Cyclotomic p-adic multi-zeta values

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by  Sinan Unver

Cyclotomic p-adic Multi-Zeta Values
7 SinanU¨nver
1
0
2

n
a Abstract. Thecyclotomicp-adicmulti-zetavaluesarethep-adicperiodsof
J π1(Gm\µM,·),theunipotent fundamental groupofthemultiplicativegroup
minustheM-throotsofunity. Inthispaper,wecomputethecyclotomicp-adic
0
multi-zetavaluesatalldepths. Thispapergeneralizestheresultsin[6]and[7].
2
Sincethemainresultgivesquiteexplicitformulasweexpectittobeusefulin

proving non-vanishingand transcendence resultsfor these p-adicperiods and
]
T also, throughtheuseofp-adicHodgetheory, inprovingnon-trivialityresults
forthecorrespondingp-adicGaloisrepresentations.
N
.
h 1. Introduction
t
a
There are not many examples of motives over Z. The most basic examples of
m
such motives are the Tate motives. Another one is the unipotent completion of
[
the fundamental group of the thrice punctured projective line π1(Gm \{1},·), at

1 a suitable tangential basepoint [2]. In fact by a theorem of F. Brown, this motive
v generates the tannakian category of mixed Tate motives over Z. The complex pe-
9
riods of π (G \{1},·) are Q-linear combinations of the multi-zeta values given
2 1 m
by
7
1
5
ζ(s ,s ,··· ,s ):= ,
0 1 2 k X ns1ns2···nsk
. 0<n1<···<nk 1 2 k
1 for s ,··· ,s ≥ 1 and s > 1. These values were defined by Euler and studied
1 k−1 k
0
by Deligne, Goncharov,Terasoma, Zagier etc.
7
Similarly, one can consider the unipotent fundamental group π (G \µ ,·) of
1 1 m M
: the multiplicative group minus the group µM of M-th roots of unity for M ≥1. If
v
O denotestheringofintegersoftheM-thcyclotomicfield,thenthisfundamental
i M
X group defines a mixed Tate motive over O [1/M]. The periods of this motive are
M
r linear combinations of the cyclotomic multi-zeta values
a
ζi1n1+···iknk
,
X ns1ns2···nsk
0<n1<···<nk 1 2 k
where i , for 1 ≤ j ≤ k, are fixed integers and ζ is an M-th root of unity. These
j
values were studied and related to modular varieties and the theory of higher cy-
clotomy in [4].
This paper concerns the p-adic periods of the motive π (G \µ ,·). We have a
1 m M
realisationmapfromthecategoryofmixedTatemotivesoveranumberfieldtothe
category of mixed Tate filtered (ϕ,N)-modules for any non-archimedean place of
the number field [1]. Also for any (framed) mixed Tate filtered (ϕ,N)-module we
associate a period. The cyclotomic p-adic multi-zeta values (henceforth cmv’s) are
the p-adic periods associated to the mixed Tate motive defined by the unipotent
fundamentalgroupofG \µ ,forp∤M.Thesevaluesweredefinedintermsofthe
m M
1
2 SinanU¨nver
actionofthe crystallinefrobenius onthe fundamentalgroupin[6], generalisingthe
notion of p-adic multi-zeta values (henceforth pmv’s) in [5]. In this paper we give
an explicit series representation of these p-adic periods. This is a generalisation of
[7] to the cyclotomic case.
We give an overview of the contents of the paper. In §2, we start with studying
certaintypesofseriesintermsofwhichthecmv’swillbeexpressed. Theseseriescan
beoftwotypes,denotedbyσorγ,andarecalledthecyclotomic p-adiciteratedsum
series (or ciss). In fact the ciss are divergent and we will need to regularise them.
The regularisation can be intuitively thought of as removing a combination of the
summands which have large p factors in the denominators that cause divergence.
