L’INSTITUTFOURIER DE ANNALES

A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

DE

L’INSTITUT FOURIER

Kenichi BANNAI & Shinichi KOBAYASHI Integral structures onp-adic Fourier theory Tome 66, n^o2 (2016), p. 521-550.

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INTEGRAL STRUCTURES ON p-ADIC FOURIER THEORY

by Kenichi BANNAI & Shinichi KOBAYASHI (*)

Abstract. — In this article, we give an explicit construction of the p-adic Fourier transform by Schneider and Teitelbaum, which allows for the investigation of the integral property. As an application, we give a certain integral basis of the space of K-locally analytic functions on the ring of integers O_K for any finite extensionKofQp, generalizing the basis constructed by Amice for locally analytic functions onZp. We also use our result to prove congruences of Bernoulli-Hurwitz numbers at non-ordinary (i.e. supersingular) primes originally investigated by Katz and Chellali.

Résumé. — Dans cet article, nous donnons une construction explicite de la transformation de Fourier p-adique de Schneider et Teitelbaum, qui nous permet d’étudier son integralité. Comme application, pour toute extension finieKdeQp

nous donnons une certaine base entière de l’espace deK-fonctions localement ana- lytiques sur l’anneau des entiersO_K, en généralisant la base construite par Amice pour les fonctions localement analytiques sur Zp. Nous utilisons également notre résultat pour démontrer certaines relations de congruence étudiées initialement par Katz et Chellali entre nombres de Bernoulli-Hurwitz aux places non-ordinaires (c’est-à-dire supersingulières).

1. Introduction

One important method in studying the congruences and p-adic proper- ties of important invariants in number theory is the use ofp-adic measures interpolating such values. Such theory was applied to obtain the Kummer congruence between special values of Riemann zeta function as well as the

Keywords:p-adic distribution,p-adic Fourier theory, Amice transform, integrality, con- gruence, Lubin-Tate group, Bernoulli-Hurwitz number,p-adic periods.

Math. classification:11S40.

(*) The first and second authors were supported by the JSPS Postdoctoral Fellowships for Research Abroad 2005-2007/2004-2006. This work was also supported in part by KAKENHI (21674001, 25707001, 26247004).

construction of thep-adicL-functions for elliptic curves with ordinary re- duction atp. When dealing with the non-ordinary case, it is necessary to use the theory of p-adic analytic distributions, which is a generalization of the theory ofp-adic measures. For suchp-adic distributions onZp, the Amice transform gives a one-to-one correspondence betweenC^p-valued dis- tributions on Zp and rigid analytic functions on the open unit disc. The general idea is to study the congruences andp-adic properties of the inter- polated invariants through thep-adic property of the rigid analytic function corresponding to thep-adic distribution. However, contrary to the case of p-adic measures, the Amice transform is not well-behaved integrally for generalp-adic distributions, hence it is necessary to investigate in detail the precise integral structure of this transform. Amice [1, §10] investigated the precise integral structure of the Amice transform.

Let OK be the ring of integers of a finite extensionK of Qp. In [8, §4], Schneider and Teitelbaum constructed thep-adic Fourier transform, which is a one-to-one correspondence betweenCp-valued distributions onOK and rigid analytic functions on an open unit disc. The purpose of this article is to give an explicit and elementary construction of thep-adic Fourier transform of Schneider-Teitelbaum, which allows investigation of the precise integral structure of this correspondence. We then determine an integral structure on the ring of locally analytic functions onOK. The integrality of thep- adic Fourier transform for general K is even less well behaved than for the case of Qp; even if the rigid analytic function corresponding to a p- adic distribution has bounded coefficients, thep-adic distribution may not necessarily be ap-adic measure. As an application of our result, we obtain the congruences originally proved by Katz [7, Theorem 3.11] and Chellali [4, Théorème 1.1] of Bernoulli-Hurwitz numbers, which are essentially special values ofp-adicL-functions of CM elliptic curves at non-ordinary primes.

