On integrality of $p$-adic iterated integrals

0.29 MB·English

by  Andre Chatzistamatiou

ON INTEGRALITY OF p-ADIC ITERATED INTEGRALS
5 ANDRECHATZISTAMATIOU
1
0
2 Abstract. The purpose of this paper is to prove integrality for certain p-
adic iterated Coleman integrals. As underlying geometry we will take the
v
complementofadivisorD⊂Xwithgoodreduction,whereXistheprojective
o
lineoranellipticcurveovertheWittvectorsofaperfectcharacteristicpfield.
N
As a corollary we prove a lower bound for the valuations of p-adic multiple
zetavalues.
9

]
T
N
Introduction
.
h
The basic idea for analytic continuation in real analytic spaces fails for classical
t
a p-adic spaces due to their totally disconnected nature. On such spaces it is not at
m
all evident how to patch local primitives of a function together into an essentially
[ unique global primitive. Guided by Dwork’s principle, that the p-adic analogue of

analyticcontinuationalongapathis”analyticcontinuationalongFrobenius”,Cole-
2
v mandevelopedin[Col82,Col85,CdS88]atheoryforsolvingiteratedintegrals. The
0 solutionsform a subring ofthe locally analytic functions whose elements are called
6 Coleman functions. As an application, Coleman constructed the p-adic version of
7
the polylogarithm functions. The relationship between values of ”the” p-adic zeta
5
function at positive integers and the value of Coleman polylogarithms at 1 is the
0
. same as in the complex setting.
1
Inourpaper,wewillusethegeneralizationofColeman’stheorybyBesser[Bes02]
0
and,independently,Vologodsky[Vol03],whichisbuiltontheTannakianformalism.
5
1 The notion ofa setof paths betweentwo points is replacedby the space ofnatural
: isomorphisms between two fibre functors on a suitable category of unipotent con-
v
i nections. The Frobenius action on the space of natural isomorphisms singles out
X
a distinguished path, which is called the Frobenius invariant path. Therefore ”an-
r alytic continuation along Frobenius” admits a precise formulation as continuation
a
along the Frobenius invariant path.
The main tool in our paper is an extension of the Tannakian formalism to very
simple unipotent categories over any commutative ring (Corollary 1.7). Broadly
speaking,these categoriesareequivalentto unipotent representationsofa freeten-
soralgebrawithfinitelymanygenerators. Wedevelopthetheoryinthefirstsection
toadegreesufficientforourapplication;amoregeneralstudyofTannakianduality
inthe unipotentcasewouldbe veryworthwhile. Ourapplicationis tothe category
U of unipotent connections on Xˆ with logarithmic singularities in D 6=∅, where X
is a geometrically connected smooth projective curve over the ring of Witt vectors
W of a perfect characteristic p field, Xˆ denotes the completion along the special
fibre, and D is finite ´etale over W.
Theauthor issupportedbytheHeisenbergprogram(DFG).
1
2 ANDRECHATZISTAMATIOU
In Sections 2 and 3, we recall some properties of U. In particular, we describe
the absolute Frobenius functor, and we give integral versions (that is, defined over
W) of the fibre functors attached to W-points of X with good or bad reduction
with respect to D, and give an account of tangential fibre functors.
Section4containsthetwomaintheorems. Theorem4.2assertsthatitissufficient
to construct an integral version of the Frobenius invariant path between two fibre
functorsx andy for unipotentconnectionsonXˆ. Then thereis a unique extension
to connections with logarithmic singularities. In the case X = P1 , this implies
W
integrality immediately. For an elliptic curve X, we can only prove integrality
under the additional assumption that the reductions x ,y ∈ X(k) of the points
0 0
underlying x,y, yield a point x −y of order prime to p (Theorem 4.3).
0 0
In Section 5, we give the application to p-adic multiple zeta values (Corollary
5.5): ζp(k1,...,km) ∈ p[Pmi=1ki], for all positive integers k1,...,km, all primes p,
and where p[Pmi=1ki] ⊂Zp is the ideal generated by all pjj! with j ≥ mi=1ki.