More precisely, we extend the algebra of M-power series functions by adding some
highly divergent functions which we denote by σ and we show in Proposition 2.9
p
that the ciss are contained in this algebra. In Corollary 2.6, we show that the
{σ (s;i)}’s form a basis for this extended algebra as a module over the algebra of
p
M-powerseriesfunctions. Thesetwofactshelpustodefinetheregularisedversions
of the ciss, denoted by σ˜ and γ˜, in Definition 2.10. The limits of these regularised
series are called the cyclotomic p-adic iterated sums (or cis), and denoted by σ
and γ. Let ζ be a primitive M-th root of unity. Let P denote the Q(ζ)-algebra
M
generatedbythecis,andZ thealgebrageneratedbythecmv. Themaintheorem
M
is
Theorem 1.1. We have the inclusion Z ⊆P .
M M
The proof of this theorem occupies the whole of §3. The proof expresses in
an inductive way every cmv as a series and should be thought of as an explicit
computation of these values.
Finally, wewouldliketo mentionthatFurusho definedin[3]anotherp-adicver-
sionofmulti-zetavaluesthatisessentiallyequivalenttooursin[5]. Moreprecisely,
the two versions generate the same algebra and each version can be obtained from
the other one by elementary linear algebraic manipulations. This is explained in
detail in [7, Lemma 3.13]. One can also define a version of cyclotomic version of
Furusho’s p-adic multi-zeta values which will again be essentially equivalent to the
above version by the proof of [7, Lemma 3.13].
Acknowledgements. This paper was written while the author was visiting H.
Esnault at Freie Universit¨at Berlin supported by the fellowship for experienced
researchersof the Humboldt Foundation. The author thanks Prof. H. Esnault and
the Humboldt Foundation for this support.
2. Cyclotomic p-adic iterated sum series
Fix a prime p and M ≥1, with p∤ M. Let ζ be a primitive M-th root of unity,
K = Q (ζ) and q, the cardinality of the residue field of K. For s := (s ,··· ,s ),
p 1 k
with 0 ≤ s ; i := (i ,··· ,i ) with 0 ≤ i < M; and m := (m ,··· ,m ), with
i 1 k j 1 k
0≤m <p, let
i
ζi1n1+···iknk
σ(s;i;m)(n):= ,
X ns1···nsk
1 k
where the sum is over 0 < n < n < ··· < n < n with p|(n −m ). If we let
1 2 k i i
n := (n ,··· ,n ) we will also write the numerator of the above summand as ζi·n
1 k
and the denominator as ns.
Cyclotomic p-adic Multi-Zeta Values 3
Similarly, we let γ(s;i;m)(n):= ζikn ·σ(s′;i′;m′)(n), if p|(n−m ) and 0 other-
nsk k
wise, with s′ = (s ,··· ,s ), i′ := (i ,··· ,i ), and m′ = (m ,··· ,m ). Let
1 k−1 1 k−1 1 k−1
σ (s;i)(n) := σ(s;i;0)(n), where 0 = (0,··· ,0). We define the depth as d(s) = k
p
and the weight as w(s):= s .
i
P
Wecallasequenceoftheformσ(s;i;m)orγ(s;i;m)acyclotomic p-adic iterated
sum series (or ciss).
Definition 2.1. Let n ∈ N and let f : N → K be any function. We say that
≥n
f is an M-power series function, if there exist power series p (x) ∈ K[[x]], which
i
converge on D(0,r ) for some r >|p|, for 0<i≤pM, such that f(a)=p (a−i),
i i i
for all a≥n and pM|(a−i).
Clearly there is a unique M-power series function with domain N and which
extendsf.Weidentify twoM-powerseriesfunctionsiftheyagreeontheircommon
domains ofdefinition. By the Weierstrasspreparationtheorem, the powerseries p
i
in the above definition are unique. Fix 0 < l ≤ pM, and let f be as above. Then
there is a power series p(x) ∈K[[x]] which converges on some D(0,r) with r > |p|
and f(lqN)=p(lqN), for N sufficiently large.