We now give the exact statements of our theorems. Let pbe a rational prime and let | · |be the absolute value of Cp such that |p| =p⁻¹. Let π be an uniformizer ofO_K, and letFq be the residue field ofO_K. We define LAN(OK,Cp) to be the space of locally analytic functions onOK of order Nwhich take values inC^p. That is,f(x)∈LAN(OK,C^p) if and only iff(x) is defined as a convergent power seriesP∞

n=0an(x−a)ⁿona+π^NOKfor any a∈ OK. We letkfka,N := maxn{|anπ^nN|}. The spaceLAN(OK,Cp) is ap- adic Banach space induced by the norm max_a∈OK{kfka,N}and we denote byLAN(OK,Cp)0the submodule of elements whose absolute values are less than or equal to 1. We letGbe a Lubin-Tate group ofK corresponding to

π, and let$_p∈Cp be ap-adic period ofG. We let ρ(k) = max

k6m{|m!/$_p^m|}, ρ(k) = min

06m6k{|m!/$^m_p|}.

See Proposition 3.1 for the properties of these numbers.

Letϕ(t) be a rigid analytic function on the open unit disc. In other words, ϕ(t) is a power series of the formϕ(t) =P∞

n=0c_ntⁿsuch that|cn|rⁿ₀ →0 for any 0< r0 <1. Let µϕ be the distribution on OK corresponding to ϕ(t) given by Schneider-Teitelbaum’s p-adic Fourier theory [8, Theorem 2.3].

Then we have the following:

Theorem 1.1. — Letf ∈LAN(OK,C^p). Then we have

Z

a+π^NOK

f(x)dµϕ

6 ρ(0) π q

N

kfka,NkϕkN

(1.1) where

(1.2) kϕkN := max

k

|ck|ρ k

q^N and[x]is the integral part ofx.

The crucial difference from the case whenK=Qpis the fact that|π/q|>

1 whenK6=Q^p. A finer version of the above is given as Theorem 4.3. Since ρh

k q^N

i∼p^−krwherer= 1/eq^N(q−1), the valuekϕkN is approximated by

kϕk_B(p−r)= max

x∈B(p^−r)

{ |ϕ(x)| },

whereB(p^−r)⊂Cp is the closed disc of radiusp^−r centered at the origin.

As an application of our main theorem, we obtain an estimate of the Fourier coefficients of Mahler like expansion of functions inLAN(OK,Cp).

Let λ(t) be the formal logarithm of G, and following [8], we define the polynomialP_n(x) by

exp(xλ(t)) =

∞

X

n=0

P_n(x)tⁿ.

Note that whenG is the multiplicative formal groupG=Gb^m, thenλ(t) = log(1 +t) and the above expansion is simply

(1 +t)^x=

∞

X

n=0

x n

tⁿ.

Hence the polynomialPn(x) is a generalization of the binomial polynomial x

n

= x(x−1)· · ·(x−n+ 1)

n! .

Then we have the following.

Theorem 1.2(Theorem 4.7). — The seriesP∞

n=0anPn(x$p)converges to an element ofLAN(OK,Cp)0 foran satisfying

|an|6ρ n

q^N

, lim

n→0|an|/ρ n

q^N

= 0.

Conversely, iff(x)∈LAN(OK,Cp)0, then it has an expansion f(x) =

∞

X

n=0

anPn(x$p) of the form

|an|6c π q

N

ρ n

q^N

, lim

n→0|an|/ρ n

q^N

= 0, wherec= 1 ife6p−1, and c=ρ(0)otherwise.

Corollary 1.3 (Corollary 4.8). — Suppose e_N,n:=γ

n q^N

P_n(x$_p), (n= 0,1,· · ·),

where γ(u)is an element in Cp such that ρ(u) = |γ(u)|. If we denote by L_N theO_Cp-module topologically generated bye_N,n, then

ρ(0)⁻² q π

N

LAN(OK,Cp)0 ⊂ LN ⊂ LAN(OK,Cp)0.

In particular,L_N⊗Qp =LA_N(OK,Cp). In other words, the functionse_N,n form ap-adic Banach basis ofLAN(OK,Cp). Moreover, ife6p−1, then

q π

N+1

LAN(OK,Cp)0 ⊂ LN ⊂ LAN(OK,Cp)0.

This result for the caseOK =Z^p gives the result of Amice[1, Théorème 3], namely that the functions

n p^N

! x

n

(n= 0,1,· · ·)

form a topological basis ofLA_N(Zp,Cp)₀(actually, we can show that it is a basis ofLAN(Zp,Qp)0).

As another application, in Theorem 5.8, we derive from our estimate of the integral the congruence of Bernoulli-Hurwitz numbers BH(n) at supersingular primes established by Katz [7, Theorem 3.11] and Chellali [4, Théorème 1.1].