P
1. A simple case of unipotent Tannaka duality for arbitrary base
rings
Let R be a commutative ring, and let C be an R-linear exact category. Let 1 be
an object of C such that Hom(1,1)=R.
Definition 1.1. For any non-negative integer r, we define U =U (C,1) to be the
r r
fullsubcategoryconsistingofobjectsV ∈C thathaveanadmissiblefiltration(that
is, the inclusions are admissible morphisms)
0⊂V ⊂V ⊂···⊂V =V,
0 1 r
such that the quotients V /V and V are isomorphic to a finite sum with sum-
i+1 i 0
mands 1. We set U = U .
r r
S
Note that U is closed under extensions in C and is an R-linear exact category.
Let U be the idempotent completion of U. We have its derived category D(U ) at
c c
disposal [Nee90], and may set
Exti(X,Y):=Hom (X,Y[i]),
D(Uc)
forX,Y ∈U. Asforabeliancategories,thisconstructionisthesameastheYoneda
construction where an element in Exti(X,Y) is represented by an exact sequence
of admissible morphisms in U , which starts with Y and ends with X.
c
Notation 1.2. We say that U is free if Ext1(1,1) is a free and finitely generated
R-module and Ext2(1,1)=0.
Definition 1.3. Let r ≥0. An R-linear functor ω :U −→(R-modules) is called a
r
fibre functor if ω respects exact sequences and ω(1)∼=R. We define fibre functors
with source category U in the same way.
By definition, we have U ∼= (free finitely generated R-modules). For a free
0
finitelygeneratedR-moduleV,wewilldenotebyV⊗1∈U therepresentingobject
0
of the functor U −→ (free finitely generated R-modules);T 7→ Hom(T,1)⊗ V. It
0 R
comes equipped with an isomorphism Hom(1,V ⊗1)−→V.
Proposition 1.4. Suppose that U is free, and let r ≥0.
ON INTEGRALITY OF p-ADIC ITERATED INTEGRALS 3
(1) There is an object E of U such that Hom(E ,−):U −→(R-modules) is a
r r r r
fibre functor.
(2) Let E be an object with the properties of (1), let ǫ:E −→1 be a generator
r r
for Hom(E ,1). For every fibre functor ω : U −→ (R-modules), and every
r r
e∈ω(E ) such that ω(ǫ)(e) generates ω(1), ω is represented by (E ,e).
r r
Proof. For r =0, we can take E =1.
0
Set Γ := Ext1(1,1)∨ and T⊗r := Γ⊗r ⊗ 1. For r > 0, we will proceed by
induction and construct an extension
(1.4.1) 0−→T⊗r −→E −→E −→0,
r r−1
such that the induced map
(1.4.2) Hom(T⊗r,1)−→Ext1(E ,1)
r−1
isanisomorphism. Notethatinviewoftheembeddingtheorem[TT90,A.7.1,A.7.16]
(cf.[Bu¨h10, Theorem A.1]), the category U is closed under extensions in U .
c
Obviously,Ext2(1,1)=0impliesExt2(V,W)=0foranytwoobjectsV,W ∈U .
r
Suppose we have constructed E . Then
r−1
Ext1(E ,1)−∼→= Ext1(T⊗r−1,1)
r−1
is an isomorphism. Let E correspond to the preimage of the identity via the map
r
Ext1(E ,T⊗r)−→Ext1(T⊗r−1,T⊗r)−→Ext1(1,1)⊗Γ∨⊗r−1⊗Γ⊗r
r−1
=Hom(Γ⊗r,Γ⊗r).
Letus show (1.4.2) holds. This mapis induced by the class ofE andthe first row
r
of the commutative diagram
Hom(T⊗r,1)⊗Ext1(E ,T⊗r) // Ext1(E ,1)
r−1 r−1
∼= ∼=
(cid:15)(cid:15) (cid:15)(cid:15)
Hom(T⊗r,1)⊗Ext1(T⊗r−1,T⊗r) // Ext1(T⊗r−1,1),
which proves the claim.
