Formal group exponentials and Galois modules in Lubin-Tate extensions

0.27 MB·English

by  Erik Jarl Pickett

Formal group exponentials and Galois modules in
Lubin-Tate extensions
Erik Jarl Pickett and Lara Thomas
2
1
0 January 20, 2012
2

n
a
J
Abstract

9
1 ExplicitdescriptionsoflocalintegralGalois modulegeneratorsincertainextensions

of p-adic fields due to Pickett have recently been used to make progress with open
]
T questions on integral Galois module structure in wildly ramified extensions of number
N fields. In parallel, Pulita has generalised the theory of Dwork’s power series to a set of
. power series with coefficients in Lubin-Tate extensions of Q to establish a structure
h p
t theorem for rank one solvable p-adic differential equations.
a
m In this paper we first generalise Pulita’s power series using the theories of formal
[ group exponentials and ramified Witt vectors. Using these results and Lubin-Tate

theory, we then generalise Pickett’s constructions in order to give an analytic represen-
1
v tation of integral normal basis generators for the square root of the inverse different
3 in all abelian totally, weakly and wildly ramified extensions of a p-adic field. Other
2
applications are also exposed.
0
4
.
1 Introduction
0
2
1 The main motivation for this paper came from new progress in the theory of Galois
:
v module structure. Indeed, explicit descriptions of local integral Galois module gener-
i
X ators due to Erez [5] and Pickett [16] have recently been used to make progress with
r open questions on integral Galois module structure in wildly ramified extensions of
a
number fields (see [18] and [23]).
Precisely, let p be a prime number. Pickett, generalising work of Erez, has con-
structed normal basis generators for the square root of the inverse different in degree p
extensions of any unramified extension of Q . His constructions were obtained by us-
p
ing special values of Dwork’s power series. Moreover, they have recently been used by
Pickett and Vinatier [18] to prove that the square root of the inverse different of E/F
is free over Z[G] under certain conditions on both the decomposition groups of G and
the base field F, when E/F is a finite odd degree Galois extension with group G.
In parallel, Pulita has generalised the theory of Dwork’s power series to a set of
power series with coefficients in Lubin-Tate extensions of Q in order to classify rank
p
one p-adic solvable differential equations [19].
1
Our main goal was to generalise Erez and Pickett’s construction in order to give
explicitdescriptionsofintegralnormalbasisgeneratorsforthesquarerootoftheinverse
different in all abelian totally, weakly and wildly ramified extensions of a p-adic field.
In this paper, our goal is totally achieved using a combination of several tools : formal
group exponentials, Lubin-Tate theory, and the theory of ramified Witt vectors. This
leads us to generalise Pulita’s formal power series to power series with coefficients in
Lubin-Tate extensions of any finite extension of Q .
p
At the same time, we also get explicit generators for the valuation ring over its
associated order, in maximal abelian totally, weakly and wildly ramified extensions of
any p-adic field.
Notation. Let p be a rational prime, and let Q be the field of p-adic numbers,
p
Q¯ be a fixed algebraic closure of Q and C be the completion of Q¯ with respect
p p p p
to the p-adic absolute value. We let v and | | be the normalised p-adic valuation
p p
and absolute value on C such that v (p) = 1 and |x| = p−vp(x). As v and | | are
p p p p p
completely determined byeach other, either can beusedinthestatement of results; we
will use the valuation v as this is the convention in the literature on Galois modules
p
in Lubin-Tate extensions.