Acknowledgment. — Part of this research was conducted while the first author was visiting the Ecole Normale Supérieure in Paris, and the second author Institut de Mathématiques de Jussieu. The authors would like to thank their hosts Yves André and Pierre Colmez for hospitality. The au- thors would also like to thank Seidai Yasuda for reading an earlier version of the manuscript and pointing out mistakes.

2. Schneider-Teitelbaum’s p-adic Fourier theory.

Let K be a finite extension of Qp and k = Fq the residue field. Let e be the absolute ramification index ofK. We fix a uniformizerπof K and letG be a Lubin-Tate formal group of K associated to π. For a natural numberN and an elementaofOK, we define the spaceA(a+π^NOK,Cp) ofK-analytic functions ona+π^NO_K by

A(a+π^NOK,Cp) :=

(

f:a+π^NOK→Cp|f(x) =

∞

X

n=0

a_n(x−a)ⁿ, a_n∈Cp, π^nNa_n→0 )

.

We equip the spaceA(a+π^NO_K,Cp) with the norm

kfka,N := maxn {|π^nNan|}= max_x∈a+πNO_Cp {|f(x)|}.

We also define the spaceLA_N(OK,Cp) of locallyK-analytic functions on OK of orderN by

LAN(OK,Cp) :=

f:O_K→Cp | f|_a+πNOK∈A(a+π^NO_K,Cp) for anya∈ O_K , which is ap-adic Banach space by the norm max_a {kfka,N}. We denote by LAN(OK,Cp)0 the submodule of elements whose absolute values are less than or equal to 1. We put

LA(O_K,Cp) =[

N

LA_N(O_K,Cp)

and equip it with the inductive limit topology. A continuousC^p-linear func- tionLA(OK,Cp)→Cpis called aCp-valued distribution onOK. We denote the space ofCp-valued distributions onOK byD(OK,Cp), i.e.

D(O_K,Cp) = lim

←−N

Hom^cont_C

p (LA_N(O_K,Cp),Cp).

We write an element ofD(OK,Cp) symbolically as Z

dµ:LA(OK,C^p)→C^p, f 7→

Z

f dµ= Z

OK

f(x)dµ(x).

The space D(OK,Cp) has a product structure given by the convolution product. For a compact open setU ofOK, we let

Z

U

f(x)dµ(x) :=

Z

OK

f(x)·1U(x)dµ(x), where 1U is the characteristic function ofU.

The structure of D(OK,Cp) is well-known for the case K = Qp and described through the so-called Amice transform. We denote by R^rig the ring of rigid analytic functions on the open disc of radius 1, that is, the ring of power series of the formϕ(T) =P∞

n=0c_nTⁿsuch that|c_n|rⁿ₀ →0 for any 0< r0<1. Then there exists an isomorphism of topological Cp-algebras (2.1) D(Zp,Cp)∼=R^rig, µ7→ϕ

that is characterized by the equation c_n =

Z

Zp

x n

dµ(x) or equivalently

ϕ(T) = Z

Zp

(1 +T)^xdµ(x).

For the Mahler expansion

f(x) =

∞

X

n=0

an

x n

of f ∈ LA(Zp,Cp), Amice showed that |an|rⁿ → 0 for some r > 1 and hence we can compute the integral as

(2.2)

Z

Zp

f(x)dµ=

∞

X

n=0

ancn.

Schneider-Teitelbaum [8, Theorem 2.3] constructed an isomorphism analo- gous to (2.1) for a general local fieldK.

Let $p be a p-adic period of G. By Tate’s theory of p-divisible groups and Lubin-Tate theory, we have

Hom_O

Cp(G,Gbm)∼= Hom_Zp(T_pG, T_pGbm)∼=O_K.

(The last isomorphism is non-canonical.) Hence there exists a generator of the O_K-module Hom_OC

p(G,Gbm), which is written in the form of the integral power series exp($pλ(t))∈ O_Cp[[t]] whereλ(t) is the logarithm of

G. The element$_p∈ O_Cp is determined uniquely up to an element ofO^×_K. We fix such a$p and call it the p-adic period of G (if the height of G is equal to 1, then theinverse of$_p is often called a p-adic period of G. For example, see [9]). It is known that|$p|=p^−s, wheres=_p−1¹ −_e(q−1)¹ (see Appendix of [8] or an elementary proof in [3] whenK/Qp is unramified).

