Ramification filtration of the Galois group of a local field via deformations

0.32 MB·

by  Victor Abrashkin

RAMIFICATION FILTRATION OF THE GALOIS
GROUP OF A LOCAL FIELD VIA DEFORMATIONS
VICTOR ABRASHKIN
7 Abstract. Let K be a field of formal Laurent series with co-
1 efficients in a finite field of chracteristic p, G — the maximal
<p
0
quotient of Gal(K /K) of period p and nilpotent class < p and
sep
2
{G<(vp)}v>0 — its filtration by ramification subgroups in the up-
r
a per numbering. Let G<p = G(L) be the identification of nilpo-
M tent Artin-Schreier theory: here G(L) is the group obtained from
a suitable profinite Lie F -algebra L via the Campbell-Hausdorff
p
7 composition law. We give a new proof of the construction of gen-
2 eratorsofthe ideals L(v) suchthat G(L(v))=G(v) from[1]. Given
<p
] v0 > 0 we construct epimorphism of Lie algebras η¯† : L −→ L¯†
T and the action Ω of the formal group α = Spec F [U], Up = 0,
U p p
N of order p on L¯†. Then dΩ = B†U, where B† ∈ DiffL¯†, and
U
. Lv0) = (η¯†)−1L¯†[v0], where L¯†[v0] is the ideal of L¯† generated by
h
t the elements of B†(L¯†).
a
m
[

2
Introduction
v
7
Everywhere in the paper v is a fixed positive real number.
0 0
2 Suppose s ∈ N. For any topological group G, we denote by C (G)
s
2
theclosureofthesubgroupofG generatedbythecommutators oforder
0
. > s. If L is a toplogical Lie algebra then C (L) is the closure of the
1 s
0 ideal generated by commutators of degree > s. For any topological
7 A-modules M and B we use the notation M = M⊗ˆ B.
B A
1
If L is any Lie algebra over F of nilpotent class < p (i.e. C (L) = 0)
: p p
v
then we use the notation G(L) for the p-group obtained from the set
i
X L via the Campbell-Hausdorff composition law (l ,l ) 7→ l ◦ l , cf.
1 2 1 2
r Subsection 1.2.
a
Let k = F . Consider a free pro-finite Lie algebra L over k with
pN0 k
the set of free generators {D | a ∈ Z+(p),n ∈ Z/N }∪{D }, where
an 0 0
Z+(p) is the set of all natural numbers prime to p. e
If σ is the absolute Frobenius automorphism of k (for any α ∈ k,
σ(α) = αp) we agree to use the same symbol σ for its continuation to
an automorphism of L such that σ(D ) = D and σD = D .
k an a,n+1 0 0
Then L := L | is a free Lie algebra over F . Clearly, L⊗ k = L .
k σ=id p Fp k
e
Datee: Marceh 27, 2017. e e
2010 Mathematics Subject Classification. 11S15, 11S20.
Key words and phrases. local field, Galois group, ramification filtration.
1
2 VICTOR ABRASHKIN
Set L = L/C (L). Note that the images of D and D in L (which
p an 0 k
will be denoted by the same symbols) give a fixed system of generators
of L . Wee alsoeset for n ∈ Z/N , D = σn(α )D , where α is a
k 0 0n 0 0 0
fixed element from k such that Tr α = 1. From time-to-time we
k/Fp 0
allow in the definition of D that n ∈ Z under the agreement that
an
D = D .
an a,nmodN0
Let A be the enveloping algebra of L with augmentation ideal J.
Consider an element e ∈ L such that in A
K K
exp(e) ≡ 1+ η(a ,...,a )t−(a1+···+as)D ...D modJp
1 s a10 as0 K
1X6s<pa1X,...,as
(hgere exp is the truncated exponential) where the “structural con-
stants” η(a ....,a ) ∈ k are such that exp(e) is “diagonal modulo
1 s
degree p” , cf. Subsection 1.3. Fix f ∈ G(L ) such that σf = e◦f
g Ksep
and consider the map η0 from G := Gal(K /K) to G(L) such that
e gsep
for any τ ∈ G, η0 : τ 7→ (−f)◦τ(f). This map (due to the nilpotent
e
version of the Artin-Schreier theory, cf. [2]), provides us with the in-
duced identification η : G ≃ G(L), where G := G/GpC (G) is the
e <p <p p
maximal quotient of G of period p and nilpotent class < p.