Throughout this paper, for any extension K/Q considered we will always assume
p
K is contained in Q¯ and we will denote by O , P and k its valuation ring, maximal
p K K
ideal and residue field respectively. We identify the residue field of Q with the field
p
of p elements, F . For any n ∈ Z , we denote by µ the group of nth roots of unity
p >0 n
contained in Q¯×.
p
Presentation of the paper. Let γ ∈ Q¯ be a root of the polynomial Xp−1 + p.
p
Dwork’s exponential power series with respect to γ is defined as
E (X) = exp(γ(X −Xp)) ∈ 1+XZ [[X]],
γ p
where the right hand side is the composition of the two power series γ(X −Xp) and
exp(X). Dwork’s power series is over-convergent, in the sense that it converges with
respect to | | on an open disc {x ∈C : v (x) > c} for some c < 0 ([14], Chap. 14, §2,
p p p
Remark after Lem. 2.2); it also has the property that E (1) is equal to a primitive pth
γ
root of unity in Q¯ ([14], Chap. 14, §3, Thm 3.2).
p
Dwork’s power series was recently generalised by Pulita [19] to a set of power series
withcoefficientsinLubin-TateextensionsofQ : Letf(X) ∈ Z [X]besomeLubin-Tate
p p
polynomial with respect to the uniformising parameter p, i.e.,
P(X) ≡ Xp mod pZ [X] and P(X) ≡ pX mod X2Z [X] .
p p
Let {ω } be a coherent set of roots associated to P(X), namely a sequence of el-
i i>0
ements of Q¯ such that ω 6= 0, P(ω ) = 0 and P(ω ) = ω ; we refer to ω as an
p 1 1 i+1 i n
nth Lubin-Tate division point with respect to P. For n ∈ Z , Pulita defines the
>0
exponentials
n−1ω (Xpi −Xpi+1)
E (X) = exp n−i ∈ 1+XZ [ω ][[X]] .
P,n pi p n
!
i=0
X
2
Thisgeneralises Dwork’s powerseries as E (X) = E (X) whenP(X) = Xp+pX.
P,1 γ
For all choices of P(X) and n, the power series E (X) is over-convergent and has the
n
property that E (1) is a primitive pnth root of unity ζ in Q¯ . Comparing degrees
n pn p
then shows us that Q (ω ) = Q (ζ ) for all n and all choices of P(X). We remark
p n p pn
that this result is also a consequence of basic Lubin-Tate theory, see [15] or [20] for
details of this theory.
In this paper, we first generalise Pulita’s exponentials to power series with coef-
ficients in Lubin-Tate extensions of any finite extension K of Q , in particular by
p
combining Fro¨hlich’s notion of a formal group exponential ([7], Chap. IV, §1) with the
theory of ramified Witt vectors. Note that we impose no other restrictions on our
base field K and no restrictions on the uniformising parameter used to construct the
Lubin-Tate extensions of K. Inspired by the methods of Pulita, we prove the following
core result of the paper :
Theorem 1 Let K be a finite extension of Q , with valuation ring, maximal ideal and
p
residue field denoted by O , P and k respectively. Let q = card(k) for some power q
K K
of p.
Let π and π′ be two uniformising parameters for O , and let P,Q ∈ O [[X]] be
K K
Lubin-Tate polynomials with respect to π and π′ respectively, i.e.,
P(X) ≡ Xq mod πO [X] and P(X) ≡ πX mod X2O [X] .
K K
We write F ∈ O [[X,Y]] for the unique formal group that admits P as an endo-
P K
morphism and exp ∈ XK[[X]] for the unique power series such that
FP
exp (X +Y) = F (exp (X),exp (Y)).
FP P FP FP
Let {ω } be a coherent set of roots associated to Q.
i i>0
1. For every n≥ 1, the formal power series
n−1ω (Xqi −Xqi+1)
Q n−i
E (X) := exp
P,n FP πi
!
i=0
X
lies in XO [ω ][[X]], and is over-convergent if π ≡ π′ mod Pn+1.
K n K
2. Moreover, we have the congruence EQ (X) ≡ ω X mod ω2XO [ω ][[X]].
P,n n n K n
To compare to Pulita’s result, we remark that if K = Q , π = π′ = p, and P(X) =
p
(X +1)p−1, then EQ (X) = E (X)−1.
