x)\B[x;d]

defined by

m₂(B)ˆlim

e m_2;e(B);

for all the open ballsBofO(x)\B[x;d]. In the same manner for alln3 one can define a measurem_nonO(x) by:

m_n(B)ˆlim

e m_n;e(B):

Next, by an easy computation it follows that for all n2, one has m_n;e Me^{n 1}, and so m_nˆ0 for all n2. In what follows we shall

prove that

F₁(z)ˆ Z

O(x)\B[x;d]

1

z tdm(t); z2E[x;d]:

…22†

Indeed, with the above notations and using (17) one has:

F₁(z) Z

O(x)\B[x;d]

1

z tdm_1;e(t) ˆ

F₁(z) X

s(x)2B(x;d)s2Se

a^(e)_1;s z s(x)

X

s(x)2B(x;d)s2Se

X

n2

a^(e)_n;s (z s(x))ⁿ

; …23†

for anyz2E(x;d). Ifdˆdist(z;O(x)\B(x;d))>0, for any 05e5dfrom (23) and (17) one has

F1(z) Z

O(x)\B[x;d]

1

z tdm_1;e(t) sup

n2

1 d

e d

_{n 1}

ek kF _E[x;e;d]; …24†

which tends to zero whenetends to zero.

By the definition ofmˆm₁ (see (21)) one has Z

O(x)\B[x;d]

1

z tdm(t)ˆlim

e!0

Z

O(x)\B[x;d]

1

z tdm_1;e(t):

By (24) and the analytic continuation principle one obtains (22). Finally one has the following result, which is similar to a theorem of Barsky (see [B] and [APVZ1]):

THEOREM2. Let x be a transcendental element of C_pand letd be a fixed positive real number. Let F:E[x;d]!Cpbe a rigid analytic func- tion such that the set of real numbersfekFk_E[x;e;d]gd>e>0is bounded. There exists a unique p-adic measurem_F on O(x)\B[x;d]such that F has the following integral representation

F(z)ˆG(z)‡ Z

O(x)\B[x;d]

1

z tdm_F(t); z2E[x;d];

…25†

where G(z)is an entire function on B[x;d], and the representation of F is unique.

REMARK. It is easy to see that ifFhas a singular pointt2O(x)\B[x;d]

thenk kF _E[x;e;d]! 1;e!0 so one has lim sup_z!tjF(z)j ˆ 1and moreoverF is transcendental overCp(z). Indeed, if the reverse is true one hasm_Fˆ0 andFhas no singular points onO(x)\B[x;d], which is false.

One has the following result.

THEOREM3. Let x be a transcendental element of Cpand letd be a fixed positive real number. Let F:E[x;d]!Cpbe a rigid analytic func- tion such that all the points of O(x)\B[x;d]are singular points of F. Then lim inf_z!xjF(z)j ˆ0.

PROOF. Let us suppose the reverse. One has lim infz!xjF(z)j>0 so there exists M>0 and d⁰>0 such that jF(z)j>M for any z2B[x;d⁰]nO(x). For the sake of simplicity, let us supposedˆd⁰. (We can work with the restriction of F toE[x;d⁰].) By definition, F is a limit (in uniform convergence on eachE[x;e;d]) of rational functions of the form

F(z)ˆ lim

n!1

P^(e)_n(z) Q^(e)_n(z); …26†

without poles inE[x;e;d]. Fix 05e5dand letn_Mbe a natural number such that

F P^(e)_n Q^(e)_n

E[x;e;d]

5M

for any nn_M. Because jF(z)j>M, for any z2E[x;e;d] one obtains P^(e)_n(z)

Q^(e)n (z)

>M, for any z2E[x;e;d], so F is a uniform limit of rational functions without zeros and poles inE[x;e;d]. DenoteG^(e)_n(z)ˆP^(e)_n (z)

Q^(e)n(z). One has

1 F

1 G^(e)n

E[x;e;d]

F G^(e)_n

E[x;e;d] 1 FG^(e)n

E[x;e;d]

5 1

M²F G^(e)_n

E[x;e;d]

for anynn_Mso 1

Fis rigid analytic onE[x;d]. Two cases may appear:

(I) The function 1

Fhas analytic continuation onO(x)\B[x;d] and then 1

Fˆ0 onO(x)\B[x;d]. Sincexis transcendental overQp,O(x) has limit points insideB[x;d] (see [APZ1]), and this forces 1

Fto be identically zero, which is a contradiction.

(II) 1

F has singular points onO(x)\B[x;d]. Let us suppose that xis a singular point. As in the above considerations one obtains a sequence (y_n)_n1 such that y_n!x and 1

F(y_n)! 1 i.e. jF(y_n)j !0 that is also

false. p

REMARK. The hypothesis that x is transcendental over Qp is really needed in the statement of Theorem 3. Situation is completely different ifx is algebraic. Indeed, ifxˆais algebraic overQp, one can takeFˆ1=Pa, whereP_ais the minimal polynomial ofaoverQ_p, and then the conclusion of the theorem is obviously false for suchF.

