Specialization of the
p-adic Polylogarithm to p-th Power Roots of UnityDedicated to Professor Kazuya Kato
for his fiftieth birthday
Kenichi Bannai
Received: November 11, 2002 Revised: April 28, 2003
Abstract. The purpose of this paper is to calculate the restriction of thep-adic polylogarithm sheaf top-th power torsion points. 2000 Mathematics Subject Classification: 14F30,14G20
Keywords and Phrases: p-adic polylogarithm, syntomic cohomology, rigid cohomology
1 Introduction
Fix a rational primep. The classical polylogarithm sheaf, constructed by Beilin-son and Deligne, is a variation of mixed Hodge structures on the projective line minus three points. Thep-adic polylogarithm sheaf is itsp-adic analogue, and is expected to be thep-adic realization of the motivic polylogarithm sheaf. In our previous paper [Ban1], we explicitly calculated the p-adic polylogarithm sheaf on the projective line minus three points, and calculated its specializa-tions to the d-th roots of unity for dprime to p. The purpose of this paper is to extend this calculation to the d-th roots of unity forddivisible by p. In particular, we prove that the specialization of thep-adic polylogarithm sheaf to d-th roots of unity is again related to special values of thep-adic polylogarithm function defined by Coleman [Col].
Let K = Qp(µd), with ring of integers OK. Let Gm = SpecOK[t, t−1] be
the multiplicative group overOK. Denote byS(Gm) the category ofsyntomic coefficients onGm. This category is a rough p-adic analogue of the category
will use the definition in [Ban2], which is a generalization of the definition in [Ban1] to the case whenpis ramified inK.
In order to describe the polylogarithm sheaf, it is first necessary to introduce the logarithmic sheafLog, which is a pro-object inS(Gm). The first property
we prove for this sheaf is that it satisfies thesplitting principle, even at roots of unity whose order is divisible byp.
Proposition (= Proposition 5.1) Let z6= 1 be ad-th root of unity inK, and letiz: SpecOK ֒→Gm be the closed immersion defined byt7→z. Then
i∗
zLog=
Y
j≥0 K(j).
Let U = Gm\ {1}. In our previous paper, following the method of [HW1]
Definition III 2.2, we constructed the polylogarithm extension pol∈Ext1Ssyn(U)(K(0),Log).
We first consider the case whenzis ad-th root of unity, wheredis an integer of the form d=N prwith (N, p) = 1 andN >1. In this case, we have a natural
mapiz: SpecOK→U. Leti∗zpol be the image of pol in
Ext1S(OK)(K(0), i
∗
zLog) =
Y
j≥0
Ext1S(OK)(K(0), K(j))
with respect to the pull-back map Ext1S(U)(K(0),Log)
i∗
z
−→Ext1S(OK)(K(0), i
∗
zLog).
Our main result is concerned with the explicit shape ofi∗
zpol.
For integers j ≥ 1, let Lij(t) be the p-adic polylogarithm function defined
by Coleman ([Col] VI, the function denoted ℓj(t)). It is a locally analytic
function defined on P1(C
p)\ {1,∞} satisfying Lij(0) = 0. On the open unit
disc{z∈Cp| |z|p<1}, the function is given by the usual power series
Lij(t) =
∞ X
n=1 tn
nj.
To deal with the specialization at points in the open unit disc around one, we also consider the locally analytic function
Lij,c(t) = Lij(t)−c1−jLij(tc),
wherec is an integer>1.
Theorem 1 (= Theorem 7.3) Let z be a d-th root of unity, where d is an
Remark 1 The above is compatible with the results of Somekawa [So] and also Besser-de Jeu [BdJ] on the calculation of the syntomic regulator.
Remark 2 In [Ban1], we proved that whendis prime top, on the open unit disc around 0is given by
ℓ(jp)(t) =
X
n≥1,(n,p)=1 tn
nj.
The difference between this formula and the formula of the previous theorem comes from the choice of the isomorphism (1). (See Remark 7.2 for details.)
For the case when z is a pr-th root of unity, let c > 1 be an integer and let
[c] :Gm→Gmbe the multiplication bycmap induced fromt7→tc. We denote
by [c]∗the pull back morphism of syntomic coefficients. We define the modified
polylogarithm to be
We note that this modification, which removes the singularity around one, is standard in Iwasawa theory.