By induction, we easily obtain Hom(E ,1)∼=R.
r
Let us prove by induction on r that
(1.4.3) 0−→Hom(E ,V )−→Hom(E ,V)−→Hom(E ,V )−→0
r 0 r r r−1
is an exact sequence if V ∈ U , and V ⊂ V is such that V ∈ U and V :=
r 0 0 0 r−1
V/V ∈U .
0 r−1
First, we note that, for every V ∈U , the map
r−1
Hom(E ,V)−→Hom(E ,V),
r−1 r
induced by E −→ E , is an isomorphism. Certainly, this holds for V ∈ U . For
r r−1 0
the general case, we can use an exact sequence
0−→V −→V −→V −→0,
0 r−2
with V ∈U ,V ∈U , and induction.
0 0 r−2 r−2
4 ANDRECHATZISTAMATIOU
For η ∈Hom(E ,V )=Hom(E ,V ) consider
r−1 r−1 r r−1
0 //V // V //V //0
OO0 OO r−OO 1
= η
0 //V // E // E // 0,
0 η r−1
with E being the induced extension. Since Hom(T⊗r,V )−→Ext1(E ,V ) is an
η 0 r−1 0
isomorphism,E definesamorphismτ :T⊗r −→V ,andtheclassofE istheimage
η 0 η
of the class of E via τ. We obtain a commutative diagram
r
0 //V // E // E // 0,
0 η r−1
OO OO OO
τ ψ =
0 // V // E //E // 0.
0 r r−1
The composition η˜:= (E −→ V)◦ψ : E −→ V yields a lifting of η and proves the
η r
exactness of (1.4.3).
As a consequence, we conclude that Hom(E ,V) is a free and finitely generated
r
R-module. Furthermore, the morphism Hom(E ,V)⊗E −→ V is an admissible
r r
epimorphism,whereHom(E ,V)⊗E is the obviousdirectsumofE terms. That
r r r
is, we have an exact sequence in U:
0−→K −→Hom(E ,V)⊗E −→V −→0.
r r
Let the following be an exact sequence in U:
0−→X −→Y −→Z −→0.
We will show by induction on r that, for all s ≥ r and Y ∈ U ,Z ∈ U , the map
s r
Hom(E ,Y)−→Hom(E ,Z) is surjective. Suppose first that r =0. Since
s s
Hom(E ,Y)⊗E −→Y −→Z
s s
is an admissible epimorphism and factors through Hom(E ,Y) ⊗ E , the claim
s 0
follows.
For r ≥1, we consider:
0 // X //Y′ //Zr(cid:127)_−1 //0
=
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 //X // Y // Z //0
❊
❊❊
❊
❊❊
❊❊"" (cid:15)(cid:15)(cid:15)(cid:15)
Z .
0
Since the r = 0 case is proved, we only need to lift morphisms contained in
Hom(E ,Z )toY. WeknowY′ ∈U forsometandHom(E ,Y′)−→Hom(E ,Z )
s r−1 t t t r−1
is surjective by induction. Since every morphism E −→ Y factors through E , we
t r
are done with the first part of the proposition. The second part follows easily. (cid:3)
Proposition 1.5. Assumptions and E as in Proposition 1.4. We set S =
r r
Hom(E ,E )op and I :=ker(S −→S ). The functor
r r r 0
Hom(E ,−):U −→U ((S -modules),R)
r r r r
ON INTEGRALITY OF p-ADIC ITERATED INTEGRALS 5
is an equivalence of categories. Moreover, In =ker(S −→S ), for all r ≥n≥1,
r n−1
and In/In+1 ∼= Ext1(1,1)∨ ⊗n.