P,n Q,n
WethenapplyTheorem1togivetwoexplicitresultsinLubin-Tatetheory. Foreach
integer n ≥ 1, wedenotebyK then-thLubin-TateextensionofK withrespecttoπ.
π,n
Thisextensionisabelianandtotally ramified,withdegreeqn−1(q−1)andconductorn.
Let P ∈ O [X] be a Lubin-Tate polynomial with respect to π. The extension K /K
K π,n
is generated by any primitive n-th Lubin-Tate division point with respect to P, i.e.,
any element ω ∈Q¯ such that P(n)(ω) = 0 whereas P(n−1)(ω) 6=0.
p
3
As a first application of Theorem 1, we give an analytic representation of Lubin-
Q
Tate division points as values of the power series E (X) for all n ≥ 0. Precisely,
P,n
keeping the same notation, we prove :
Proposition 1 1. If π ≡ π′ mod Pn+1, then EQ (1) is a primitive mth Lubin-
K P,m
Tate division point with respect to P, for all integers m with 0 < m ≤ n.
2. If π′ = π, then {EQ (1)} is a coherent set of roots associated to P(X).
P,i i>0
Another application of Theorem 1 is concerned with an explicit description of the
action of the Galois group Gal(K /K) over the Lubin-Tate extension K for all
π,n π,n
Q
n ≥ 1. Indeed, since E (1) is a primitive nth Lubin-Tate division point with respect
P,n
to P, from standard theory (see [11], §6-7, specifically Theorem 7.1) we know that the
Q Q
elements of Gal(K /K) are those automorphisms such that E (1) 7→ [u] (E (1)),
π,n P,n P P,n
where u runs over a set of representatives of O×/(1+Pn) — here [u] (X) is a specific
K K P
power series in XO [[X]], see Section 1.1 for full details. Therefore, the following
K
proposition gives a complete description of Gal(K /K) in terms of values of our
π,n
power series.
Proposition 2 Let π ≡ π′ mod Pn+1. For 0 ≤ i ≤ n−1, let z ∈ µ ∪{0} with
K i q−1
z 6= 0. Then,
0
n−1
[ z πi] (EQ (1)) = EQ (z )+ EQ (z )+ ...+ EQ (z ) .
i P P,n P,n 0 FP P,n−1 1 FP FP P,1 n−1
i=0
X
Note that, when z = ··· = z = 0, this proposition implies the relation
1 n−1
Q Q
E (z )= [z ] (E (1)) , for all z ∈ µ .
P,n 0 0 P P,n 0 q−1
This is a generalisation of ([13], Chap. 14, Thm. 3.2).
Finally, we shall prove our second main result, as a consequence of Theorem 1 and
Q
Proposition 1 : we use the power series E (X) to construct explicit normal basis
P,2
generators in abelian, weakly and wildly ramified extensions of any p-adic field. In this
manner, we generalise the constructions of Erez and Pickett, and give supporttowards
the resolution of open questions in Galois module structure theory :
Theorem 2 Let K be a finite extension of Q , with valuation ring O , residue field
p K
k and residue cardinality q. Fix a uniformising parameter π of K and let P(X) =
Xq + q−1a Xi +πX be some Lubin-Tate polynomial of degree q with respect to π.
i=2 i
Let M be a maximal abelian totally, weakly and wildly ramified extension of K. Let
π,2
P
A denote the unique fractional ideal in M whose square is equal to the inverse
Mπ,2/K π,2
different of M /K (see Section 3.3).
π,2
Let π′ ∈ K be another uniformising element of K, with π ≡ π′ mod p3 ; and let
K
Q ∈ O [X] be a Lubin-Tate polynomial with respect to π′.