Our theorem in this case is:
Theorem 2 (= Theorem 8.3) Let z be apr-th root of unity. Then we have
z is the pull back of syntomic coefficient by the natural inclusion iz :
Contents
1 Introduction 73
2 Review of the p-adic polylogarithm function 76
3 The Category of Syntomic Coefficients 78
4 Morphisms of Syntomic Data 82
5 The splitting principle 84
6 The specialization of pol to torsion points 88
7 The main result (Case N >1) 91
8 The main result (Case N = 1) 94
Acknowledgement. I would like to wish Professor Kazuya Kato a happy fiftieth birthday, and to thank him for all that he has contributed to mathe-matics. I would also like to thank the referee for carefully reading this paper, and for giving helpful comments.
Notation Let p be a rational prime. In this paper, we let K be a finite extension of Qp with ring of integersOK and residue fieldk. We denote byπ
a generator of the maximal ideal of OK. We let K0 the maximal unramified
extension ofQpinK, andW its ring of integers. We denote byσthe Frobenius
morphism onK0andW.
2 Review of the p-adic polylogarithm function
In this section, we will review the theory of p-adic polylogarithm functions due to Coleman [Col]. Since we will mainly deal with the value of thep-adic polylogarithm function at units in OCp, we will not need the full theory of
Coleman integration.
As in [Col], we call any locally analytic homomorphism log :C×
p →C+p, such
that dzd log(1) = 1, a branch of the logarithm. Throughout this paper, we fix once and for all a branch of the logarithm. Since we will only deal with the values of p-adic analytic functions at points outside the open unit disc where the functions have logarithmic poles, the results of this paper isindependent of the choice of the branch.
We define thep-adic polylogarithm function ℓ(jp)(t) for|t|<1 by
ℓ(jp)(t) = X (n,p)=1
tn
By [Col] Proposition 6.2, this function extends to a rigid analytic function on Cp\ {z;|z−1|p< p(p−1)
−1
}.
Proposition 2.1 ([Col] Section VI) The p-adic polylogarithm function
Lij(t) (Denotedℓj(t)in [Col])is a locally analytic function onP1(Cp)\ {1,∞} satisfying
(i) Li0(t) =t/(1−t)
(ii) d
dtLij+1(t) =
1
tLij(t) (j ≥0). (iii) ℓ(jp)(t) = Lij(t)−p−jLij(tp) (j≥1).
Definition 2.2 (i) For any integerj, we define the functionuj(t)by
uj(t) =
(
1
j!log
j(t) (j≥0)
0 (j <0).
Note that ifz is a root of unity inCp, thenuj(z) = 0 (j6= 0). (ii) For any integern≥1, we define the functionDn(t)by
Dn(t) = n−1
X
j=0
(−1)jLi
n−j(t)uj(t).
Ifz is a root of unity inCp, thenDn(z) = Lin(z).
To deal with the torsion points ofp-th power order, we need modified versions of the above functions.
Definition 2.3 Letc >1 be an integer prime top. We let: (i) ℓ(j,cp)(z) =ℓ
(p)
j (z)−c1−nℓ
(p)
j (zc) (j≥1). (ii) Lij,c(z) = Lij,c(z)−c1−nLij,c(zc) (j≥1). (iii)
Dn,c(z) = nX−1
j=0
(−1)jLin−j,c(t)uj(t).
3 The Category of Syntomic Coefficients
In this section, we will review the construction of the category of syntomic coefficients given in [Ban2]§4. Note that since we need to deal with the case when the primepis ramified inK, the theory of [Ban1] is not sufficient.
Definition 3.1 A syntomic datum X = (X, X, j,PX, φX, ι) consists of the following:
(i) A proper smooth scheme X, separated an of finite type overOK, and an open immersion j : X ֒→ X, such that the complement D is a relative simple normal crossing divisor overOK.
(ii) A formal schemePX overW.
(iii) For the formal completionX ofX with respect to the special fiber, a closed immersionι:X → PX⊗W OK, such that bothPX and the morphismι are smooth in a neighborhood ofXk.
(iv) A Frobenius map φX :PX → PX, which fits into the diagram
Xk ι
−−−−→ PX −−−−→ SpfW F
y φX
y σ∗
y
Xk ι
−−−−→ PX −−−−→ SpfW,
(2)
whereF is the absolute Frobenius ofXk.
We will often omitj andιfrom the notation and write
X= (X, X,PX, φX).