(cid:0) (cid:1)
Proof. Since Hom(E ,V) ⊗ E −→ V is an admissible epimorphism for all V ∈
r r
U , the functor is faithful. In order to show that the functor is full, let f ∈
r
Hom (Hom(E ,V),Hom(E ,W)). We need to prove that it induces a morphism
Sr r r
of exact sequences
0 // K // Hom(E ,V)⊗E //V // 0
V r r
f⊗idEr
(cid:15)(cid:15)
0 //K // Hom(E ,W)⊗E // W //0.
W r r
Again,wehaveanadmissibleepimorphismHom(E ,K )⊗E −→K ,forsomes≥
s V s V
r. Therefore, it suffices to show that for every a∈Hom(E ,V)⊗ Hom(E ,E )=
r R s r
Hom(E ,V)⊗ Hom(E ,E )withtrivialimageinHom(E ,V)underthecomposi-
r R r r r
tionmap,(f⊗id )(a) hasalsotrivialimageinHom(E ,W). Thisfollows
Hom(Er,Er) r
immediately, because f is a morphism of S -modules.
r
Finally, let us prove the essential surjectivity. It is sufficient to show that the
map
Ext1(1,1)−→Ext (R,R),
Sr
isanisomorphismfor r≥1. The claimiseasilyprovedforr =1. We wouldlike to
prove ker(S −→S ) =I2 for r ≥ 2, which implies Ext (R,R)= Ext (R,R) and
r 1 Sr S1
reduces the assertion to the r = 1 case. Let us show Hom(E ,T⊗r) = Ir. Let X
r
be the following pullback
0 // T⊗r //X //T⊗r−1 //0
=
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 //T⊗r // E //E // 0,
r r−1
and set J := Hom(E ,X) ⊂ S . Since X = Γ⊗r−1 ⊗ E , we conclude I ·J =
r r 1
Hom(E ,T⊗r). By induction, J maps to Ir−1 in S , hence Ir =Hom(E ,T⊗r).
r r−1 r
(cid:3)
1.6. For a fibre functor ω :U −→(R-modules), we have
End(ω)=limHom(ω ,ω ),
←− |Ur |Ur
r≥0
where Hom(ω ,ω ) are the R-linear natural transformations; we set
|Ur |Ur
I :=ker(End(ω)−→End(ω )).
ω |U0
Corollary 1.7. Suppose U is free. A fibre functor ω : U −→ (R-modules) induces
an equivalence of categories
U −→U(End(ω),R).
Moreover, In = ker(End(ω) −→ End(ω )), and there is a natural isomorphism
ω |Un−1
In/In+1 ∼= Ext1(1,1)∨ ⊗n.
ω ω
(cid:0) (cid:1)
6 ANDRECHATZISTAMATIOU
Proof. The equivalence of categories is an immediate consequence of Proposition
1.5. The natural isomorphism In/In+1 ∼= Ext1(1,1)∨ ⊗n is induced by
ω ω
(cid:0) (cid:1)
Hom (I /I2,R)∼=Ext1 (R,R)=Ext1(1,1),
R ω ω End(ω) U
and the isomorphism (I /I2)⊗n −∼=→In/In+1. (cid:3)
ω ω ω ω
Definition 1.8. (cf. Definition [BF06]) Let σ : R −→R be an endomorphism. Let
F : U −→U be a σ-linear exact functor, where U is a category as in Definition 1.1.
A pair (ω,η), where ω :U −→(R-modules) is a fibre functor, and
η :ω⊗ R−∼=→ω◦F
R,σ
is an R-linear natural isomorphism, is called an F-fibre functor.
There is a unique isomorphism F(1)∼=1 compatible with η and ω(1)∼=R.
1.9. For two F-fibre functors x=(ω ,η ),y =(ω ,η ), we define
1 1 2 2
ρ :Hom (ω ,ω )−→Hom (ω ⊗ R,ω ⊗ R)
y,x R 1 2 R 1 R,σ 2 R,σ
to be the R-module morphism rendering commutive the following diagram:
ω ⊗ R η1 //ω ◦F
1 R,σ 1
ρy,x(a) a◦F
ω ⊗(cid:15)(cid:15) R η2 //ω ◦(cid:15)(cid:15) F.