Forv > 0,letG(v) bethesubgroupoframificationwithuppernumber
v. Denote by G(v) the image of G(v) in G . Then η (G(v)) = L(v) is an
<p <p e <p
ideal in L.
In this paper we give a substantially new proof of the main result
from [1]. This result can be stated as follows.
For α > 0 and N ∈ N, introduce F0 ∈ L such that
α,−N k
F0 = a η(a¯,n¯)[...[D ,D ],...,D ].
α,−N 1 a1n1 a2n2 asns
16s<p
aXi,ni
Here:
— a¯ = (a ,...,a ) and n¯ = (n ,...,n ) with n = 0;
1 s 1 s 1
— α = α(a¯,n¯) = a pn1 +a pn2 +···+a pns;
1 2 s
— if 1 6 s < s < ··· < s = s and n = ··· = n > ··· >
1 2 r 1 s1
n = ··· = n > −N then
sr−1+1 sr
η(a¯,n¯) := σns1η(a1,...,as1)...σnsrη(asr−1+1,...,asr),
otherwise, η(a¯,n¯) = 0.
Theorem 0.1. For any v > 0, there is N(v) such that if N > N(v) is
fixed then the ideal L(v) is the minimal ideal in L such that its extension
of scalars L(v) contains all F0 with αe> v. e
k α,−N
Our method can be briefly described as follows.
First, we introduce a decreasing central filtration by the ideals L =
L(1) ⊃ ··· ⊃ L(s) ⊃ ... andprove that L(p) ⊂ L(v0). This gives us the
RAMIFICATION VIA DEFORMATIONS 3
projection η : G −→ G(L¯) := G(L/L(p)) and it remains to describe
e¯ <p
the ideal L¯(v0) of L¯ such that G(L¯(v0)) = η (G(v0)).
e¯ <p
Then we:
a) introduce a lift V : L¯† −→ L¯ where L¯† is a Lie algebra over F of
p
nilpotent class < p with a central filtration L¯†(s) such that V(L¯†(s)) =
L¯(s) and L¯†(p) = 0;
b) prove the existence of a group epimorphism η : G −→ G(L¯†)
e¯† <p
such that Vη = η ;
e¯† e¯
c) introduce an action of the group Z/p on L¯†;
d) prove that L¯(v0) is the minimal ideal in L¯ such that for any γ ∈
Z/p, L¯(v0) ⊃ V(γ(KerV)) and the induced action of Z/p on L¯/L¯(v0)
is trivial (Note that this condition is very difficult to study because
it involves very complicated group composition law coming from the
Campbell-Hausdorff formula.);
e) the above action of the group Z/p comes from the co-action
Ω : L¯† −→ L¯† ⊗ F [U] of the group scheme α = F [U] such that
U p p p
Up = 0 and the coaddition is such that ∆U = U ⊗1+1⊗U;
f) if dΩ = B†U is the differential of Ω , where B† ∈ DiffL¯†, then
U U
the conditions from above item d) mean that VB†(L¯†) ⊂ L¯(v0);
g) the above linearization procedure allows us to find explicit expres-
sions for the generating elements F0 from Theorem 0.1.
α,−N
There is the following non-formal aspect of our construction.