Example 3.2 1. LetP1be the projective line overW with coordinatet, and letP1O
K =P
1⊗ OK. We let G
m be the syntomic datum given by
Gm=
³
GmOK,P
1
OK,Pb
1, φ´,
where
(a) GmOK is the multiplicative group over OK, with natural inclusion
j :GmOK ֒→P
1
OK.
(b) Pb1 is thep-adic formal completion of P1.
(c) ι:Pb1
OK→bP
1⊗ O
K is the identity.
2. We letUbe the syntomic datum given by
U=³UOK,P1O
K,bP
1, φ´,
whereUOK=P
1
OK\{0,1,∞}, with the natural inclusionj:UOK ֒→P
1
OK. 3. We letOK be the syntomic datum given by
OK = (SpecOK,SpecOK,SpfW, σ), wherej andι are the identity.
Throughout this section, we fix a syntomic datumX. We will next review the definition of the category of syntomic coefficients S(X) on X. We will first define the categories SdR(X), Srig(X) and Svec(X). Let XK = X ⊗K and
XK=X⊗K.
Definition 3.3 We define the categorySdR(X) to be the category consisting of objects the triple MdR:= (MdR,∇dR, F•), where:
(i) MdR is a coherent OX
K module. (ii) ∇dR : MdR → MdR⊗Ω1(logD
K) is an integrable connection on MdR with logarithmic poles alongDK =D⊗K.
(iii) F• is the Hodge filtration, which is a descending exhaustive separated
filtration onMdR by coherent sub-OXK modules satisfying
∇dR(FmMdR)⊂Fm−1MdR⊗Ω1
XK(logDK).
LetXk =X ⊗kbe the special fiber of X andX the formal completion of X
with respect to the special fiber. We denote byXKthe rigid analytic space over
K associated toX ([Ber1] Proposition (0.2.3)) and by Xan
K the rigid analytic
space overK associated toXK (loc. cit. Proposition (0.3.3)). We will use the
same notations forX.
Definition 3.4 We say that a setV ⊂ XK is a strict neighborhood of XK in
XKan, if V ∪(XKan\ XK)is a covering of XKan for the Grothendieck topology.
For any abelian sheafM onXan
K, we let
j†M := lim−→
V
αV∗α∗VM,
where the limit is taken with respect to strict neighborhoodsV ofXK in XKan
with inclusion αV : V ֒→ XK. If M has a structure of a OXan
K-module, then
j†M has a structure of aj†O
Xan
Definition 3.5 We define the category Svec(X) to be the category consisting of objects the pair Mvec:= (Mvec,∇vec), where:
(i) Mvec is a coherentj†O
Xan
K module. (ii) ∇vec:Mvec→Mvec⊗Ω1
Xan
K is an integrable connection onMvec.
LetpdR:Xan
K →XK be the natural map. Definition 3.6 We define the functor
FdR:SdR(X)→Svec(X)
by associating toMdR:= (MdR,∇dR, F•)the modulej†(p∗
dRMdR)with the
con-nection induced from∇dR. The functor FdR is exact, since it is a composition
of exact functors ([Ber1] Proposition 2.1.3 (iii)).
LetPK0 be the rigid analytic space overK0 associated toPX ([Ber1] (0.2.2)).
As in loc. cit. D´efinitions (1.1.2)(i), we define the tubular neighborhoodof Xk
(resp. Xk) inPK0 by
]Xk[P:= sp−1(Xk)
¡
resp. ]Xk[P:= sp−1(Xk)
¢
,
where sp :PK0 → PX is thesp´ecialization[Ber1] (0.2.2.1). The tubular
neigh-borhoods are rigid analytic spaces overK0 with structures induced from that ofPK0.
Definition 3.7 We say that a setV ⊂]Xk[P is a strict neighborhood of]Xk[P
in ]Xk[P, if
V ∪(]Xk[P\]Xk[P)
is a covering of ]Xk[P for the Grothendieck topology.
For any abelian sheafM on ]Xk[P, we let
j†M := lim−→
V
αV∗α∗VM,
where the limit is taken with respect to strict neighborhoods V of ]Xk[P in
]Xk[P with inclusion αV : V ֒→]Xk[P. If M has a structure of a O]Xk[P
-module, thenj†M has a structure of aj†O
]Xk[P-module.