2 R,σ 2
For all r≥0, we obtain induced maps
ρ :Hom (ω ,ω )−→Hom (ω ⊗ R,ω ⊗ R)
y,x R 1|Ur 2|Ur R 1|Ur R,σ 2|Ur R,σ
Remark 1.9.1. If U is free then
limHom (ω ,ω )⊗ R−→Hom (ω ⊗ R,ω ⊗ R),
←− R 1|Ur 2|Ur R,σ R 1 R,σ 2 R,σ
r≥0
isanisomorphism,becausebothmodulescanbeidentifiedwithlim ω (E )⊗
←−r≥0 2 r R,σ
R, where (E ,e ) is representing ω as in Proposition1.4.
r r 1|Ur
1.9.2. If x=y, then ρ :=ρ is a morphism of R-algebras. If ω =ω , then
x x,x 1 2
(1.9.1) ρ =(η−1◦η )◦ρ .
y,x 2 1 x
WecanconsiderHom(ω ,ω )asanEnd(ω )-module. Sincethereexistsanatural
1 2 2
isomorphism ω −→ω , it is a free End(ω )-module of rank 1. Obviously,
2 1 2
(1.9.2) ρ (ab)=ρ (a)·ρ (b)
y,x y y,x
for all a∈End(ω ) and b∈Hom(ω ,ω ).
2 1 2
Remark 1.9.3. Suppose U is free, and let (ω,η) be an F-fibre functor. Let (E ,e )
r r
berepresentingω ,weobtainaprojectivesystemE =(E ) withe 7→e ,and
|Ur r r r+1 r
an isomorphism
τ :End(ω)−∼=→limω(E ), 17→(e ) =:e.
←− r r r
r≥0
ON INTEGRALITY OF p-ADIC ITERATED INTEGRALS 7
Let φ : E −→ F(E) be the unique morphism of projective systems in U such that
ω(φ)(e)=η(e⊗1). Via the isomorphism τ, we have
ω(φ) η−1
ρ =lim ω(E)−−−→ω(F(E))−−→ω(E)⊗ R .
(ω,η) ←−(cid:20) R,σ (cid:21)
r≥0
Definition 1.10. For two F-fibre functors x=(ω ,η ),y =(ω ,η ), we set
1 1 2 2
Hom(x,y):={a∈Hom (ω ,ω )|ρ (a)=a⊗id },
R 1 2 y,x R
Isom(x,y):={a∈Isom (ω ,ω )|ρ (a)=a⊗id }.
R 1 2 y,x R
1.10.1. SupposethatExt1(1,1)isfreeandfinitelygenerated;wesetΓ=Ext1(1,1)∨.
We have a σ-linear map
Ext1(F,F):Ext1(1,1)−→Ext1(F(1),F(1))=Ext1(1,1),
and let φ:Γ∨⊗ R−→Γ∨ be the associated R-linear map. Define
R,σ
φ∨ :Γ−→(Γ∨⊗ R)∨ =Γ⊗ R
R,σ R,σ
as dual of φ.
Lemma 1.11. Suppose U is free. Set I = ker(End(ω ) −→ End(ω )) and H :=
2 2|U0
Hom(ω ,ω ).
1 2
(1) The following map is an isomorphism
In InH
Hom (ω (1),ω (1))⊗ −→ , b⊗a7→a·˜b,
R 1 2 R In+1 In+1H
where ˜b∈H is a lifting of b.
(2) For all n, ρ induces a map
y,x
InH InH
ρ(n) : −→ ⊗ R,
y,x In+1H In+1H R,σ
which equals ρ(0) ⊗(φ∨)⊗n via the isomorphism of (1).
y,x
Proof. Assertion (1) is obvious. For (2), we may use (1.9.2) to reduce to the case
x = y = (ω,η). Since ρ is a morphism of R-algebras, it suffices to compute for
x
n=1. Recall that we have an extension
ǫ
0−→T −→E −→1−→0.