Example 2.2. (i) Let s ∈ Z and f(k) := ζikks, for p ∤ k and f(k) = 0 for p|k.
Then f is an M-power series function.
(ii) Clearly the sums and products of M-power series functions are M-power
series functions.
(iii) Let f be an M-power series function. For any 0<l≤pM, with p|l let
f := lim f(n),
l
n→0
pM|(n−l)
with n rangingover positive integers such that pM|(n−l), and tending to 0 in the
p-adic metric.
Let f[1] be defined by
f(k)−f
f[1](k)= l,
k
ifp|k andpM|(k−l);andf[1](k)=0, if p∤k. We then see thatf[1] is anM-power
seriesfunction. Infact,ifp|l,andpisapowerseriesaround0suchthatf(n)=p(n)
for all pM|(n−l) then f[1](n)=q(n), for all pM|(n−l), where
p(x)−p(0)
q(x)= .
x
Inductively, we let f[k+1] :=(f[k])[1].
(iv) Using the notationas above,let f(1) be defined by f(1)(k):=f[1](k), if p|k;
and f(1)(k)= f(k), if p6|k. Then f(1) is also an M-power series function.
k
Proposition 2.3. Let f :N →K be an M-power series function. If we define
≥n0
F :N →K by
≥n0
F(n):= f(k)
X
n0≤k≤n
then F is also an M-power series function.
The following lemma on power series will be essential while we are proving the
linear independence of the σ ’s.
p
4 SinanU¨nver
Lemma 2.4. Let f,g ∈K[[z]] be two power series which are convergent on D(a),
for some a > 1. Suppose that g 6= 0, and let h := f/g. If there exist a ∈ K and
ij
n≥1 such that
a
ij
h(z+M)−h(z)=
X (z+j)i
1≤i≤n
0≤j<M
for infinitely many z ∈D(a) then h is constant and a =0, for all i and j.
ij
Proof. The proof is a generalizationof the proofof [7, Lemma 2.0.2]. Note that by
the Weierstrass preparation theorem the number of poles of h on the closed unit
disc D(1) is finite. This set is nonempty if at least one a 6= 0. Assume that this
ij
is the case and let this set be {α ,··· ,α }. Arrange α so that α is a pole of
1 k i 1
h(z), and hence α ∈ {0,−1,··· ,−(M −1)}. Since α −M is not in the last set,
1 1
it cannot be a pole of h(z+M)−h(z), but since it is a pole of h(z+M), it also
has to be a pole of h(z). Let α = α −M. Continuing in this manner we will get
2 1
α = α −(i−1)M, and that α −kM is a pole of h(z+M)−h(z) and hence is
i 1 1
in {0,··· ,−(M −1)}. This is a contradiction. (cid:3)
Let P denote the algebra of M-power series functions which are 0 on N\pN.
M
We will consider these as functions on pN. They are functions f : pN → K such
thatthereexistpowerseriesp ,for1≤i≤M,around0withradiusofconvergence
i
greater than |p| and which satisfy f(pk) = p (pk) for M|(k−i). Let us consider
i
σ (s;i) as functions on pN as well and let P denote the module over P
p M,σ M
generated by the σ (s;i) in F(pN,K). This is an algebra as can be seen by using
p
the shuffle product formula for series.
Proposition 2.5. The algebra P is free with basis {σ (s,i)|(s,i)∈∪ (N×n×
M,σ p n
[0,M −1]×n)} as a module over P .
M
Proof. We will provethe linear independence of the set S :={σ (s,i)|d(s)≤m},
m p
by induction on m. For any function f : pN → K, we let δ(f) denote the function
defined by δ(f)(n):=f(n+p)−f(n). Note that
ζikn
(2.1) δσ (s;i)(n)= σ (s′;i′)(n).
p nsk p
Let δ (f)(n)=f(n+pM)−f(n). Then
M
ζik(n+pl)
(2.2) δ (σ (s;i))(n)= σ (s′;i′)(n+pl).