We define the polynomialsPn(X)∈K[X] by the formal expansion exp(Xλ(t)) =

∞

X

n=0

P_n(X)tⁿ.

Note that in the caseG=Gbm,π=pandλ(t) = log(1 +t), the polynomial Pn(X) is simply the binomial polynomial ^X_n

. By construction, Pn(x$p) is inO_Cp ifx∈ O_K.

Theorem 2.1 (Schneider-Teitelbaum [8, §4]). i) The series

∞

X

n=0

anPn(x$p)

converges to an element of LA(OK,Cp) if limn|an|ⁿ¹ < 1. Con- versely, any locally K-analytic function f(x)onOK has a unique expansion

f(x) =

∞

X

n=0

anPn(x$p)

for some sequence(an)n inCp such thatlimn|an|ⁿ¹ <1.

ii) There exists an isomorphism of topologicalCp-algebras (2.3) D(OK,Cp)∼=R^rig.

having the following characterization property: ifϕ(T) =P∞ n=0cnTⁿ corresponds to a distributionµ, then

cn= Z

OK

Pn(x$p)dµ(x) or equivalently

ϕ(t) = Z

OK

exp(x$_pλ(t))dµ(x).

Schneider and Teitelbaum called the power seriesϕ(t)correspond- ing toµthe Fourier transform ofµand denoted it byFµ(t).

3. Power sums

In this section, we give an estimate of the absolute value of the power sum

SN,n,k:=∂ⁿ_G X

t_N∈G[π^N]

(t⊕tN)^k|t=0,

where x⊕y = G(x, y), ∂_G is the differential operator λ⁰(t)⁻¹(d/dt), and G[π^N] is the kernel of the multiplication [π^N] ofG. This estimate is crucial for everything in this paper. We use Newton’s method to compute this value.

We defineρ[l, n] andρ[l, n] by ρ[l, n] = max

l6m6n{|m!/$^m_p |}, ρ[l, n] = min

l6m6n{|m!/$^m_p|}

forl6n. Forl > n, we putρ[l, n] = 0 andρ[l, n] =∞. Thenρ(k) =ρ[k,∞]

andρ(k) =ρ[0, k] are the constants appearing in the introduction.

Proposition 3.1.

i) The valuesρ(k)andρ(k)are decreasing withk.

ii) We have

ρ(k)6ρ(k), ρ(k)6ρ(0)ρ(k).

iii) We have

ρ(k1+· · ·+kn)6ρ(k1)· · ·ρ(kn).

iv) We have

p^{p−11 −e(q−1)k} 6 ρ(k)61.

Proof. — i) is clear. For ii), first we haveρ(k)>|k!/$^k_p|>ρ(k). Suppose ρ(k) =|k1!/$^k_p¹|andρ(k) =|k2!/$_p^k2|. Thenk1>k>k2 and

k1!

$^kp¹

/ k2!

$^kp²

=

k1

k₂

(k1−k2)!

$p^k1−k2

6ρ(0).

For iii), suppose thatρ(k_i) =|li!/$_p^li|forl_i6k_i. Then the assertion for ρ follows from

ρ(k₁+· · ·+k_n)6

(l₁+· · ·+ln)!

$p^l1+···+ln

6

(l₁+· · ·+ln)!

l₁!· · ·l_n!

l₁!

$^lp¹

· · ·

ln!

$^lpⁿ

.

For iv), suppose that ρ(k) =|l!/$_p^l|forl6k. Then p^{p−11 −e(q−1)k} 6 p^{p−11 −e(q−1)l} 6

l!

$^l_p

=ρ(k).

Ife6p−1, then we can determineρ(k) andρ(k) explicitly.

Lemma 3.2. — Let k be a non-negative integer and let q be a power ofp.

i) For any integer06r < q, we have ^kq+r_r

≡1 modp.

ii) We have ^k_q

∈[k/q]Zp.

Proof. — i) is clear. For ii), we writek=aq+rwith 06r < q. We put (1 +x)^q = 1 +x^q+pf(x) for some integral polynomialf(x). Then

(1 +x)^k = (1 +x^q+pf(x))^a(1 +x)^r≡(1 +x^q)^a(1 +x)^r modapZp[x].

Hence the coefficient ofx^q in the above is inaZp. Proposition 3.3. — Leti,eandhbe natural numbers. We putq=p^h. Then we have

vp(i!)> i

p−1 − i

e(q−1)−h+1 e +

i q

1 e− 1

p−1+ 1 e(q−1)

+vp

i q

!