1
τ
Choose an isomorphism R −→ ω(1) and e ∈ ω(E ) such that ω(ǫ)(e) = τ(1). Via
1
τ, we may identify ω(T), Γ and I/I2. For a ∈ I/I2, a·e ∈ ω(T) corresponds to
a. Moreover, we identify 1 and F(1) via the isomorphism t : 1 −∼→= F(1) satisfying
ω(t)(τ(1))=η(τ(1)⊗1). We have
◦t
Hom(1,T)⊗ R−→Hom(F(1),F(T))−→Hom(1,F(T)),
R,σ
which yields F(T)=(Γ⊗ R)⊗1.
R,σ
Let ψ : E −→ F(E ) be the unique morphism such that ω(ψ)(e) = η(e⊗1).
1 1
Then
ρ (a)=η−1(ω(ψ)(a·e))∈ω(T)⊗ R=I/I2⊗ R.
x R,σ R,σ
TherestrictionofψtoT isgivenbytheclassf ofF(E )inExt1(1,(Γ⊗ R)⊗1)=
1 R,σ
Hom (Γ,Γ⊗ R). By definition, we have (γ ⊗1)◦f = φ(γ ⊗1) for all γ ∈ Γ∨,
R R,σ
which proves f =φ∨. (cid:3)
8 ANDRECHATZISTAMATIOU
Definition 1.12. SupposeU isfree. AfullsubcategoryU′ ofU iscalledadmissible
if the following conditions are satisfied:
• 1∈U′,
• there exists a fibre functor ω : U −→ (R-modules), and a two-sided ideal
K ⊂End(ω), such that
ω
E ∈U′ ⇔K ·ω(E)=0,
ω
andEnd(ω )/K ·End(ω )isanobjectinU(End(ω),R)foreveryr ≥0.
|Ur ω |Ur
We could replace the second condition by requiring the existence of K for
ω
every ω. An admissible subcategory is automatically an exact category. In view
of Corollary 1.7, the ideal K ·End(ω ) depends only on U′, and ω induces an
ω |Ur
equivalence of categories
(1.12.1) U′ −→U(End(ω)/K ,R).
ω
Remark 1.12.1. The R-module defined by
Kω∩Iωn ⊂ Iωn = Ext1(1,1)∨ ⊗n
K ∩In+1 In+1
ω ω ω (cid:0) (cid:1)
is independent of the choice of ω.
Lemma 1.13. Suppose F(U′) is contained in U′. For every F-fibre functor (ω,η),
and every n≥0,
ρ (K ·End(ω ))⊂K ·End(ω )⊗ R,
(ω,η) ω |Un ω |Un R,σ
and
K ∩In K ∩In
(φ∨)⊗n ω ω ⊂ ω ω ⊗ R.
(cid:18)K ∩In+1(cid:19) K ∩In+1 R,σ
ω ω ω ω
Proof. By Corollary 1.7, there is E′ in U′∩U with ω(E′)∼=End(ω )/K . For
n n n |Un ω
everya∈K ·End(ω ),wehaveρ (a)·ω(E′)⊗ R=0,whicheasilyimplies
ω |Un (ω,η) n R,σ
ρ (a)∈K ·End(ω )⊗ R.