M p X (n+pl)sk p
0≤l<M
We know the linear independence for the set S ={1}. Assuming that we know
0
the linear independence for S , we will prove it for S . Let us suppose that
m−1 m
{σ (s;i)} ∪ S is linearly dependent over P . Then there exists an l′ with
p m−1 M
0≤l′ <M such that we have an expression of the form
σ (s;i)= a σ (t;j),
p X t,j p
(t,j)
d(t)≤m−1
whichisvalidforallnwhichsatisfiespM|(n−pl′)andwitha aquotientofpower
t;j
series which converge on an open disc containing |z|≤|p|.
Applying δ to the last equation we get
M
ζik(n+pl)
σ (s′;i′)(n+pl)=
X (n+pl)sk p
0≤l<M
Cyclotomic p-adic Multi-Zeta Values 5
δ (a )σ (t;j)+ b σ (t;j) (n),
(cid:16) X M t,j p X t,j p (cid:17)
(t,j) (t,j)
d(t)=m−1 d(t)<m−1
fornwhichsatisfiespM|(n−pl′).Fromtheidentity(2.1)weseethatσ (s′;i′)(n+pl)
p
is equal to σ (s′;i′)(n) plus a linear combination of the terms σ (t;j)(n), with
p p
d(t)≤m−2andwithcoefficientswhicharequotientsofpowerseries. Thistogether
with the induction hypothesis implies that
ζik(p(l′+l))
=δ (a )(n),
X (n+pl)sk M s′;i′
0≤l<M
which contradicts the lemma above.
Next we do an induction on the number of elements σ (s,i) with d(s)=m, and
p
a 6=0. Suppose that we have a non-trivial equation
s,i
a σ (s,i)=0.
X s,i p
(s,i)
d(s)≤m
By the induction assumption on m, there is an (s,i) with d(s) = m such that
a 6= 0. In particular, there exists an 0 ≤ l′ < M such that a is not the zero
s,i s,i
functionwhenrestrictedtopl′+pMN.Intheremainderoftheproofwewillconsider
all the functions as functions on pl′+pMN. Dividing by a and rearranging we
s,i
get
σ (s,i)+ b σ (t,j)= b σ (t,j),
p X t,j p X t,j p
(t,j)6=(s,i) (t,j)
d(t)=m d(t)<m
where b are quotients of power series. Applying δ to this equation and using
t,j M
induction on the number of b 6= 0 with d(t) = m we obtain δ (b ) = 0 for all
t,j M t,j
(t,j) with d(t)=m, hence these b are constant and equal to, say c .
t,j t,j
So the last equation can be rewritten as
σ (s,i)+ c σ (t,j)= b σ (t,j).
p X t,j p X t,j p
(t,j)6=(s,i) (t,j)
d(t)=m d(t)<m
Applying δ and using the induction hypothesis to compare the coefficients of
M
σ (s′;i′) we obtain that
p
ζikp(l+l′) ζbp(l+l′)
p−sk X (z+l)sk + X c(s′,a;i′,b) X p−a(z+l)a =δM(b(s′;i′)),
0≤l<M (a,b)6=(sk,ik) 0≤l<M
where we put pz = n. The previous lemma then implies that the left hand side is
equal to 0. Putting α := c and looking at the coefficient of 1 we find
b (s;i′,b) (z+l)sk
that
ζikp(l+l′)+ Xαbζbp(l+l′) =0,
b6=ik
for every 0 ≤ l < M. Rephrasing we see that there exist β ∈ K, for 0 ≤ b < M
b
with β =1 such that
0
β ζlb =0,
X b
0≤b<M
for every 0 ≤ l < M. This contradicts the non-vanishing of the Vandermonde
determinant for {1,ζ,··· ,ζM−1}. (cid:3)
6 SinanU¨nver
Let F denote the algebraof M-powerseries functions and ι∈F denote the
M M
functionthatsendsnton.LetF (1)bethealgebraobtainedbyinvertingι.Note
M ι
that ι is already invertible on the components i+pN with 0 < i < p. Let F
M,σ
be the module over F generated by the σ (s;i)’s. Then by the shuffle product
M p
formula for series, FM,σ is an algebra. Let FM,σ(1ι)=FM,σ⊗FM FM(1ι).