We could have started with the Lie algebra L¯ = L/L(p) and the
¯ ¯
image e¯= e¯(t) ∈ L of e ∈ L to fix the projection η : G −→ G(L)
K K e¯ <p
via methods of nilpotent Artin-Schreier theory. In order to define a
deformation of η we choose auxiliary parameters N∗ = N∗(v ) ≡
e¯ 0
0modN and b∗ ∈ Z+(p) such that r∗(v ) := b∗/(q − 1) is “close” to
0 0
v (here q = pN∗). If we apply σN∗ to e¯ we obtain σN∗e¯= e¯(tq) and η
0 e¯
will be not changed. On the other hand, we can use a “deformation”
of σN∗ such that t 7→ tqexp(tb∗). This operation gives us an element
e¯′ ∈ L¯ and the nilpotent Artin-Schreier theory provides us with a new
K
projection η : G −→ G(L¯). The initial expectation is that (after
e¯′ <p g
a careful choice of the parameters N∗ = N∗(v ) and r∗ = r∗(v )) the
0 0
ramification ideal L¯(v0) will appear as the minimal ideal in L¯ such that
ηe¯ coincides with its “deformation” ηe¯′ modulo L¯(v0). When following
this idea we introduced a lift V : L¯† −→ L¯ and defined a deformation
of σN∗ as an automorphism of some subquotient of L¯† . An interesting
K
aspect of this construction is that this deformation induces a group
automorphism of the quotient G(L¯†) of G which is not induced by an
<p
automorphism of K. As a matter of fact, it comes in the form exp(B†)
where B† is a differentiation of L¯† induced by a higher derivation taq 7→
ataq−b∗ (on the above subquotient of L¯† ). Finally, we prove that the
K g
4 VICTOR ABRASHKIN
ramificationideal L¯(v0) isgeneratedbytheelements ofV(B†(KerV)), or
equivalently, L¯(v0) is the minimal ideal in L¯ such that the deformation
B† descends to a trivial deformation of L¯/L¯(v0).
The methods from this paper admit a generalisation to the “modulo
pM” situation as well as to the case of higher dimensional local fields
in the characteristic p case (in preparation). They also provide us with
much better background for the paper [8] and its upcoming generalisa-
tions to the “modulo pM” situation and the case of higher dimensional
local fields in the mixed characteristic case.
1. Identification of nilpotent Artin-Schreier theory
Suppose K is a field of characteristic p, K is a separable closure of
sep
K and G = Gal(K /K). We assume that G acts on K as follows:
sep sep
if g ,g ∈ G and a ∈ K then g (g a) = (g g )a. Denote by σ the
1 2 sep 1 2 1 2
p-power morphism.
In [1, 2] we developed a nilpotent analogue of the classical Artin-
Schreier theory of cyclic extensions of fields of characteristic p. We are
going to use the covariant analogue of this theory, cf. the discussion in
[7], for explicit description of the group G = G/GpC (G) as follows.
<p p
1.1. Lie algebra L. Suppose K = k((t)) where t is a fixed uniformizer
and k ≃ F with N ∈ N. Fix α ∈ k such that Tr (α ) = 1.
pN0 0 0 k/Fp 0
Let Z+(p) = {a ∈ N | (a,p) = 1} and Z0(p) = Z+(p)∪{0}.
LetLbeaprofinitefreeLieF -algebrawiththe(topological)module
p
of generators K∗/K∗p and L = L/C (L). Then we can take as a system
p
of topoelogical generators of L the set
k
e e
{D }∪{D | a ∈ Z+(p),n ∈ Z/N }
0 an 0
due to the following identifications:
(K∗/K∗p)⊗ˆ k = Hom (K/(σ −id)K,k) =
Fp Fp
Hom (⊕ kt−a ⊕F α ,k) = Hom (kt−a,k)×kD
Fp a∈Z+(p) p 0 Fp 0
a∈Z+(p)
Y
and Hom (kt−a,k) = kD , where for any α ∈ k and a,b ∈
Fp n∈Z/N0 an
Z+(p), D (αt−b) = δ σn(α). Note also that the first identification
an aQb
uses the Witt pairing [9, 6] and D comes from t⊗1 ∈ (K∗/K∗p)⊗ˆ k.
0 Fp
For any n ∈ Z/N , set D = t⊗(σnα ) = (σnα )D .