The Frobenius map φX : PX → PX induces a natural morphism of rigid
analytic spacesφX: ]Xk[P→]Xk[P.
Definition 3.8 We define the category Srig(X) to be the category consisting of objects the triple Mrig:= (Mrig,∇rig,ΦM), where:
(i) Mrig is a coherent j†O
(ii) ∇rig:Mrig→Mrig⊗Ω1
]Xk[P is an integrable connection on Mrig.
(iii) ΦM is the Frobenius morphism, which is an isomorphism
ΦM :φ∗XMrig
∼
= −→Mrig
ofj†O
]Xk[P-modules compatible with the connection.
The mapι:X → PX⊗W OK induces a map of rigid analytic spaces
prig :XKan→]Xk[P. (3)
Definition 3.9 We define the functor
Frig:Srig(X)→Svec(X)
by associating to the object Mrig:= (Mrig,∇rig,ΦM)the object
Frig(Mrig) := (p∗
rigMrig, p∗rig∇rig)
in Svec(X). This functor is exact by definition.
Definition 3.10 We define the category of syntomic coefficients to be the cat-egory S(X)such that:
(i) The objects ofS(X)consists of the tripleM:= (MdR, Mrig,p), where: (a) Mtyp is an object inStyp(X)fortyp∈ {dR,rig}.
(b) pis an isomorphism
p:FdR(MdR)−∼=→Frig(Mrig)
in Svec(X).
(ii) A morphism f : M → N in S(X) is given by a pair (fdR, frig), where
ftyp :Mtyp →Ntyp are morphisms in Styp(X) for typ∈ {dR,rig} com-patible with the comparison isomorphismp.
Example 3.11 For each integer n ∈ Z, we define the Tate object K(n) in
S(X)to be the setK(n) := (K(n)dR, K(n)rig,p), where:
(i) K(n)dR in SdR(X) is given by the rank one free OXK-module generated byen,dR, with connection ∇dR(en,dR) = 0 and Hodge filtration
(
(ii) K(n)rig in Srig(X)is given by the rank one free j†O
]Xk[P-module gener-ated byen,rig, with connection∇rig(en,rig) = 0 and Frobenius
Φ(en,rig) :=p−nen,rig.
(iii) pis the isomorphism given by p(en,dR) =en,rig.
Example 3.12 (See [Ban1] Definition 5.1) We define the logarithmic sheaf
Log(n):= (L(n) dR, L
(n) rig,p)
in S(Gm) by:
(i) L(dRn) inSdR(Gm)is given by the ranknfree OP1
K-module
L(dRn)=
n
Y
j=0 OP1
Kej,dR,
with connection ∇dR(ej,dR) = ej+1,dR ⊗dlogt for 0 ≤ j ≤ n−1 and ∇(en,dR) = 0, and Hodge filtration given by
F−mL(n) dR =
m
Y
j=0 OP1
Kej,dR.
(ii) L(rign) inSrig(Gm)is given by the ranknfree j†O]P1
k[bP1-module
L(rign)=
n
Y
j=0 j†O]P1
k[bP1ej,rig,
with connection ∇rig(ej,rig) = ej+1,rig ⊗dlogt for 0 ≤ j ≤ n−1 and ∇(en,rig) = 0, and Frobenius
Φ(ej,rig) :=p−jej,rig.
(iii) pis the isomorphism given by p(ej,dR) =ej,rig.
4 Morphisms of Syntomic Data
Definition 4.1 Define a morphism between syntomic datau:X→Y to be a pair (udR, urig) such that:
(ii) urig : PX → PY is a morphism of formal schemes over W compatible with the Frobenius, such that the diagram
X⊗k −−−−→ Pι X⊗k udR
y urig
y
Y ⊗k −−−−→ Pι Y ⊗k
(4)
is commutative.
Remark 4.2 Notice that in (4), contrary to [Ban2] Definition 4.2 (iii), we do not impose the commutativity of the diagram
X −−−−→ Pι X
udR
y urig
y
Y −−−−→ Pι Y.
(5)
Example 4.3 Letz be an element in O×
K, and letGmbe the syntomic datum defined in Example 3.2.1. We denote by z0 the Teichm¨uller representative of
z. In other words,z0 is a root of unity inW such thatz≡z0 (mod π). Then
iz= (idR, irig) :OK →Gm
is a morphism of syntomic data, where idR : SpecOK → GmOK and irig :
SpfOK →bP1W are morphisms defined respectively byt7→z andt7→z0.