. In the above, equality holds if and only ifi≡ −1 modq. In particular, if e6p−1 ori < q, then we have

v_p(i!)> i

p−1− i

e(q−1)−h+1 e

and equality holds if and only ifi=q−1. In this case, we haveρ(0) =|π/q|.

Proof. — First, we assume that i < q. We prove the inequality by in- duction onh. Ifh= 1, theni < p. Hence the left-hand side v_p(i!) is equal to zero, and the right-hand side takes the maximum value wheni=p−1, which is also equal to zero. We assume that the inequality holds for natural numbers less thanh. Since the right-hand side is strictly increasing fori, andvp(i!) strictly increases only whenpdividesi, we may assume thatiis of the formi=kp−1 for some natural numberk6p^h−1. We have

vp(i!) =vp((kp)!)−vp(kp) =k−1 +vp((k−1)!).

On the other hand, we have i

p−1 − i

e(q−1)−h+1 e

= (k−1) +k−1

p−1 − k−1

e(p^h−1−1)−(h−1) +1

e + k−1

e(p^h−1−1) − kp−1 e(q−1) 6k−1 +vp((k−1)!).

In the last inequality, we used the inductive hypothesis andk6p^h−1. Hence we have the desired inequality, and equality holds only whenk=p^h−1, i.e.

wheni=q−1. For i>q, by Lemma 3.2 ii) and by induction, we have vp(i!)>vp((i−q)!) +vp(q!) +vp

i q

> i

p−1 − i

e(q−1)−h+1 e +

i q

1 e− 1

p−1+ 1 e(q−1)

+v_p

i q

!

. From the above argument and by induction, to have the equality, i must be congruent to−1 modulo q. On the other hand, ifi≡ −1 modq, then

direct calculations give the equality.

Proposition 3.4. — Suppose thate6p−1, and thate >1or h >1.

i) We have|n!/$_pⁿ|>1for0< n < q.

ii) For any non-negative integern,ρ(n) =|n0!/$_pⁿ⁰|wheren0= [n/q]q.

iii) Forn≡ −1 modqand a natural number i6=q, we have

n!

$ⁿ_p

>

(n+q)!

$p^n+q

>

(n+i)!

$ⁿ⁺ⁱp

In particular, for any non-negative integern, we haveρ(n) =|n1!/$ⁿ_p¹| wheren1= [n/q]q+q−1.

Proof. — We prove i) by induction onhof q=p^h. Ifh= 1, thenn! is a p-adic unit and the assertion is clear. Assume that h >1. We write as n=kp+r with 06r < p. Then

n!

$ⁿ_p = n

r (kp)!

$^kpp

r!

$^r_p.

Hence by Lemma 3.2 i) and the induction onn, we may assume thatr= 0 andk>1. Then

v_p (kp)!

$^kpp

!

=v_p((kp)!)− kp

p−1+ kp

e(q−1) < v_p(k!)− k

p−1+ k e(p^h−1−1). By the inductive hypothesis forh, the right-hand side is negative or 0.

Next we prove ii). Suppose that m < n0. Then

n₀!

$ⁿ⁰/ m!

$^m_p

=

n₀

$p

n₀−1 m

(n₀−m−1)!

$ⁿp^0−m−1

6

n₀

$p

ρ(0) =

n₀π q$p

<1.

Suppose thatn >m > n₀. We write as m = [n/q]q+rwith 0 < r < q.

Then i) and Lemma 3.2 i) show that

n0!

$pⁿ⁰

/ m!

$^m_p

=

m r

−1$^r_p r!

<1.

Finally, we show iii). Letnbe such that n≡ −1 modq. We have (n+i)!

$pⁿ⁺ⁱ

/(n+q)!

$^n+qp

= (n+i)!

(n+q)!$_p^q−i=uq π

(i−1)!

$pⁱ⁻¹

π$^q−1_p q! , whereu= ^n+q_q−1⁻¹ n+i

i−1

is ap-adic integer by Lemma 3.2 i). By Proposi- tion 3.3, thep-adic (additive) valuation of the right-hand side is positive.

Sincev_p(π/$_p)>0, thep-adic (additive) valuation of (n+q)!

$^n+qp

/ n!

$ⁿ_p = n+q

q

q!