K
If v (a ) = v (π), then
p q−1 p
4
Q
1. the trace element Tr (E (1)) is a uniformising parameter of M and
Kπ,2/Mπ,2 P,2 π,2
a generator of the valuation ring O of M over its associated order in the
Mπ,2 π,2
extension M /K ;
π,2
2. if p is odd, then the elements
Q Q
Tr (E (1)) Tr (E (1))+q
Kπ,2/Mπ,2 P,2 and Kπ,2/Mπ,2 P,2
π π
are both generators of A over O [Gal(M /K)] .
Mπ,2/K K π,2
Furthermore, Part 2 of this theorem will enable us to give explicit integral normal
basis generators for the square root of the inverse different in every abelian totally,
weakly and wildly ramified extension of any p-adic field (see Corollary 3.2).
We also remark that in this second part, the first element seems the more natural,
however the second is in fact the generalisation of Erez’s basis generator for the square
root of the inverse different. If these basis generators can be used in local calculations
in a similar way to those of Erez and Pickett, it should be possible to solve the case of
whether A is free over Z[G] whenever the decomposition groups at wild places are
E/F
abelian and in particular whenever E/F itself is abelian. We hope this will be possible
in the future, however, so far these calculations have eluded us.
Organisation of the paper. This paper is organised into three sections. In Sec-
tion 1, we give the background to the theory we need to prove our results. Precisely,
we first introduce Lubin-Tate formal groups in their original setting and also in terms
of Hazewinkel’s so called functional equation approach. We also introduce Fro¨hlich’s
notion of formal group exponentials and logarithms. Following work of Ditters, Drin-
feld,andFontaine andFargues, wefinallyintroducethetheoryoframifiedWittvectors
needed to generalise Pulita’s methods. In Section 2, we study the properties of the
Q
power series E (X) and prove Theorem 1. We also improve on Fro¨hlich’s original
P,n
bound of the radius of convergence of any formal group exponential coming from a
Lubin-Tate formal group over a non-trivial extension of Q . In Section 3, we explore
p
theapplications describedabove andprove Propositions 1 and2, as well as Theorem2.
1 Background
1.1 Formal groups
Let A be a commutative ring with identity.
Definition 1.1 We define a 1-dimensional commutative formal group over A to be a
formal power series F(X,Y) ∈ A[[X,Y]] such that
1. F(X,Y)= F(Y,X)
2. F(X,F(Y,Z)) = F(F(X,Y),Z)
5
3. F(X,0) = X = F(0,X)
Throughout, all the formal groups we consider will be 1-dimensional commutative for-
mal groups. For brevity we will now refer to these simply as formal groups.
Properties 1-3 can beusedto prove that thereexists a uniquej(X) ∈ XA[[X]] such
that F(X,j(X)) = 0 (see Appendix A.4.7 of [9]). This means that the formal group
F(X,Y) endows XA[[X]], among other sets, with an abelian group structure.
Notation 1.2 When considering a set endowed with such a group structure we write
the group operation as + . We will also use the notation − to determine this group
F F
operation composed with the group inverse, for example
F(X,Y) = X + Y and F(X,j(Y)) = X − Y .
F F
Definition 1.3 Let F(X,Y) and G(X,Y) be two formal groups over A. A homomor-
phism over the ring A, f : F(X,Y) → G(X,Y), is a formal power series f(X) ∈
XA[[X]] such that
f(F(X,Y)) = G(f(X),f(Y)) .
Moreover, we say that the homomorphism f is an isomorphism if there exists a homo-
morphism f−1 :G(X,Y)→ F(X,Y) such that f(f−1(X)) = f−1(f(X)) = X.
Lubin-Tate formal groups
Wenowdescribeaspecialtypeofformalgroup,dueoriginally toLubinandTate. Such
formal groups are used in local class field theory to construct maximal totally ramified
abelian extensions of a p-adic field K. For full details see, for example, [15] or [20].
Let K be a finite extension of Q , fix a uniformising parameter π of O and let
p K
q = |O /P | be the cardinality of the residue field of K.