Let u = (udR, urig) : X → Y be a morphism of syntomic data. By [Ber1] (2.2.16), we have a functoru∗
rig:Srig(Y)→Srig(X).
Lemma 4.4 Let u : X → Y be a morphism of syntomic data, and let
M := (MdR, Mrig,p) be an object in S(Y). Then there exists a canonical and functorial isomorphism
u∗(p) :FdR(u∗dRMdR)→Frig(u∗rigMrig)
in Svec(X).
The above lemma is trivial if we assume the commutativity of (5).
Proof. Let uvec : X → Y be the morphism of formal schemes induced from udR, and denote again byuvecthe map induced on the associated rigid analytic space. Then we have
FdR(u∗dRMdR) =u∗vecFdR(MdR).
Then u∗
vecFrig(Mrig) = u∗1K(Mrig⊗K) andFrig(u∗rigMrig) = u∗2K(Mrig⊗K).
Since (4) is commutative,u1 andu2 coincide on Xk. Hence by [Ber1]
Propo-sition (2.2.17), we have a canonical isomorphism ǫ1,2:u∗1K(Mrig⊗K)
≃
→u∗
2K(Mrig⊗K). (6)
The isomorphism of the lemma is the composition of the isomorphism
FdR(u∗
dRMdR) =u∗vecFdR(MdR)
p∼= −−→u∗
vecFrig(Mrig).
withǫ1,2.
Definition 4.5 Letu:X→Y be a morphism of syntomic data. Then
u∗:S(Y)→S(X)
is the functor defined by associating to any object M := (MdR, Mrig,p) the object
u∗M= (u∗dRMdR, u∗rigMrig, u∗(p))
in S(X).
5 The splitting principle
Let Log(n) be the logarithmic sheaf defined in Example 3.12. In this section, we will extend the splitting principle of [Ban1] Proposition 5.2 to the points defined in Example 4.3.
Proposition 5.1 (splitting principle) Let d be a positive integer, and let
z=ζd be a primitived-th root of unity inK. Let
iz= (idR, irig) :OK →Gm
be the morphism of syntomic data of Example 4.3 corresponding toz. Then we have an isomorphism
i∗
zLog(n)∼= n
Y
j=0 K(j)
in S(OK).
The proof of the proposition will be given at the end of this section. In order to prove the proposition, it is necessary to explicitly calculate the mapi∗
z(p) of
We assume for now thatzis an arbitrary element inO×
K. We denote byz0 the
root of unity in W such thatz ≡z0 (modπ). Let A= Γ(GmOK,OGmOK) =
OK[t, t−1]. We fix a presentation
OK[x1,· · ·, xn]/I∼=A
overOK, which defines a closed immersion
GmOK ֒→A n
OK.
Then the intersections Uλ ofGanmK with the ball B(0, λ+)⊂AnKanforλ→1+
form a system of strict neighborhoods (Definition 3.4) of GbmK in GanmK. For
λ >1, we let Aλ= Γ(Uλ,OUλ). Then limλ→1+Aλ=A
†⊗K, whereA† is the
weak completion ofA.
LetMvec= (Mvec,∇vec) be an object inSvec(Gm). By [Ber1] Proposition 2.2.3,
Mvecis of the formj†(M0,∇0), whereM0is a coherent module with integrable
connection ∇0 on a strict neighborhood Uλ. LetMλ = Γ(Uλ, M0). Then for
Suppose the connection ∇vec isoverconvergent. By [Ber1] Proposition 2.2.13, for anyη <1, there existsλ >1 such that
OKK→K are ring homomorphisms
given respectively byt7→z andt7→z0. By [Ber1] 2.2.17 Remarque, ǫ1,2:M⊗ivecK
∼
=
−→M ⊗irigK
The existence of the Frobenius ΦM on Mrig insures that the connection ∇rig
(hence ∇vec) is overconvergent ([Ber1] Theorem 2.5.7). Since |z−z0|<1, the above series converges by (8).
Next, let
Log(n):= (LdR(n), L(rign),p)
be the logarithmic sheaf of Example 3.12. As in (7), we L= Γ(Gan
mK, L
(n) vec) for L(vecn)=L(rign)⊗K0K. Then
L=
n
Y
j=0
(A†⊗K)e
j
for the basis ej =ej,rig⊗1, and the connection is given by ∇(ej) =ej+1⊗
dt
t (0≤j≤n−1). (10) Letuj(t) be the function defined in Definition 2.2.