Corollary2.6. ThealgebraF (resp. F (1))isfreewithbasis{σ (s;i)|(s,i)∈
M,σ M,σ ι p
∪ (N×n×[0,M −1]×n)} as a module over F (resp. F (1)).
n M M ι
Proof. For a set S, let F(S,K) denote the algebra of functions from S to K. We
have the following decomposition
F(N,K)=⊕ F(pN,K),
1≤i≤p
where we send f ∈ F(N,K) to the element on the right hand side whose i-th
component is f ∈F(pN,K), defined by
i
f (k)=f(k−p+i),
i
for k ∈ pN. We have σ (s;i) = σ (s;i), for all 1 ≤ i ≤ p, where we abuse the
p i p
notationanddenotebyσ (s;i)boththefunctiononthelefthandsideoftheequality
p
whose domain is N and also the function on the right hand side of the equation
which is its restriction to pN. By the definition of the power series functions, the
above decomposition gives the following decompositions:
F =⊕ P
M 1≤i≤p M
and
F =⊕ P .
M,σ 1≤i≤p M,σ
Usingthis,thefreenessofF overF followsfromProposition2.0.3andthe
M,σ M
statement for F (1) follows by localization. (cid:3)
M,σ ι
Definition 2.7. Let r : F → F denote the projection with respect to the
M,σ M
above basis. We will denote the projection F (1) → F (1) by the same no-
M,σ ι M ι
tation. Similarly, let s : F (1) → F denote the projection that has the effect
M ι M
of deleting the principal parts of the Laurent series expansions around 0 for the
components pN, and is identity on the components i+pN with 0<i<p.
Let s := (s ,··· ,s ), and t := (t ,··· ,t ). We write t ≤ s if there exists an
1 k 1 l
increasing function j :{1,··· ,l}→{1,··· ,k} such that t ≤s , for all i.
i j(i)
Lemma 2.8. Let f be an M-power series function and let g be defined as
g(n)= f(a)σ (s;i)(a)
X p
0<a<n
for some s:=(s ,··· ,s ) and i:=(i ,··· ,i ). Then
1 k 1 k
g = f σ (t,j),
X t,j p
(t,j)
t≤s
for some M-power series functions f . Similarly, if h is defined as
t,j
f(a)
h(n):= σ (s;i)(a),
X as p
0<a<n
p|a
Cyclotomic p-adic Multi-Zeta Values 7
for some s≥1 then
h= f σ (t;j),
X t,j p
(t,j)
t≤s′
for some M-power series functions f , where s′ :=(s ,··· ,s ,s).
t,j 1 k
Proof. We will prove this by induction on d(s). Suppose that d(s) = 0 and hence
σ (s;i)=1. Then for g the assertion follows from Proposition 2.3. For 0≤l <M,
p
let f be the power series in K[[z]] which has the property that f(n)=f (n) for n
l l
such that p|n and M|(n−l). Write f (z)= b zi, for |z|≤|p| then
l P0≤i il
b
il
h(n)= + f(a),
X X X as−i X
0≤l<M0≤i<s 0<a<n 0<a<n
p|a,M|(a−l)
where f is the unique M-power series function which satisfies f(n) = 0 if p ∤ n
and f(n) = b ni−s if p|n and M|(n−l). Then Proposition 2.3 implies that
Ps≤i il
the second sum defines an M-power series function. In order to see that h is an
M-power series functions it suffices to show that the function
1
t(n):= ,
X at
0<a<n
p|a,M|(a−l)
for any 0 ≤ l < M, is a K-linear combination of the σ (t;i)’s for 0 ≤ i < M. This
p
follows immediately from the fact that the characters χ : Z/M → K defined by
i
χ (α)=ζiα are distinct for 0≤i<M and hence are K-linearly independent.