K K
Definition 1.4 We define F as the set of formal power series P(X) over O such
π K
that
P(X) ≡ πX mod X2O [[X]] and P(X) ≡ Xq mod πO [[X]] .
K K
Such power series are called Lubin-Tate series with respect to π.
For each P ∈ F there exists a unique formal group F (X,Y) ∈ O [[X,Y]] which
π P K
admits P as an endomorphism. Such formal groups are known as Lubin-Tate formal
groups. For each P ∈ F and each a ∈ O , there exists a unique formal power series,
π K
[a] (X) ∈XO [[X]], such that P([a] (X)) = [a] (P(X)) and
P K P P
[a] (X) ≡ aX mod X2O [[X]] .
P K
Further,themapa 7→ [a] (X)isaninjectiveringhomomorphismO → End (F )
P K OK P
and for any P,Q ∈ F , the formal groups F (X,Y) and F (X,Y) are isomorphic over
π P Q
O .
K
6
Let P = {x ∈ C : v (x) > 0}. For P(X) ∈ F and a ∈ O , the formal power
Cp p p π K
series F (X,Y) and [a] (X) converge to limits in P when evaluated at elements of
P P Cp
P . We can thus use the abelian group operation + and the injective ring homo-
Cp F
morphism a 7→ [a] (X) to endow P with an O -module structure. For every n≥ 1,
P Cp K
we then let
T = {x ∈P : [πn] (x) = 0}
P,n Cp P
be the set of πn-torsion points of this module and refer to it as the set of the nth
Lubin-Tate division points with respect to P. If x ∈ T if and only if m ≤ n, then
P,m
we say x is a primitive nth division point.
We let
K = K(T ) and K = ∪ K .
π,n P,n π n π,n
The set T depends on the choice of the polynomial P(X) butthe field K depends
P,n π,n
only on the uniformising parameter π. The extensions K /K are totally ramified,
π,n
abelian and of degree qn−1(q−1). We have Kab = K Kun and Kun∩K = K, where
π π
Kab and Kun are the maximal abelian and unramified extensions of K respectively.
Hazewinkel’s approach to Lubin-Tate formal groups
We now describe a different approach to the construction of Lubin-Tate formal groups
due to Hazewinkel. This approach enables us to use Hazewinkel’s functional equation
lemma to prove the integrality of various power series relating to formal groups, which
will be essential in the sequel. For full details see [9] (Chap. I, §2).
Recall that p is a rational prime, K is a finite extension of Q , π is a fixed uni-
p
formising parameter of O and q is the cardinality of the residue field of K. For any
K
series g(X) ∈ XO [[X]] we construct a new power series f (X) ∈ XK[[X]] by the
K g
recursion formula (or functional equation) :
f (Xq)
g
f (X) = g(X)+ .
g
π
We denote by f−1(X) ∈ XK[[X]] the unique power series such that
g
f (f−1(X)) = X = f−1(f (X)) .
g g g g
Note that if f (X) ∈ XO [[X]] and the coefficient of X in f (X) is invertible in O ,
g K g K
then f−1(X) ∈ XO [[X]], see [9, A.4.6].
g K
We now state two parts of the functional equation lemma for this special setting.
For the full statement in all generality and its proof, see [9, §2.2-2.4].
Theorem 1.5 (Hazewinkel, [9, 2.2]) Letg(X),h(X) ∈ XO [[X]]andsuppose that
K
the coefficient of X in g(X) is invertible in O . Then,
K
1. f−1(f (X)+f (Y)) ∈ O [[X,Y]].
g g g K
2. f−1(f (X)) ∈XO [[X]].
g h K
7
It is routine to check that f−1(f (X)+f (Y)) is a formal groupand from part1. of
g g g
the previous theorem we know that it has coefficients in O . In fact, it is a Lubin-Tate
K
formal group and every Lubin-Tate formal group can be constructed in this manner.
This link is described in the following proposition.