0 0n 0 0 0
1.2. Groups and Lie algebras of nilpotent class < p. The basic
ingredient of the nilpotent Artin-Schreier theory is the equivalence of
the category of p-groups of nilpotent class s < p and the category of
0
Lie Z -algebras of the same nilpotent class s . In the case of objects
p 0
killed by p, this equivalence can be explained as follows.
Let L be a Lie F -algebra of nilpotent class < p, i.e. C (L) = 0.
p p
RAMIFICATION VIA DEFORMATIONS 5
Let A be the enveloping algebra of L. Then there is a natural em-
bedding L ⊂ A, the elements of L generate the augmentation ideal J
of A and we have a morphism of algebras ∆ : A −→ A⊗A uniquely
determined by the condition ∆(l) = l⊗1+1⊗l for all l ∈ L (in other
words all l ∈ L are primitive). Applying the Poincare-Birkhoff-Witt
Theorem we obtain that:
— L∩Jp = 0;
—LmodJp = {amodJp | ∆(a) ≡ a⊗1+1⊗amod(J⊗1+1⊗J)p};
— the set exp(L)modJp is identified with the set of all ”diagonal
elements modulo degree p“, i.e. with the set of a ∈ 1+J modJp such
that ∆(a) ≡ a⊗amod(J ⊗1+1⊗J)p (here exp(x) = xi/i! is
g 06i<p
the truncated exponential).
P
g
In particular, there is a natural embedding L ⊂ A/Jp and in terms
of this embedding the Campbell-Hausdorff formula appears as
1
(l ,l ) 7→ l ◦l = l +l + [l ,l ]+..., l ,l ∈ L,
1 2 1 2 1 2 1 2 1 2
2
where exp(l )exp(l ) ≡ exp(l ◦l )modJp. This composition law pro-
1 2 1 2
vides the set L with a group structure and we denote this group by
G(L). Note that I is an ideal in L iff G(I) is a normal subgroup
g g g
in G(L). Clearly, G(L) has period p and nilpotent class < p. Then
the correspondence L 7→ G(L) is the above mentioned equivalence of
the categories of p-groups of period p and nilpotent class s < p and
Lie F -algebras of the same nilpotent class s. This equivalence can
p
be naturally extended to the categories of pro-finite Lie algebras and
pro-finite p-groups.
1.3. Epimorphism η0 : G −→ G(L). Let L := L . Then the
e sep Ksep
elements of G = Gal(K /K) and σ act on L through the second
sep sep
factor, L | = L and (L )G = L . The covariant nilpotent Artin-
sep σ=id sep K
Schreier theory states that for any e ∈ G(L ), the set
K
F(e) = {f ∈ G(L ) | σ(f) = e◦f}
sep
is not empty and for any fixed f ∈ F(e), the map τ 7→ (−f)◦τ(f) is a
continuous group homomorphism π (e) : G −→ G(L). The correspon-
f
dence e 7→ π (e) has the following properties:
f
a) if f′ ∈ F(e) then f′ = f ◦l, where l ∈ G(L), and π (e) and π (e)
f f′
are conjugated via l;
b) for any continuous group homomorphism π : G −→ G(L), there
are e ∈ G(L ) and f ∈ F(e) such that π (e) = π;
K f
c) for appropriate elements e,e′ ∈ G(L ) and f,f′ ∈ G(L ), we
K sep
have π (e) = π (e′) iff there is an x ∈ G(L ) such that f′ = x◦f and,
f f′ K
therefore, e′ = σ(x)◦e◦(−x).
6 VICTOR ABRASHKIN
In[1,2,3]weappliedthistheorytotheLiealgebraLfromSubsection
1.1 via a special choice of e ∈ L such that
K
(1.1) expe ≡ 1+ η(a ,...,a )t−(a1+···+as)D ...D modJp .
1 s a10 as0 L,K
16s<p
X
Here Jgis the augmentation ideal in the envelopping algebra A of L.