COROLLARY2. If F is as in the above Theorem and a2Cpthen there exists a sequence zn2CpnO(x)such thatlimn!1F(zn)ˆa.

PROOF. We apply Theorem 3 forF_aˆF a. p

REMARK. One could say that the points ofO(x)\B[x;d] are essential singular points forF.

COROLLARY 3. Let x2Cp be a transcendental element and d be a fixed positive real number. Denote Dˆ [_s2GB(s(x);d)nO(x). Let F2 A^G(D;C_p)be a rigid analytic function that is equivariant and such that x is a singular point of F, i.e. F does not have analytic continuation to [_s2GB(s(x);d). Then lim sup_z!xjF(z)j ˆ 1 and lim infz!xjF(z)j ˆ0.

Moreover, for any a2Qp there exists a sequence (zn)_n12CpnO(x) such that lim_n!1F(z_n)ˆa, so all the points of O(x) are essential singular points for F.

4. A few examples of rigid analytic functions with and without zeros 1) In this section we give an example of a rigid analytic function such that it has no zeros on P¹(C_p)nO(x). For, letx2C_pnQ_p, and let (en)_n1 be a strictly decreasing sequence with limit 0. Denote Hnˆ fs2G:js(x) xj5eng and Snˆ(Hn=Hn‡1)_left be a complete system of representatives on the left for H_n=H_n‡1. Let us define the following sequence of rational functions

fn(z)ˆYⁿ

iˆ1

Y

s2Si

1 x s(x) z s(x)

ˆYⁿ

iˆ1

Y

s2Si

z x

z s(x); n1:

…27†

For any d>0 the sequence (fn)n1 is uniform convergent on Cpn [_s2GB(s(x);d). It is easy to see that

fn‡1(z)ˆfn(z) Y

s2Sn‡1

1 x s(x) z s(x)

ˆfn(z)hn(z);

whereh_n(z)ˆ Q

s2Sn‡1

z x

z s(x). One hasf_n‡1(z) f_n(z)ˆf_n(z)(h_n(z) 1). For anyd>0 there existsndsuch thaten5dfor anyn>nd. In such a way one hasjfn(z)j ˆ jfnd(z)jas soon as dist(z;O(x))d. Moreover

hn(z) 1ˆ Y

s2Sn‡1

1 x s(x) z s(x)

1

ˆ X

s2Sn‡1

x s(x) z s(x)‡ X

s;t2Sn‡1

x s(x)

z s(x)x t(x) z t(x)

‡( 1)^#Sn‡1 Y

s2Sn‡1

x s(x) z s(x): …28†

By (28) one obtains

jhn(z) 1j sup

s2Sn‡1

x s(x) z s(x) en‡1

d ; …29†

for any n>n_d and any z with dist(z;O(x))d. Clearly, (29) implies limn!1jhn(z) 1j ˆ0 uniformly inz. We have

jfn‡1(z) fn(z)j ˆ jfnd(z)j jhn(z) 1j jfnd(z)j e_n‡1 d ; …30†

that tends to 0 as soon asn! 1uniformly inz, dist(z;O(x))d. By (27)- (30) one defines

f(z)ˆ lim

n!1fn(z);

…31†

that is a rigid analytic function onP¹(C_p)nO(x). Moreoverxis a singular point off, sof does not have analytic continuation to the entireO(x).

It remains to prove that f has no zeros in C_pnO(x). For, let z2 CpnO(x) such thatf(z)ˆlimn!1 fn(z)ˆ0. Let us supposedˆ jz xj. If e_n5d one has x s(x)

z s(x)51 so 1 x s(x)

z s(x)ˆ1 and jh_{n 1}(z)j ˆ1. It follows that for anyn>n_d

jfn(z)j ˆ jfnd(z)j;

wherendˆmaxfk:ekdg. It is clear thatjfn(z)jcannot be small enough as soon asjz xj ˆd, withdfixed, which means thatf has no zeros.

2) Now, we give an example of a rigid analytic function that has in- finitely many zeros aroundO(x). For, let us consider the same notations as above and let (a_n)_n1be a sequence fromQ_psuch thatjx a_nj e_nfor any n1, via the theorem of Ax-Sen, see [Ax]. Then we define

F_n(z)ˆYⁿ

iˆ1

Y

s2Si

1 a_i s(x) z s(x)

ˆYⁿ

iˆ1

Y

s2Si

z a_i

z s(x); n1:

…32†

Using the same arguments as above it is easy to see that (F_n(z))_n1 is uniformly convergent onC_pn [_s2GB(s(x);d), for anyd>0, and denote its limit byF(z), which is rigid analytic onP¹(C_p)nO(x) and has (a_n)_n1as the set of zeros.