i
Now assumethe statementfor all(s,i)with d(s)≤k andfix s:=(s ,··· ,s )
1 k+1
and i:=(i ,··· ,i ). Let F be as in Proposition2.3, then
1 k+1
ζik+1nk+1σp(s′;i′)(nk+1)
g(n)=F(n−1)σ (s;i)(n)− F(n )
p X k+1 nsk+1
0<nk+1<n k+1
p|nk+1
and the statement follows from the induction hypothesis on h.
On the other hand, to prove the statement on h, we write h(n)=
b
il
σ (s;i)(a)+ f(a)σ (s;i)(a),
X X X as−i p X p
0≤l<M0≤i<s 0<a<n 0<a<n
p|a,M|(a−l)
using the notation above. The second summand defines a function which is of the
form as in the statement of the lemma because of the induction hypothesis on g.
To finish the proof, it suffices to show that the function which sends n to
1
σ (s;i)(a)
X at p
0<a<n
p|a,M|(a−l)
is a K-linear combination of the functions σ (s,t;i,j), for 0 ≤ j < M. We prove
p
this exactly as above. (cid:3)
Proposition 2.9. For any s and m, σ(s;i;m)∈F .
M,σ
Proof. Wewillprovethisbyinductionond(s).Ifd(s)=1,thenσ(s;i;m)=σ (s;i)
p
if m = 0; and σ(s;i;m) ∈ F otherwise, by Proposition 2.3. Suppose we know
1 M
the result for d(s)≤k, and fix s with d(s)=k+1.
8 SinanU¨nver
Since
ζaik+1σ(s′;m′)(a)
σ(s;i;m)(n)= ,
X ask+1
0<a<n
p|(a−mk+1)
using the induction hypothesis we realize that we only need to show that functions
of the form
f(a)
σ (t;j)(a),
X as p
0<a<n
p|(a−m)
with f an M-power series function, are in F and this is exactly the statement
M,σ
of the previous lemma. (cid:3)
In fact, from the proofabove it follows that σ(s;i;m) is an F -linear combina-
M
tion of σ (t;j) with t≤s.
p
Definition 2.10. For a function f ∈ F , let f˜:= r(f) ∈ F . We call f˜the
M,σ M
regularization of f. Since by the previous proposition σ(s;i;m) ∈ F , we let
M,σ
σ˜(s;i;m) ∈ F be its regularization and for 0 < l ≤ M, we let σ(s;i;m)[l] :=
M
lim σ˜(s;i;m)(lqN) and σ(s;i;m):=σ(s;i;m)[1].
N→∞
For a function f : N → K and 0 ≤ m < p, let f denote the function which
[m]
is equal to f for values n which are congruent to m modulo p and is 0 other-
wise. Recall that γ(s;i;m)(n) := ζnikn−sk ·σ(s′;i′;m′)[mk](n). We will define the
regularized version γ˜(s;i;m) of γ(s;i;m) as follows. If m 6= 0, then it is de-
k
finedasγ˜(s;i;m)(n)=ζnikn−sk·σ˜(s′;i′;m′)[mk](n). Ifmk =0,andfor0≤l <M,
p (z)=a +a z+··· issuchthatσ˜(s′;i′;m′)(n)=p (n)forp|nandM|(n−l),then
l 0l 1l l
γ˜(s;i;m)(n):=ζnik(askl+ask+1,ln+···),ifp|nandM|(n−l)and0ifp∤n.Finally,
weletγ(t;i;m)[l]=limN→∞γ˜(t;i;m)(lqN)=ζlikasklandγ(t;i;m):=γ(t;i;m)[1].
Another way to describe this is as follows. For any s,i and m, γ(s;i;m) ∈
F (1), and γ˜(s;i;m):=s◦r(γ(s;i;m)).