L L
In the above sum the indices a ,...,a run over Z0(p) and the “struc-
1 s
tural constants” η(a ,...,a ) ∈ F satisfy the following identities:
1 s p
I ) η(a ) = 1;
e 1
II ) if 0 6 s 6 s < p then
e 1
η(a ,...,a )η(a ,...,a ) = η(a ,...,a ),
1 s1 s1+1 s π(1) π(s)
πX∈Is1s
where I consists of all permutations π of order s such that the se-
s1s
quences π−1(1),...,π−1(s ) and π−1(s +1),...,π−1(s) are increasing
1 1
(i.e. I is the set of all “insertions” of the ordered set {1,...,s } into
s1s 1
the ordered set {s +1,...,s}).
1
Under the assumption I ) the map π (e)modGpC (G) induces a
e f 2
group isomorphism of Gab⊗ˆF and G(L)/C (G(L)) = Lab = K∗/K∗p,
p 2
which coincides with the inverse to the reciprocity map of local class
field theory, cf. [6]. This also implies that η0 := π (e) (when taken
e f
modulo GpC (G)) induces a group isomorphism η : G ≃ G(L). In
p e <p
particular, we can work with the elements of L = “G ⊗ˆk” in terms
k <p
of our explicit generators {D | a ∈ Z+(p),n ∈ Z/N }∪{D }.
an 0 0
Assumption II ) means that
e
∆(exp(e)) ≡ exp(e)⊗exp(e)mod(J ⊗ˆ1+1⊗ˆJ )p,
L,K L,K
i.e. exp(e) is diagonal modulo Jp .
L,K
g g g
Under a fixed choice of the structural constants η(a ,...,a ) the
1 s
element e depends on the uniformiser t ∈ K. We underline this fact
g
in some cases by writing e = e(t). Note that in most examples or
interpretations of results from [1, 2, 3] we used the identification G ≃
<p
G(L) coming from the simplest choice e = t−aD (in this
a∈Z0(p) a0
case all η(a ,...,a ) = 1/s!). But as a matter of fact, all proofs in the
1 s
P
papers [1, 2, 3] were adjusted to the above more general choice of e.
¯
1.4. Lie algebra L and epimorphism η . Introduce a weight func-
e¯
tion wt : L −→ N on L by setting on its generators wt(D ) = s if
k k an
(s−1)v 6 a < sv . Then we obtain a central decreasing filtration of
0 0
ideals L(s) = {l ∈ L | wt(l) > s}. This weight function gives us also a
decreasing filtration of ideals J (s) in A such that J (1) = J and for
L L L L
any s, (J (s)+Jp)∩L = L(s) (use the Poincare-Birkhoff theorem).
L L
Let L¯ := L/L(p) and consider the natural projection p¯r : L −→ L¯.
ThenL¯(s) = p¯r(L(s))isadecreasing filtrationinL¯such thatL¯(p) = 0.
RAMIFICATION VIA DEFORMATIONS 7
Consider an F -submodule N in L¯ generated by all t−bl where l ∈
p K
L¯(s) and b < sv . Then N has a natural structure of a Lie algebra
k 0
over F . For any i > 0, let N(i) be the F -submodule in L¯ generated
p p K
by all t−bl where l ∈ L¯(s) and b < (s−i)v . Then N(i) is an ideal in
k 0
N and we can introduce the Lie F -algebra N¯ = N/N(p). Note that
p
N(p) ⊂ L¯ , where m = tk[[t]].
m
Lemma 1.1. e := p¯r e ∈ N.
L¯ K
Proof. Let N be an F -submodule in L generated by the elements of
L p K
L(p) and all t−bl such that l ∈ L(s) , b < sv and 1 6 s < p. It will
K k 0
be sufficient to prove that e ∈ N .