π$^q−1p

π

$p

is positive.

Next we investigate the absolute values of the coefficients of a power of the logarithm and the exponential map of the Lubin-Tate group. The case k= 1 in the proposition below is obtained in [10].

Proposition 3.5. — We put∂=d/dt. Then we have

$^k_p∂ⁿλ(t)^k k!n! |t=0

6ρ[k, n]⁻¹,

∂ⁿexp^k_G(t)|t=0

6|$ⁿ_p|ρ[k, n].

Proof. — The case forn < kork= 0 is trivial. Suppose thatn>k>1.

We first assume that the formal logarithm ofG is given by λ(t) =

∞

X

m=0

t^qm π^m. Then it suffices to show inequalities

∂ⁿλ(t)^k|t=0

6|k!$^n−k_p |,

∂ⁿexp^k_G(t)|t=0

6|k!$^n−k_p |.

Whenk= 1, the inequality forλ(t) is proven by direct calculations. We prove the general case by induction onk. We have

∂ⁿλ(t)^k|t=0=k∂ⁿ⁻¹(λ(t)^k−1λ⁰(t))|t=0

=k∂ⁿ⁻¹

∞

X

m=0

λ(t)^k−1q^mt^qm−1 π^m |t=0

=

∞

X

m=0

n−1 q^m−1

q^m!k

π^m ∂^n−qmλ(t)^k−1|t=0.

Hence we have|∂ⁿλ(t)^k|t=0|6|k!$_p^n−k|.

We put exp^k_G(t) = P∞

n=ka_ntⁿ. We prove that |n!an| 6 |k!$^n−k_p | by induction onn. Ifn=k, then the assertion is true sinceak = 1. We assume that the assertion is true for integers less than n. Since exp^k_G(λ(t)) = t^k, we have

t^k=a_kλ(t)^k+a_k+1λ(t)^k+1+· · ·+a_nλ(t)ⁿ+· · · . By i) and the inductive hypothesis, we have

|am∂ⁿλ(t)^m|t=0|6|k!$^n−k_p |

form < n. Since ∂ⁿλ(t)ⁿ|t=0 =n! and ∂ⁿλ(t)^m|t=0 = 0 for n < m, the assertion is also true forn.

Now we consider a general parameters. Then the logarithm and the ex- ponential forGwith parametersare of the formλ(φ(s)) andψ(exp_G(s)) for someφ(s), ψ(s)∈sOK[[s]]^×. We putλ(t)^k=P∞

n=kc^(k)n tⁿ and λ(φ(s))^k = Pd^(k)n sⁿ. Then we have shown that |c^(k)n | 6|k!$^n−k_p /n!|. Since d^(k)n is a linear sum ofc^(k)_l (k6l6n) with integral coefficients, we have

$^k_pd^(k)n

k!

6 max

k6l6n

(

c^(k)_l $_p^k k!

)

6 max

k6l6n

(

$^l_p l!

)

=ρ[k, n]⁻¹. Hence we have the inequality for the logarithm. The inequality for the

exponential is straightforward.

Lemma 3.6.

i) Suppose thatf(t)∈ OK[[t]]satisfies f(t⊕tN) =f(t)for all tN ∈ G[π^N]. Then there exists a power series g(t) ∈ OK[[t]] such that f(t) =g([π^N]t).

ii) There exists an integral power seriesg_k(t)∈ OK[[t]]such that π^−N X

t_N∈G[π^N]

(t⊕tN)^k=gk([π^N]t).

Proof. — See [5], Chapter III.

We put

F(t, X) = Y

t_N∈G[π^N]

(1−(t⊕tN)X) = 1 +α1(t)X+· · ·+α_qN(t)X^qN. For∂_X =∂/∂X, we consider the power series

(3.1) π^−N∂_XF(t, X) F(t, X) =−

∞

X

k=0



π^−N X

t_N∈G[π^N]

(t⊕t_N)^k+1



X^k.

By Lemma 3.6 and the above formula,π^−N∂_XF(t, X)∈ OK[[t]][X].