M,σ ι
Definition 2.11. Let P (resp. S , S˜ ) denote the Q(ζ)-algebra (resp. vector
M M M
space)spannedbytheσ(s;i;m)(resp. σ(s;i;m),σ˜(s;i;m))andtheγ(s;i;m)(resp.
γ(s;i;m), γ˜(s;i;m)).
We callp-adicnumbers ofthe formσ(s;i;m)or γ(s;i;m), the cyclotomic p-adic
iterated sums (or cis).
3. proof of theorem 1.1
3.1. Cyclotomic p-adic multi-zeta values. We recall notation and concepts
from [6]. Fix M ≥ 1, and p ∤ M. Let Khhe ,··· ,e ii denote the ring of non-
0 M
commutativepowerseriesinthevariablese ,e ,··· ,e .Studyingtheactionofthe
0 1 M
crystalline frobenius on the fundamental group of G \µ , we defined, for every
m M
1 ≤ i ≤ M, g ∈ Khhe ,··· ,e ii [6, §2.2.3]. For an element α ∈ Khhe ,··· ,e ii
i 0 M 0 M
and any monomial eI = e ···e , let α[eI] denote the coefficient of eI in α. If
i1 in
eI =e ···e , we call w(eI)=w(e ···e ):=n, the weight of eI. By [6, (2.2.7)],
i1 in i1 in
we see that {g [eI]|I} = {g [eI]|I}, for any i,j. Therefore it makes sense to study
i j
onlyoneoftheg ’s. Weletg :=g ,andwedefinedthecyclotomicp-adicmulti-zeta
i M
values (or cmv) as the coefficients g[e ···e ], and we used the notation
i1 in
g[esk−1e ···es1−1e ]=pPsiζ (s ,··· ,s ;i ,··· ,i ),
0 ik 0 i1 p k 1 k 1
Cyclotomic p-adic Multi-Zeta Values 9
where1≤i ,··· ,i ≤M.Wecallk thedepthofthemonomialesk−1e ···es1−1e
1 k 0 ik 0 i1
or the corresponding cmv, and denote it by d(eI).
Let U denote the affinoid that is obtained by removing discs of radius one in
M
P1 around every M-th root of unity. Let A denote the algebra of rigid analytic
K M
functions on U . Then choosing the lifting F of frobenius given by F(z) = zp,
M
defines a corresponding element g ∈ A hhe ,··· ,e ii. Let ω := dlog(z) and
F M 0 M 0
ω := dlog(z −ζi), for 1 ≤ i ≤ M. For 1 ≤ i ≤ M, let i be the unique integer
i
such that M|(i−pi). Then in [6, (2.2.10)], we proved the following fundamental
differential equation for g :
F
dg = e F∗ω ·g −g · pg−1e g ω ,
F X i i F F X i i i i
where the sums are over 0≤i≤M and g :=1. We can rewrite this as follows,
0
(3.1) dg [eI]=F∗ω g [eI′]−p (g g−1)[eJ]g [eK]ω
F a F X F i i i
i,J,K
where I = (a,I′), and the second sum runs over J,K and 0 ≤ i ≤ M such that
(J,i,K)=I.
Let us h denote g (∞). Then we proved the following equation in [6, (4.1.1)]
F
that relates h and the g ’s:
i
(3.2) h· g−1e g = e ·h,
X i i i X i
where the sums are over 0≤i≤M.
For α∈K[[z]]hhe ,··· ,e ii, and a monomial eI, note that α[eI] ∈K[[z]] is the
0 n
coefficient of eI in α. We let α{eI} denote the function from N to K that sends n
to the coefficient of zn in α[eI]. If α ∈ A hhe ,··· ,e ii, we define α{eI} by first
M 0 n
viewing α in K[[z]]hhe ,··· ,e ii, by expanding around the origin.