(ω,η) ω |Un R,σ
We know ρ In·End(ω ) ⊂In·End(ω )⊗ R. Since End(ω )/K
(ω,η) |Un |Un R,σ |Un ω
is a projective R-m(cid:0)odule for all n,(cid:1)we conclude
K ·End(ω )⊗ R ∩ In·End(ω )⊗ R =
ω |Un R,σ |Un R,σ
(cid:0) (cid:1) (cid:0) (K(cid:1)∩In)·End(ω )⊗ R,
ω |Un R,σ
which yields the claim by using Lemma 1.11. (cid:3)
Proposition 1.14. Suppose U is free and U′ is an admissible F-invariant subcat-
egory, that is, F(U′) is contained in U′. Let x = (ω ,η ),y = (ω ,η ) be F-fibre
1 1 2 2
functors on U. If
K ∩In K ∩In
(1.14.1) (φ∨)⊗n−id⊗1: ω1 ω1 −→ ω1 ω1 ⊗ R
K ∩In+1 K ∩In+1 R,σ
ω1 ω1 ω1 ω1
is surjective (resp. injective) for all n≥1, then
Hom(x,y)−→Hom(x ,y )
|U′ |U′
is surjective (resp. injective).
ON INTEGRALITY OF p-ADIC ITERATED INTEGRALS 9
Proof. In the first step we note, for all n≥0, the surjectivity of
(1.14.2) Hom(ω ,ω )−→Hom(ω ,ω ).
1|Un 2|Un 1|Un′ 2|Un′
Indeed,inordertoprovethis,wemaysupposeω =ω =ω. SincethereisE′ ∈U′
1 2 n n
with ω(E′) ∼= End(ω )/K (as End(ω )-module), we may use the equivalence
n |Un ω |Un
of categories (1.12.1) to conclude. Moreover, we note that the kernels of (1.14.2)
form a projective system with surjective transition maps. Therefore
Hom(ω ,ω )−→Hom(ω ,ω )
1 2 1|U′ 2|U′
issurjective. AfterliftinganelementinHom(x ,y )toHom(ω ,ω ),onecanuse
|U′ |U′ 1 2
thesurjectivityof (1.14.1)toalteraliftandobtainanelementinHom(x,y)(Lemma
1.11, Lemma 1.13). Clearly the injectivity of (1.14.1) implies the uniqueness of a
lift in Hom(x,y). (cid:3)
2. Unipotent connections for curves
2.1. Let k be a perfect field of characteristic p. We denote by W(k) the ring of
Wittvectorsandbyσ theFrobeniusendomorphism. Letπ :X −→Spec(W(k))bea
smooth projective curve such that W(k)=H0(X,O ), that is, X is geometrically
X
connected. LetD ⊂X beasubschemesuchthatπ isfiniteand´etale. We denote
|D
by Xˆ and Dˆ the completions along the special fibres X and D .
0 0
We denote by C (resp. Cˆ) the category of locally free coherent O -modules
X
(resp.O -modules)withlogarithmicconnectionalongD (resp.Dˆ),thatis,objects
Xˆ
are of the form (E,∇) with ∇ : E −→ E ⊗ Ω1 (logD) (resp. ∇ : E −→
OX X/W(k)
E⊗ Ω1 (logDˆ)) a connection. The categoriesC and Cˆare exact categories
OXˆ Xˆ/W(k)
intheevidentway,weset1=(O ,d)(resp.1=(O ,d)),anddefineU :=U(C,1)
X Xˆ
(resp. Uˆ := U(Cˆ,1)) with R = W(k). There is an evident completion functor
U −→ Uˆ,E 7→ Eˆ. We will only work with unipotent logarithmic connections in the
following.
We have
Ext1(1,(E,∇))−∼=→H1(X,E −∇→E⊗ Ω1 (logD))
U OX X/W(k)
=H1(X ,Eˆ −∇→Eˆ⊗ Ω1 (logDˆ))−∼=→Ext1(1,(Eˆ,∇)),
0 OXˆ Xˆ/W(k) Uˆ
sothatthecompletionfunctorisanisomorphismonExt1(1,1). Moreover,wehave
an exact sequence
0−→H0(X,Ω1 (D))−→Ext1(1,1)−→H1(X,O )−→0,
X/W(k) X
which implies that Ext1(1,1) is a free and finitely generated R-module. It is easy
to see that there is a functorial injective map
Ext2(1,(E,∇))֒→H2(X ,E −∇→E⊗ Ω1 (logDˆ)),
Uˆ 0 OXˆ Xˆ/W(k)
and the analogous statement holds for U. In particular, Ext2(1,1) = 0 if D 6= ∅,
and we are in the setup of Proposition 1.4.