3) Another example of rigid analytic function that has zeros in every neighborhood of any of its singular points is the trace series of a trans- cendental element ofC_p, for more details see page 38 of [APZ2].

REMARK. The above examples satisfy the following properties:

lim sup_z!xjf(z)j ˆ 1and lim infz!xjf(z)j ˆ0.

5. Some remarks related to zeros on a corona

Letf :P¹(Cp)nO(x)!Cpbe an arbitrary rigid analytic function that is equivariant with respect to G and f(1)ˆ0. For any positive real number e consider the Mittag-Leffler decomposition of f on E⁰(x;e):ˆ

P¹(C_p)n [_s2SeB(s(x);e), whereS_eis a complete system of representatives ofGwith respect to the subgroupsH(x;e)ˆ fs2G:js(x) xj5eg:

f(z)ˆX

s2Se

X

n1

a^(e)_n;s (z s(x))ⁿ; …33†

wherea^(e)_n;sˆs(a^(e)_n;e) andeis the neutral element ofG, see [APVZ1]. In what follows we considere_nthe fundamental sequence of all distances onO(x), more precisely the image of the following function: d:O(x)!R_‡, d(s(x))ˆ js(x) xj, which is continuous and its image is a strictly de- creasing sequence (en)_n1with limit 0. Fixe>0. There exists a uniquem such thateme5em 1,m1. (One can considere0ˆ 1.)

Denoteyˆ 1

z x. Then jz xj e if and only ifjyj 1

e. We have the

following identity X

i0

aⁱ

!_n

ˆX

k0

n‡k 1 k

a^k; jaj51:

…34†

For aˆ z x

s(x) x, s6ˆe, the condition jaj51 means that jz xj5

min_s2Se;s6ˆejs(x) xj, which means that for a fixed positive real numbere

withe_me5e_{m 1}one hase_me jz xj5e_{m 1}, so 1

em 15jyj 1 e. Using (34) in the above conditions one obtains

1

(z s(x))ⁿˆ 1

(x s(x))ⁿ

1 _s(x)^{z x}_xn

ˆ 1

(x s(x))ⁿ X

k0

n‡k 1 k

h z x s(x) x

i_k

ˆX

k0

( 1)^k n‡k 1 k

(x s(x))^n‡k (z x)^k: …35†

Let us defineef(y):ˆf(z) withyˆ 1

z xas above. From (33) and (35) one has

ef(y)ˆX

n1

a^(e)_n;eyⁿ‡X

k1

b^(e)_k y ^k‡b^(e)₀ ; …36†

where

b^(e)_k ˆX

s2Ses6ˆe

X

n1

( 1)^ka^(e)_n;s n‡k 1 k

(x s(x))^n‡k ; k0:

…37†

Becausef is equivariant by (37) we have the following upper bound jb^(e)_k j sup

n1

ja^(e)_n;ej e^n‡k_{m 1} ˆ 1

e^k_{m 1}sup

n1

ja^(e)_n;ej eⁿ_{m 1}

!

1

e^k_{m 1}A(f;e);

…38†

whereA(f;e)ˆsup

n1

ja^(e)_n;ej eⁿ

!

. The inequalities (38) give us

R_~f :ˆlim X

k!1

jb^(e)_k j^1=k 1 e_{m 1}: …39†

Now, from Cauchy's inequalities

ja^(e)_n;ej eⁿk kf _E0(x;e)

…40†

one obtainsja^(e)_n;ej¹⁼ⁿ ek kf ¹⁼ⁿ_E0(x;e)so R^‡_~f :ˆ 1

lim sup_n!1ja^(e)_n;ej¹⁼ⁿ 1 e > 1

e_{m 1}R_~f ; …41†

which means thatef(y) is a Laurent series on the coronaR_~f 5jyj5R^‡_~f . In such a way the zeros ofef(y) on the above corona are the same as the zeros of f(z) in the coronaeme jz xj5em 1.

REMARK. In the previous examples 1)-3) from Paragraph 4, ~f associated tof has no zeros in the above corona. Moreover, in example 2) the zeros of~fare precisely onjzj ˆR^‡_~f orjzj ˆR_~f . In the casex2Q_pit is easy to see thatR^‡_~f ˆ 1so the finite zeros of~f are precisely onjzj ˆR_~f . Acknowledgments. We are grateful to the referee of this paper for valuable suggestions and comments.

The fifth author's research was partially supported by NSF grant DMS-0901621.

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Manoscritto pervenuto in redazione il 21 giugno 2011.