Proposition 3.7. — Letk, nbe non-negative integers andN a natural number. Then we have

(3.2)

π^−N X

t_N∈G[π^N]

∂_Gⁿ(t⊕t_N)^k|t=0

6

π^{N n+k0(1−q−11 )}$ⁿ_p ρ

k q^N

ρ(0),

wherek₀= max{[k/q^N]−n,0}. We also have (3.3)

π^−N X

t_N∈G[π^N]

∂_Gⁿ(t⊕tN)^k|t=0

6

π^{N n}$ⁿ_p

ρ[0, n]. Moreover, ife6p−1, we have

(3.4)

π^−N X

tN∈G[π^N]

∂_Gⁿ(t⊕tN)^k|t=0

6

π^{N n}$ⁿ_p ρ

k q^N

.

Proof. — We putG(t, X) =F(0, X)−F(t, X), thenG(0, X) =G(t,0) = 0. We have

1

F(t, X)= 1

F(0, X)−G(t, X) =

∞

X

l=0

G(t, X)^l

F(0, X)^l+1 ∈ OK[[t, X]].

SinceG(0, X) = 0 andG(t, X) is invariant for the translationt7→tN, it is of the form

(3.5) G(t, X) = ([π^N]t)H([π^N]t, X)

for some elementH in OK[[t]][X]. Since F(0, X) ≡1 modπ, the power seriesF(0, X)^−l−1 is equal to

∞

X

m=0

−l−1 m

(F(0, X)−1)^m=

∞

X

m=0

l+m m

π^m

1−F(0, X) π

^m . Hence we have

π^−N∂_XF(t, X) F(t, X) =

∞

X

l=0

π^−N∂_XF(t, X)·G(t, X)^l·F(0, X)^−l−1

=

∞

X

l=0

∞

X

m=0

l+m m

π^m(π^−N∂XF(t, X))G(t, X)^l

1−F(0, X) π

^m . (3.6)

To show the assertion fork+ 1, we look the coefficient ofX^k in the last term of (3.6). We consider the coefficients of the terms X^a, X^b and X^c with a+b+c = k of π^−N∂_XF(t, X), G(t, X)^l and (1−F(0, X))^mπ^−m respectively. Since deg∂XF(t, X) =q^N−1, degG(t, X) =q^N and deg (1−

F(0, X)) = q^N −1 as polynomials for X, we have a 6 q^N −1, b 6 lq^N andc6m(q^N −1). Then by (3.5), the product of these coefficients is an integral linear combination of the terms of the form

l+m m

π^mG_l([π^N]t)

whereGl(t) is a power series in t^lOK[[t]] andl,msatisfies (3.7) a+lq^N+m(q^N −1)>a+b+c=k.

We estimate the absolute value of (3.8)

l+m m

π^m∂ⁿ_GG_l([π^N]t)|_t=0. By Proposition 3.5, we have

∂_Gⁿ([π^N]t)^d|t=0

=

π^{N n} dⁿ

dzⁿ exp^d_G(z)|z=0

6

π^{N n}$_pⁿ ρ[d, n].

Therefore, we have

∂_GⁿGl([π^N]t)|t=0

6

π^{N n}$ⁿ_p ρ[l, n].

Hence we have (3.3). If n < l, then (3.8) is zero and there is nothing to prove. We assume thatn>l. We let l⁰ >l be such that ρ(l) =|l⁰!/$_p^l0|.

Then

l+m m

π^m∂_GⁿGl([π]^Nt)|t=0

6

l+m m

π^{m+N n}$_pⁿ

ρ[l, n]

(3.9)

6

π^{N n}$ⁿ_p(l+m)!

$^l+mp

(l⁰−l)!

$p^l0−l

l⁰ l

$_p^mπ^m m!

. (3.10)

First we consider the case a6q^N−2 orm6= 0. Then by (3.7) we have l+m>

k+ 1 q^N

.

In particular, m > [(k+ 1)/q^N]−n and the value (3.10) is less than or equal to

|π^{N n+k0(1−q−11 )}$ⁿ_p|ρ

k+ 1 q^N

ρ(0)

where k0 = max{[(k+ 1)/q^N]−n,0}. Hence in this case we have (3.2).

Suppose thate6p−1. Ifl⁰ < l+m, then

$_p^m <

$_p^l0−l

and hence the value (3.10) is less than|π^{N n}$ⁿ_p|ρh

k+1 q^N

i

.Ifl⁰>l+m, then ρ(l) =

l⁰!

$^l_p⁰

6ρ(l+m)6ρ

k+ 1 q^N

.

Hence the value (3.9) is also less than or equal to|π^{m+N n}$_pⁿ|ρh

k+1 q^N

i . Hence in this case we have (3.4).