Proposition 5.2 For integers i, m≥0, leta(mi)be elements inA†K such that
∇(∂ti)(e0) = n
X
j=0 a(ji)ej.
Then
∂ti(um) = n
X
j=0
a(jn)um−j.
In particular, we have
a(mi)(z0) =∂it(um)(z0). (11) Remark 5.3 The definition ofa(ji) implies
∇(∂ti)(em) = nX−m
j=0
a(ji)em+j.
Proof. We will give the proof by induction on i ≥ 0. Since a(0)0 = 1, the statement is true fori= 0. Suppose for an integeri≥0, we have
∂ti(um) = n
X
j=0
a(ji)um−j. (12)
By comparing the definition ofa(ji+1)with the equality
∇(∂i+1
t )(e0) =∇(∂t)◦ ∇(∂ti)(e0) = n
X
j=0
³
(∂ta(ji))ej+t−1a(ji)ej+1
´
we obtain the equality
This together with (13) gives the desired result. (11) follows from the fact that sincez0is a root of unity, um(z0) = 0 unless m= 0.
Proof. The assertion follows immediately from Remark 5.3
Proposition 5.5 We have
ǫ1,2(em⊗ivec1) =
the Taylor expansion ofuj(t) att=z0gives the equality
uj(z) =
The proposition now follows from the definition ofǫ1,2(9) and Corollary 5.4. Let us now return to the case when z=ζd is a primitived-th root of unity. Proof of Proposition 5.1. Since the connection is the only structure preventing L(dRn) andL(rign) from splitting, we have
It is sufficient to prove that the comparison isomorphism i∗
is given byej,dR7→ej,rig. Sincezis a torsion point,uj(z) = 0 forj6= 0. Hence
Remark 5.6 The calculation of Proposition 5.5 shows that ifzis an arbitrary element in O×
6 The specialization of pol to torsion points
In this section, we will first introduce thep-adic polylogarithmic extension pol calculated in [Ban1]. Then we will calculate its restriction to d-th roots of unity, wheredis an integer of the form d=N pr with (N, p) = 1 andN >1.
The case N= 1 will be treated in Section 8.
LetUbe the syntomic datum correspoinding to the projective line minus three points, as defined in Definition 3.2. Thep-adic polylogarithm sheaf is an ex-tension inS(U) of the trivial objectK(0) by the logarithmic sheafLoghaving a certain residue. In our previous paper, we determined the explicit shape of this sheaf.
Theorem 6.1 ([Ban1] Theorem 2) The p-adic polylogarithmic extension
pol(n)is the extension
with connection∇dR(edR) =e1,dR⊗dlog(t−1)and Hodge filtration given
(ii) Prig(n) inSrig(U)is given by
Prig(n)=j†O]U
k[bP1erig M
L(rign),
with connection∇rig(erig) =e1,rig⊗dlog(t−1) and Frobenius
Φ(erig) :=erig+
n
X
j=1
(−1)j+1ℓ(p)
j (t)ej,rig. (14)
(iii) pis the isomorphism given by p(edR) =erig⊗1.
Remark 6.2 In [Ban1] Theorem 2, the Frobenius is written as
Φ(erig) :=erig+
n
X
j=1
(−1)jℓ(jp)(t)ej,rig.
This is due to an error in the calculation of the proof. The correct Frobenius is the one given in (14).
Letz be a d-th root of unity, wheredis an integer of the form d=N pr with
(N, p) = 1 andN >1, and letz0∈W such thatz≡z0 (mod π). The purpose of this section is to prove the following theorem.
Theorem 6.3 The specialization of the polylogarithm at z is explicitly given as follows:
(i) i∗
zP
(n)
dR =KedR⊕
Ln
j=0Kej,dR with the natural Hodge filtration.
(ii) i∗
zP
(n)
rig =K0erig⊕
Ln
j=0K0ej,rig with Frobenius
Φ(erig) :=erig+
n
X
j=1
(−1)j+1ℓ(p)
j (z0)ej,rig.
(iii) pis the isomorphism given by
p(edR) =erig⊗1 +
n
X
j=1
ej,rig⊗(−1)j(Dj(z)−Dj(z0)),
whereDj(t)is the function defined in Definition 2.2.