Proposition 1.6 (Hazewinkel, [9, 8.3.6]) Let g(X) ∈ XO [[X]] with
K
g(X) ≡ X mod X2O [[X]].
K
Then,
1. f−1(πf (X)) ∈ F .
g g π
2. If we let P(X) = f−1(πf (X)), then F (X,Y)= f−1(f (X)+f (Y)).
g g P g g g
3. These relations give a one to one correspondence between the Lubin-Tate formal
groups obtained frompower seriesP(X) ∈ F andpower seriesg(X) ∈XO [[X]]
π K
with g(X) ≡ X mod X2O [[X]].
K
We also observe that for any a ∈ O , substituting h(X) = ag(X) into part 2 of
K
Theorem 1.5 then gives us f−1(af (X)) ∈ O [[X]], and so if P(X) = f−1(πf (X)),
g g K g g
then
[a] (X) = f−1(af (X)) .
P g g
Formal group exponentials
Hazewinkel’s power series f (X) andf−1(X) can bethought of as special formal group
g g
isomorphisms which were first studied by Fro¨hlich in ([7], Chap. IV, §1).
Let E be any field of characteristic 0, let F(X,Y) be a formal group over E and let
G (X,Y) = X +Y be the additive formal group. There exists a unique isomorphism
a
log : F → G over E such that log (X) ≡ X mod X2E[[X]], known as the formal
F a F
group logarithm (loc. cit., Prop. 1). The inverse of log (X) is known as the formal
F
group exponential and is denoted by exp (X); we note that necessarily we also have,
F
exp (X) ≡ X mod X2E[[X]].
F
Now let F(X,Y)= F (X,Y) = f−1(f (X)+f (Y)) be a Lubin-Tate formal group
P g g g
for K as in Prop. 1.6. We then have
f (X) = log (X) and f−1(X) = exp (X) (1)
g F g F
and these power series are uniquely determined by the following equivalent identities:
F(X,Y) = exp (log (X)+log (Y))
F F F
log (F(X,Y)) = log (X)+log (Y) (2)
F F F
exp (X +Y) = F(exp (X),exp (Y))
F F F
We also observe that,
[a] (X) = exp (alog (X)) . (3)
P F F
8
Remark 1.7 The reason these power series are referred to as formal group exponen-
tials and formal group logarithms is that if K = Q , then P(X) = (X +1)p −1 ∈ F
p p
and F (X,Y) = X +Y +XY = G , the multiplicative formal group. We then have
P m
exp (X) = exp(X)−1 and log (X) = log(X−1) where log and exp are the standard
FP FP
logarithmic and exponential power series.
1.2 Witt vectors
This section is concerned with the notion of ramified Witt vectors, generalising the
classical theory of Witt vectors introduced by Witt in his original paper [26]. This
notion was first developed independently by Ditters [3] and Drinfeld [4], and then by
Hazewinkel [10] from a formal group approach in a more general setting. The reader
is also referred to Section 5.1 of the current preprint [12] of Fontaine and Fargues.
Standard Witt vectors
We first briefly recall the construction of “standard” Witt vectors. Let p be a prime
number, and let X ,X ,... be a sequence of indeterminates. The original Witt poly-
0 1
nomials are defined by :
n
∀n≥ 0, W (X ,...,X ) = piXpn−i ∈ Z[X ,...X ] .
n 0 n i 0 n
i=0
X
The standard Witt vectors can be constructed as a functor W : A 7→ W(A) from
the category of commutative rings to itself. Precisely, if A is a commutative ring, we
first define W(A) as the set of infinite sequences AZ≥0. The elements of W(A) are
called Witt vectors, and to each Witt vector x = (a ) ∈ W(A), one can attach a
n n
sequence ha(n)in ∈ AZ≥0 whose coordinates are called the ghost components of x and
are defined by the Witt polynomials : a(n) = W (a ,...,a ), for all n ≥ 0.
n 0 n
The set W(A) is then uniquely endowed with two laws of composition that satisfy
the axioms of a commutative ring, in such a way that the ghost map Γ : (a ) ∈
A n n
W(A) 7→ ha(n)in ∈AZ≥0 becomes a ring homomorphism.