0 n
3.2. ProofofTheorem 1.1. InordertoproveTheorem1.1,weneedtoshowthat
g [eI] ∈ P , for every monomial eI and 1 ≤ i ≤ M. We will prove this together
i M
with the statement that g {eI}∈P ·S˜ . The proof will be by induction on the
F M M
weight of eI. We will first show that g {eI} ∈ K ·S , then we will prove in fact
F M
that it lies in K·S˜ and finally in P ·S˜ .
M M M
We will prove the following statements together by induction on w :
(i) g {eI}∈P ·S˜ , for w(I)≤w
F M M
(ii) h[eJ]∈P if w(J)≤w−1.
M
and
(iii) g [eJ]∈P if w(J)≤w−1
i M
• Let us look at the statements (i), (ii) and (iii) for w =1.
Fromdg [e ]=0,weseethatg [e ]=0.Similarly,fromdg [e ]=F∗ω −pω ,
F 0 F 0 F a a a
we see that
(ζ−az)n
g [e ](z)=p .
F a X n
0<n
p∤n
Fromthisweseethat(i)isvalidforw =1;asfor(ii)and(iii),theyaretrivially
true for w =1.
•Assumethatweknow(i),(ii)and(iii)forw.Wewillprovethemforwreplaced
with w+1.
10 SinanU¨nver
Notethatbytheinductionassumptiong {eJ}∈P ·S˜ ⊆K·S ,forw(J)≤
F M M M
w. This implies thatg {eI}∈K·S , if w(I)=w+1,by the differentialequation
F M
(3.1).
By construction [6, §2.2.4], g [eI] is a rigid analytic function on U . Therefore
F M
by [6, Corollary 3.0.4], for any 0≤l <pM, if lim lqNg {eI}(lqN) exists then
N→∞ F
it is equal to 0.
Now note that by the induction assumption g {eJ} ∈ P · S˜ ⊆ K · S˜ ,
F M M M
for w(J) ≤ w. In particular, g {eJ} is an M-power series function. Then the
F
differential equation shows that the function which sends n to n·g {eI}(n) de-
F
fines an M-power series function by Proposition 2.3. This implies that the limits
lim lqNg {eI}(lqN) exist, for any 0 ≤ l < pM, and therefore they are 0.
N→∞ F
This together with the above fact that n·g {eI}(n) is an M-powerseries function
F
then implies that g {eI}(n) is an M-power series function. Therefore, we have
F
g {eI}∈K·S˜ .
F M
Nowreinterpretingthe factthatlim qNg {eI}(qN)=0,using the differen-
N→∞ F
tialequation(3.1)fordg [eI],wesee,bytheinductionhypothesesandthedefinition
F
of P , that with eI =e eJe :
M a b
(a) if 1≤a,b≤M, then we get
ζ−ag [eJe ]−ζ−bg [e eJ]∈P
a b b a M
(b) If 1≤a≤M and b=0 then
g [eJe ]∈P
a 0 M
(c) If 1≤b≤M and a=0 then
g [e eJ]∈P .
b 0 M
(d) If a=b=0, we do not get any new information.
Using (a)-(c) we immediately see the following lemma.
Lemma 3.1. If 1≤i≤M, and R is of weight w, and such that eR contains an e
0
factor then g [eR]∈P .
i M
This lemma together with the relation (3.2) implies the statement (ii) above for
w replaced with w+1:
Proposition 3.2. If R has weight w, then h[eR]∈P .
M
Proof. Now for any eR with w(R)=w(I)−1 let us look at the coefficients ofe eR
0
on both sides of the identity
h· g−1e g = e ·h
X i i i X i
0≤i≤M 0≤i≤M
to get
h[eR]−(hg−1)[e eR′]∈P
r 0 M
bythe inductionhypotheses onh andg , where eR =eR′e . Againbythis hypoth-
a r
esis we see that
(hg−1)[e eR′]−(h[e eR′]−g [e eR′])∈P .
r 0 0 r 0 M
Noting that g [e eR′]∈P we arrive at
r 0 M
h[eR]−h[e eR′]∈P .
0 M