The proof of the theorem will be given at the end of this section. As in the case ofLog, we first consider the Monsky-Washnitzer interpretation of pol(n). LetBK† = Γ(UanK, j†OUanK),
P(n)
andP(n)= Γ(Uan In particular, we have
b(mi)(z0) =∂ti(Dm)(z0). (15) Proof. The proof is again by induction on i > 0. We first consider the case wheni= 1. In this case, b(1)1 = (1−t)−1. Since Lim−j(t) anduj(t) satisfy the
Hence the statement is true fori= 1. Suppose for an integeri≥1, we have
By comparing the definition ofb(ji+1) with the equality
we obtain the equality
bj(i+1)=∂tbj(i)−t−1b
This together with (17) gives the desired result. (15) follows from the fact that sincez0is a root of unity, um(z0) = 0 unless m= 0.
Proposition 6.5 We have
ǫ1,2(e⊗ivec1) =e⊗irig1 +
Proof. Substituting z to the Taylor expansion of Dj(t) at t = z0 gives the
equality
The proposition now follows from the definition ofǫ1,2and Proposition 6.4.
7 The main result (CaseN >1)
The following lemma is well-known.
Lemma 7.1 There is a canonical isomorphism
Proof. SupposeMf= (MdR,f Mrig,f pe) is an extension ofK(0) byK(j) inS(OK).
We have exact sequences
0→K(j)dR→MdRf →K(0)dR→0 0→K(j)rig→Mrigf →K(0)rig→0.
Denote byej,dR and ej,rig the basis of K(j)dR and K(j)rig, and lete0e,dR and
e
e0,rigrespectively be the liftings ofe0,dRande0,riginMdRf andMrig. If we mapf
e
e0,dR toe0,dR, then we have an isomorphism
f
MdR∼=K(0)dRMK(j)dR
inSdR(OK). Next, since the quotient ofM byK(j) is isomorphic toK(0), the
Frobenius andpe is given by
e
p(ee0,dR) =ee0,rig⊗1 +ej,rig⊗a φ∗(ee0,rig) =ee0,rig+cej,rig
for somea∈K andc∈K0. If we takeb∈K0 such that (1−σ/pj)b=c, then
we have an isomorphism
f
Mrig∼=K(0)rigMK(j)rig
inSrig(OK) given byee0,rig7→e0,rig−bej,rig. The above shows that we have an
isomorphism
f
M ∼=³K(0)dRMK(j)dR, K(0)rigMK(j)rig,p´
of extensions ofK(0) byK(j) inS(OK), wherepis the isomorphism given by
p(e0,dR) =ee0,rig⊗1 +ej,rig⊗a
=e0,rig⊗1 +ej,rig⊗(a+b).
The canonical map of the lemma is given by associating to Mf the element (a+b)ej,dR inK(j)dR.
The inverse of this canonical map is constructed by associating to wej,dR in K(j)dR the extension
³
K(0)dRMK(j)dR, K(0)rigMK(j)rig,p ´
, where
p(e0,dR) =e0,rig⊗1 +ej,rig⊗w.
Remark 7.2 SupposeK=K0. Then by [Ban1] Theorem 1 and Example 2.8, we have an isomorphism
Ext1S(OK)(K(0), K(j))
∼
=
−→Hsyn1 (OK, K(j)) =K(j)rig. (19) IfM is an extension inS(OK)corresponding to aej,dR in Lemma 7.1, thenM
maps by (19)to((1−p−jσ)a)e
j,rig inK(j)rig .
The following theorem is Theorem 1 of the introduction.
Theorem 7.3 Let z be a torsion point of order d = N pr, where (N, p) = 1 andN >1. Then
i∗
zpol(n)= ((−1)jLij(z)ej,dR)j≥1
in
Ext1S(OK)(K(0), i
∗
zLog(1)) = n
Y
j=0
Ext1S(OK)(K(0), K(j)), where we view (−1)jLi
j(z)ej,dR as an element in Ext1S(OK)(K(0), K(j)) through the isomorphism of lemma 7.1
Proof. By Theorem 6.3, the image ofi∗
zpol(n) in Ext1S(OK)(K(0), K(j)) is the
extensionMf= (MdR,Mrig,f pe) given as follows: MdR is the direct sum MdR=K(0)dRMK(j)dR,
f
Mrig is the extension ofK(0)rig byK(j)rig with the Frobenius given by Φ(ee0,rig) =ee0,rig+ (−1)j+1ℓ(jp)(z0)ej,rig
for the liftinge0e,rig ofe0,rig in Mrig, andf pe is the isomorphism given by
e
p(e0,dR) =ee0,rig⊗1 +ej,rig⊗(−1)j(Lij(z)−Lij(z0)).