L
Let N be an F -submodule in A generated by the elements of
AL p K
J (p) and all t−bj such that j ∈ J (s) and b < sv . It will be
L K L k 0
sufficient to prove that expe ∈ N . Indeed, N is closed under
AL AL
multiplication and, therefore, log(expe) ∈ N (here log is the trun-
AL
cated logarithm). It remagins to notice that e ≡ log(expe)modJp and
K
NAL ∩LK = NL. f g f
Verify that any term of the form t−(a1+···+ar)Daf10.g..Dar0 belongs to
N . Indeed, if wt(D ) = s for i = 1,...,r, then a + ··· + a <
AL ai0 i 1 r
(s + ...s )v = wt(D ...D )v . So, expe ∈ N and the lemma
1 r 0 a10 ar0 0 AL
is proved. (cid:3)
g
Let e¯:= e modN(p) ∈ N¯.
L¯
Let η := p¯r·η : G −→ G(L¯). Verify that η depends only on the
e¯ e <p e¯
image e¯ of e in N¯.
Proposition 1.2. Let e′ ∈ L and e′ := (p¯r )e′ ≡ e modN(p).
K L¯ K L¯
Then there is a unique f′ ∈ L modL(p) such that σf′ = e′ ◦ f′
sep sep
and p¯r·π (f′) = η .
e′ e¯
Proof. Note that σ is topologically nilpotent on N(p)modL(p) . This
K
implies the existence of a unique x ∈ N(p)modL(p) such that e′ =
K L¯
(σx)◦e ◦(−x). Let f = (p¯r )f ∈ L¯ . Then σ(x◦f ) = e′ ◦(x◦f ).
L¯ L¯ sep sep L¯ L¯ L¯
If f′ ∈ L is such that σf′ = e′ ◦ f′ and f′ := p¯r f′ then for
sep L¯ sep
g = (−f′)◦x◦f ∈ L¯ we have that σg ≡ gmodL¯ . This implies
L¯ L¯ sep sep
that g = l modL(p) , where l ∈ L. Therefore, we can replace f′ by
0 sep 0
f′ ◦l to satisfy the requirements of our proposition. From the above
0
construction it is clear that f′ is unique modulo L(p) . (cid:3)
sep
Remark. From now on we can use the construction of the morphism
¯ ¯
η : G −→ G(L) in terms related only to the Lie algebra N. Namely,
e¯ <p
there is a unique f¯ ∈ N¯ := N¯ ⊗ K (which comes from our
sep K sep
¯ ¯
fixed f ∈ L ) such that σf = e¯◦ f and for any τ ∈ G , η (τ) =
sep <p e¯
(−f¯)◦τ(f¯) ∈ N¯ | = L¯.
sep σ=id
8 VICTOR ABRASHKIN
2. Lie algebra L†
In this Section we introduce a Lie F -algebra L¯† together with epi-
p
morphism of Lie algebras V : L¯† −→ L¯ and its section j0 : L¯ −→ L¯†.
Let α = SpecF [U], Up = 0, be the group scheme over F with
p p p
the coaddition ∆(U) = U ⊗ 1 + 1 ⊗ U. We introduce the action
Ω : L¯† −→ F [U]⊗L¯† of α on L¯†.
U p p
2.1. Special choice of parameters r∗ = r∗(v ) and N∗ = N∗(v ).
0 0
Denote by δ = δ (v ) the minimum of the positive values of
0 0 0
1
v − (a +a /pn2 +···+a /pnu),
0 1 2 u
s
where 1 6 s,u < p, all n ∈ Z and a ∈ [0,(p− 1)v ) ∩Z0(p). The
i >0 i 0
existence of such δ > 0 can be proved easily by induction on u.
0
Let r∗ ∈ Q be such that r∗ = b∗/(q∗ − 1), where q∗ = pN0∗ with
0 0 0
N∗ > 2, b∗ ∈ N and gcd(b∗,p(q∗ − 1)) = 1. The set of such r∗ is
0 0 0 0
everywhere dense in R and we can fix r∗ = r∗(v ) ∈ (v −δ ,v ).
>0 0 0 0 0
For 1 6 s < p, introduce the following subsets in Q:
— A[s] is the set of all a +a /pm2 +···+a /pmu, where 1 6 u 6 s,
1 2 u
all m ∈ Z , all a ∈ [0,(p−1)v )∩Z0(p) and a 6= 0 (in particular,
i >0 i 0 1
0 ∈/ A[s]). IfM ∈ Z wedenotebyA[s,M]thesubset ofA[s]consisted
>0
of the elements satisfying the additional restrictions m ,...,m 6 M.