Finally we consider the case when a = q^N −1 and m = 0. Then the coefficient of π^−N∂XF(t, X) of degreea is (q/π)^Nα_qN(t), which is divis- ible by [π^N]t. Hence in this case the product of the coefficient of X^a in π^−N∂XF(t, X), the coefficient ofX^b in G(t, X)^land the coefficient of X^c in (1−F(0, X))^mπ^−m is an integral linear combination of terms in the form Gl+1([π^N]t) for some Gl+1(t) ∈ t^l+1OK[[t]]. In this case l satisfies l+ 1>

(k+ 1)/q^N

. Therefore ∂_GⁿGl+1([π]^Nt)|t=0

6

π^{N n}$ⁿ_p

ρ[l+ 1, n]6

π^{N n}$ⁿ_p ρ

k+ 1 q^N

. Ifn < l+ 1, then (3.8) is zero and there is nothing to prove. We assume thatn>l+ 1. In particular, by (3.7) we haven>[(k+ 1)/q^N], and hence k₀= max{[(k+ 1)/q^N]−n,0}= 0. Therefore we have (3.2) and (3.4).

4. Integral structures on p-adic Fourier theory In this section, we give an explicit construction of Schneider-Teitelbaum’s p-adic distribution associated to a rigid analytic function on the open unit disc.

Let ϕ(t) be a rigid analytic function on the open unit disc. We will construct a distributionµ_ϕ onOK such that

Z

OK

exp(x$pλ(t))dµϕ=ϕ(t).

If we were able to first prove a Mahler like expansion for K-analytic functions as in the case ofK=Qp, then it would be possible to define the integral by (2.2). However, as in [8], we will first define the integral then use this integral to prove the existence of the Mahler like expansion forK- analytic functions. Our construction of the integral is different from that of [8] in that we investigate directly the explicit power series corresponding to the moments of the integral, instead of formally reducing to the case ofZp.

We fix a Lubin-Tate formal group G associated to π, and denote its addition by⊕. Fora∈ OK and a natural number N, we let

Z

a+π^NOK

(x−a)ⁿdµ_ϕ:= 1 q^N$ⁿ_p



∂_Gⁿ X

t_N∈G[π^N]

ϕ_a(t⊕t_N)



 _t=0

where

ϕ_a(t) := exp(−a$pλ(t))ϕ(t).

We put ϕ(t) = P∞

k=0ckt^k and ϕa(t) = P∞

k=0c^(a)_k t^k. Then by Proposi- tion 3.7, we have

Z

a+π^NOK

(x−a)ⁿdµ_ϕ 6ρ(0)

π q

N

|π|^{N n}sup

k

{ |c^(a)_k |ρ k

q^N

} (4.1)

6ρ(0) π q

N

|π|^{N n}sup

k

{ |ck|ρ k

q^N

}.

Here for the last estimate, we used the facts thatc^(a)_k is an integral linear combination ofc₀, . . . , ck and the functionρ(m) form is decreasing.

We define the distributionµϕonLAN(OK,Cp) as follows. For an element f of LAN(OK,Cp), suppose f is of the form P∞

n=0an(x−a)ⁿ such that a_nπ^nN →0 ifn→ ∞ona+π^NOK. Then we define the integral of f on a+π^NOK by

(4.2) Z

a+π^NO_K

f(x)dµ_ϕ:=

∞

X

n=0

a_n Z

a+π^NO_K

(x−a)ⁿdµ_ϕ. We define

(4.3)

Z

OK

f(x)dµ_ϕ= X

a modπ^N

Z

a+π^NOK

f(x)dµ_ϕ. We have to show the well-definedness of the integral.

Proposition 4.1.

i) The integral (4.2) converges and does not depend on the choice of the representative ofamodπ^N. The integral(4.3)does not de- pend on the choice ofN. Henceµϕ gives a well-defined element of D(OK,Cp).

ii) For a polynomialf(x), we have Z

OK

f(x)dµϕ=f($⁻¹_p ∂_G)ϕ(t)|t=0.

Proof. — Since ρ([k/q^N]) 6 Ckp^{−eqN(q−1)k} for some constant C which depends only one, qandN, the value sup_k{ |ck|ρh

k q^N

i}is finite. Hence the convergence follows from (4.1). We show that the integral (4.2) depends only on the class ofamoduloπ^N. Since the integral is convergent, we may