Underthisfunctor,anyringhomomorphismϕ :A → B issenttotheringhomomor-
phism W(ϕ) : W(A) → W(B) which is defined componentwise, i.e., W(ϕ)((a ) ) =
n n
(ϕ(a )) . See ([1], Chap. IX) for more details.
n n
Ramified Witt vectors
Let p be a prime number. Let K be a finite extension of Q , with valuation ring
p
O and residue field k. We fix a uniformising parameter π of O , and write k = F
K K q
with q = pf. Ramified Witt vectors over O are constructed as a functor W :
K OK,π
A 7→ W (A) from the category of O -algebras to itself, starting with generalised
OK,π K
Witt-like polynomials and then proceeding along the lines of the construction of the
9
usual Witt vectors. For convenience, as well as to collect some useful properties of the
ramified Witt vectors, we shall briefly describe this functor.
In the case of ramified Witt vectors, the relevant polynomials are :
n
∀n ≥ 0, W (X ,...,X )= πiXqn−i ∈ O [X ,...,X ] .
n,OK,π 0 n i K 0 n
i=0
X
Let A be an OK-algebra. We first define WOK,π(A) as the set AZ≥0 as the set of
infinite sequences over A. We shall use the notation (a ) for elements in W (A),
n n OK,π
and hanin for elements in AZ≥0.
If (a ) ∈ W (A), we define its ghost components as a(n) = W (a ,...,a )
n n OK,π n,OK,π 0 n
for all n ≥ 0. The sequence ha(0),a(1),...i is called the ghost vector of (a ) . This
n n
defines a map, that we shall denote by Γ or simply by Γ when the setup is
π,OK,A A
explicit, called the ghost map of A :
ΓA : WOK,π(A) −→ AZ≥0
(a ) 7→ ha(n)i .
n n n
The following lemma is essential for what follows.
Lemma 1.8 Let A be an O -algebra with no π-torsion. Then, the ghost map Γ is
K A
injective. If, moreover, there exists an O -algebra endomorphism σ : A → A such
K
that σ(a) ≡ aq mod πA for all a ∈ A, then the image of Γ is the sub-algebra of the
A
product algebra AZ≥0 given by
{hu i ∈ AZ≥0 : σ(u )≡ u mod πn+1A} .
n n≥0 n n+1
Proof. We proceed along the lines of ([1], No 2, Sect. 1, Par. 1 & 2), replacing
multiplication by p by multiplication by π, and replacing p by q in the exponents. The
first assertion is a consequence of the equivalence
u = a
(⋆) Γ ((a ) ) = hu i ⇔ 0 0
A n n n n u = W (aq,...,aq)+πn+1a .
(cid:26) n+1 n,OK,π 0 n n+1
Therefore, for every sequence hunin ∈ AZ≥0, there exists at most one element (an)n ∈
W (A) such that Γ ((a ) ) = hu i .
OK,π A n n n n
The second assertion is a consequence of the following relation that can easily be
proved in the same way as Lemma 1 of ([1], No 2, Sect. 1), since q ∈ πO :
K
∀x,y ∈A, ∀n ≥0, ∀m ≥ 1 : x ≡ y mod πmA ⇒ xqn ≡ yqn mod πm+nA .
In particular, for m = 1 and for any sequence (a ) ∈ W (A), this implies that
n n OK,π
σ(W (a ,...,a )) = W (σ(a ),...,σ(a ))
n,OK,π 0 n n,OK,π 0 n
≡ W (aq,...,aq) mod πn+1A
n,OK,π 0 n
≡ W (a ,...,a ) mod πn+1A
n+1,OK,π 0 n+1
10