2 u
Note that A[s,M] is finite.
— B[s] is the set of all numbers r∗(b +b /pm2+···+b /pmu) where
1 2 u
1 6 u 6 s, all b ∈ Z and b +···+b < p. For M ∈ Z , let B[s,M]
i >0 1 u >0
be the subset of B[s] consisted of the elements satisfying the additional
restrictions m ,...,m 6 M. The set B[s,M] is also finite.
2 u
Lemma 2.1. For any s, A[s]∩B[s] = ∅.
Proof. Note that A[s] ⊂ Z[1/p]\{0}. Prove that B[s]∩Z[1/p] = {0}.
It will be sufficient to verify that for any m ,...,m ∈ Z and
1 u >0
b ∈ Z such that 0 < b +···+b < p, we have
i >0 1 u
b pm1 +···+b pmu 6≡ 0mod(q∗ −1).
1 u 0
Since q∗ ≡ 1mod(q∗ −1) we can assume that all m < N∗. But then
0 0 i 0
0 < b1pm1 + ··· + bupmu 6 (p − 1)pN0∗−1 < q0∗ − 1. The lemma is
proved. (cid:3)
For α,β ∈ Q, set ρ(α,β) = |α−β|.
Lemma 2.2. If α ∈/ B[s] then
ρ(α,B[s]) := inf{ρ(α,β) | β ∈ B[s]} =6 0.
Proof. Use induction on s.
If s = 1 there is nothing to prove because B[1] is finite.
Suppose s 6 p−2 and ρ(α,B[s]) > 0.
RAMIFICATION VIA DEFORMATIONS 9
Choose Ms ∈ Z>0 such that r∗(p−1)/pMs < ρ(α,B[s])/2.
If β ∈ B[s + 1] \ B[s + 1,M ] then there is β′ ∈ B[s] such that
s
ρ(β,β′) < ρ(α,B[s])/2. Then
ρ(α,β) > ρ(α,β′)−ρ(β′,β) > ρ(α,B[s])−ρ(α,B[s])/2 = ρ(α,B[s])/2,
and we obtain
ρ(α,B[s+1]) > min{ρ(α,B[s+1,M ]),ρ(α,B[s])/2} > 0.
s
The lemma is proved. (cid:3)
Lemma 2.3. If β ∈/ A[s] then ρ(β,A[s]) 6= 0.
Proof. The proof is similar to the proof of above Lemma 2.2. (cid:3)
Lemma 2.4. For all 1 6 u,s < p, ρ(A[u],B[s]) > 0.
Proof. If u = 1 it follows from Lemma 2.2 because A[1] is finite.
Suppose u 6 p−2 and ρ(A[u],B[s]) = δ > 0.
us
Choose Mu ∈ Z>0 such that (p−1)v0/pMu < δus/2.
If α ∈ A[u + 1] \ A[u + 1,M ] then there is α′ ∈ A[u] such that
u
ρ(α,α′) < δ /2. Then for any β ∈ B[s], we have
us
ρ(α,β) > ρ(α′,β)−ρ(α,α′) > δ /2.
us
Therefore, for any α ∈ A[s],
ρ(α,B[s]) > min{ρ(A[u+1,M ],B[s]),δ /2}} > 0.
u us
Lemma is proved. (cid:3)
Specify the choice of N∗ = N∗(v ) ∈ N by the following conditions:
0
C1) N∗ ≡ 0modN∗;
0
C2) pN∗ρ(A[p−1],B[p−1]) > 2r∗(p−1);
C3) r∗(1−p−N∗) ∈ (v −δ ,v ).
0 0 0
Under this choice we set q = pN∗ and b∗ = b∗(q − 1)/(q − 1) ∈ N
0 0
(note that r∗ = b∗/(q −1)).
We need below the following property of such N∗.