This implies that, in the notation of Lemma 7.1, we have a= (−1)j(Lij(z)−Lij(z0))
c= (−1)j+1ℓ(jp)(z0).
Sincez0is a root of unity prime top, the Frobenius acts byσ(z0) =zp0. Hence the Formula of Propisition 2.1 (iii) gives
ℓ(jp)(z0) =
µ
1− σ pj
¶
Lij(z0).
Again, in the notation of Lemma 7.1, we have c= (−1)j+1Li
j(z0).
Since a+b= (−1)jLi
j(z), the construction of the canonical map shows that
the image of i∗
zpol
(n) in Ext1
S(OK)(K(0), K(j)) maps to (−1) jLi
8 The main result (CaseN = 1)
In this section, we will consider the specialization of the polylogarithm sheaf to p-th power roots of unity. As mentioned in the introduction, we will consider a slightly modified version of the polylogarithm. Letc >1 be an integer prime to p, and let U0c,O
K = SpecOK[t,(1−t
c)−1]. We denote by U0
c the syntomic
data
U0c = (U0c,OK,P
1
OK,Pb
1, φ).
The multiplication by [c] map onGmOK defines a morphism of syntomic datum
[c] :U0c →U.
Definition 8.1 We define the modifiedp-adic polylogarithmic pol(cn)by
pol(cn)= pol(n)−[c]∗pol(n)∈Ext1
S(U0
c)(K(0),Log
(n)).
The explicit shape of pol(n) given in Theorem 6.1 and the definition of the pull-back [c]∗ gives the following proposition. Let
θc(t) =
1−tc
1−t .
Proposition 8.2 The modified p-adic polylogarithmicpol(cn) is the extension in S(U0
c), given explicitly bypol(cn):= (P
(n) dR, P
(n)
rig,p), where:
(i) PdR(n) inSdR(U0
c)is given by
PdR(n)=OP1
KedR
M
L(dRn),
with connection∇c,dR(edR) =e1,dR⊗dlogθc(t)and Hodge filtration given by the direct sum.
(ii) Prig(n) inSrig(U0
c)is given by
Prig(n)=j†O]U0
c,k[bP1erig M
L(rign),
with connection∇c,rig(erig) =e1,rig⊗dlogθc(t)and Frobenius
Φ(erig) :=erig+
n
X
j=1
(−1)j+1ℓ(j,cp)(t)ej,rig,
LetUc,OK = SpecOK[t, θc(t)
−1], and denote byU
c the syntomic data
Uc= (Uc,OK,P
1
OK,bP
1, φ).
The explicit shape of pol(cn)given in the previous proposition shows that pol
(n)
c
is in fact an object inS(Uc). In particular, we can specialize pol(cn) at points
on the open unit disc around one.
Similar caluclations as that of Theorem 6.3 with ℓ(jp), Dj(p) and Dj replaced
byℓ(j,cp),D
(p)
j,c and Dj,c gives the following theorem, which is Theorem 2 of the
introduction.
Theorem 8.3 Letz be apr-th root of unity, and letz0= 1. Then the special-ization of the modified polylogarithm atz is explicitly given as follows:
(i) i∗
zP
(n)
dR =KedR⊕
Ln
j=0Kej,dR with the natural Hodge filtration.
(ii) i∗
zP
(n)
rig =Kerig⊕
Ln
j=0Kej,rig with Frobenius
Φ(erig) :=erig+
n
X
j=1
(−1)j+1ℓ(j,cp)(z0)ej,rig.
(iii) pc is the isomorphism given by
pc(edR) =erig⊗1 + n
X
j=1
ej,rig⊗(−1)j(Dj,c(z)−Dj,c(z0)).
As a corollary, we obtain the following result.
Corollary 8.4 Let zbe a torsion point of orderpr. Then
i∗zpol through the isomorphism of lemma 7.1
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Kenichi Bannai
Graduate School of Mathematics Nagoya University
Furo-cho, Chikusa-ku, Nagoya 464-8602 Japan