Proposition 2.5. If α ∈ A[p−1] and β ∈ B[p−1] then
q|qα−(q −1)β| > b∗(p−1).
Proof. Indeed,
q2|α−β +β/q| > q2|α−β|−βq > q2ρ(A[p−1],B[p−1])
−r∗(p−1)q > 2r∗(p−1)q −r∗(p−1)q = r∗(p−1)q > b∗(p−1).
(cid:3)
10 VICTOR ABRASHKIN
2.2. Lie algebra L¯†. Use the above r∗ = r∗(v ) and N∗ = N∗(v ).
0 0
Definition. A = A(r∗,N∗) is the set of all ι = pm(qα − (q − 1)β),
where m ∈ Z , α ∈ A[p−1,m], β ∈ B[p−1,m] and |ι| 6 b∗(p−1).
>0
Lemma 2.6. Suppose ι = pm(qα−(q −1)β) ∈ A. Then:
a) m < N∗;
b) pmα ∈ N and pmβ/r∗ ∈ Z do not depend on the presentation of
>0
ι in the form pm(qα−(q−1)β) with α ∈ A[p−1,m] and β ∈ B[p−1,m].
Proof. If m > N∗ then by Proposition 2.5, |ι| > b∗(p −1) i.e. ι ∈/ A.
Fromthe definitionof thesets A[p−1,m] andB[p−1,m] it follows that
pmα,pmβ/r∗ ∈ Z . If ι = pm′(qα′−(q−1)β′) is another presentation
>0
of ι then pmβ/r∗ and pm′β′/r∗ are non-negative congruent modulo q
integersandthebotharesmallerthanq. Indeed, ifβ/r∗ = b +b p−m2+
1 2
···+b p−mu, where all m 6 m, then
u i
pmβ/r∗ 6 pm(b +···+b ) 6 pm(p−1) < pm+1 6 q
1 u
because m < N∗. Similarly, pm′β′/r∗ < q. Therefore, they coincide
and this implies also that pmα = pm′α′. (cid:3)
Corollary 2.7. If ι = pm(qα − (q −1)β) ∈ A then the sum of digits
b +···+b of the appropriate β/r∗ = b +b p−m2+···+b p−mu depends
1 u 1 2 u
only on ι.
Definition. With the above notation set ch(ι) := b +···+b .
1 u
Definition. a) A+(p) is the set of all ι = pm(qα−(q−1)β) ∈ A (given
in the above standard notation) such that:
— ι > 0;
— gcd(pmα,pmβ/r∗) 6≡ 0modp;
— if s ∈ N and ι ∈ [(s−1)b∗,sb∗) then ch(ι) 6 p−1−s;
b) A0(p) = A+(p)∪{0};
c) A0(p) = {ι ∈ A0(p) | ch(ι) = 0} = {qa | a ∈ Z0(p)∩[0,(p−1)v )};
0 0
d)A0 (p) = {ι ∈ A0(p)|ch(ι) > 1}, A0(p) = {ι ∈ A0(p)|ch(ι) = 1}.
>1 1
Proposition 2.8. If v is the normalized p-adic valuation then all
p
ιp−vp(ι), where ι ∈ A+(p), are pairwise different elements from Z+(p).
Proof. Suppose ι = pm(qα−(q −1)β) ∈ A+(p).
If ch(ι) = 0 then ι ∈ A0(p) and the correspondence ι 7→ ιp−vp(ι)
0
identifies A0(p) \ {0} ⊂ A+(p) with Z+(p) ∩ [0,(p − 1)v ). Indeed,
0 0
qpmα < b∗(p − 1) means that a = pmα < r∗(p − 1). According to
property C3 from Subsection 2.1 this condition and the condition a <
v (p−1)areequivalent because thebothimply thata 6 (v −δ )(p−1).
0 0 0
If ch(ι) > 1 then ιp−m ∈/ pN, i.e. m > v (ι). Indeed, ιp−m =
p
qα−(q −1)β ∈ pN implies (use that m < N∗) that p−m(b +b pm2 +
1 2