Proposition 2.2. Let U ⊂ Xˆ be an open, and (E ,∇),(E ,∇) logarithmic (for
1 2
Dˆ) unipotent connections on U. Then
Hom((E ,∇),(E ,∇))−→Hom((E ,∇),(E ,∇))
1|U 2|U 1|(U\Dˆ) 2|(U\Dˆ)
10 ANDRECHATZISTAMATIOU
is bijective.
Proof. Since E∨ ⊗E , with the induced connection, is unipotent again, we may
1 2
assume (E ,∇)=(O ,d).
1 Xˆ
Injective is evident. Hence the question is local; we may suppose U = Spf(A),
Dˆ consistsonlyofa singlepointx , Dˆ =V(t)for somet∈A,andΩ =Adt.
0 A/W(k)
Moreover,we may suppose that E has an A-basis e ,...,e such that the connec-
2 1 r
tionmatrixisstrictlysub-diagonal. WedenotebyA\[t−1]thep-adiccompletion. We
notethatiff ∈A\[t−1]satisfiest·∂ (f)∈A,thenf ∈A. Let f e beahorizontal
t i i i
sectionwithf ∈A\[t−1]. By induction, we may assumethatPf ,...,f ∈A, thus
i 1 r−1
t·∂ (f )∈A, which proves the claim. (cid:3)
t r
2.3. Let us define the Frobenius pullback F : Uˆ −→ Uˆ. Suppose U ⊂ Xˆ\Dˆ is
an open, and φ,φ′ : U −→ U two lifts of the absolute Frobenius. If (E,∇) is a
unipotent connection on U, then it is automatically nilpotent and we obtain a
canonical horizontal morphism
(2.3.1) φ∗(E)−∼→= φ′∗(E).
For (E,∇)=(O ,d), it is the identity.
U
Let {U } be an open covering of Xˆ together with φ : U −→U , a lifting of the
i i i i i
Frobenius for each i, such that
(2.3.2) φ∗(I )=Ip ,
i Dˆ|Ui Dˆ|Ui
whereI istheidealforDˆ. Condition(2.3.2)impliesthatφ∗(E )isalogarithmic
Dˆ i |Ui
unipotent connection on U for every E ∈ Uˆ. In view of (2.3.1) and Proposition
i
2.2, we can glue to a connection F(E)∈Uˆ via the natural morphisms
φ∗(E ) −→φ∗(E ) .
i |Ui |Ui∩Uj j |Uj |Ui∩Uj
This construction does not depend (up to natural isomorphisms) on the choice of
the covering and the choices for φ . We can always find such a covering, because
i
aroundeverypointx ∈Xˆ,thereisanopenneighborhoodU andan´etalemorphism
0
f : U −→ Aˆ1 with {x } = f−1(0). Any Frobenius lift on Aˆ1 can be lifted to
W(k) 0 W(k)
U.
In this way, we obtain the Frobenius pullback
(2.3.3) F :Uˆ −→Uˆ.
2.4. In fact, we do not need Proposition2.2 in order to define F. For x ∈D let
0 0
φ ,φ be two Frobenius lifts on the local ring O , such that (2.3.2) is satisfied.
1 2 Xˆ,x0
In the following we will give an explicit description of the natural map
(2.4.1) f :φ∗E −→φ∗E ,
1 x0 2 x0
whereE is the stalkof alogarithmicunipotent connectiondefined ina neighbor-
x0
hood of x . Let t∈O be a generator of I , we have Ω1 =O dt. Let us
0 Xˆ,x0 Dˆ Xˆ,x0 Xˆ,x0
write E [t−1] for the induced (regular) connection on O [t−1]. We denote by
x0 Xˆ,x0
O \[t−1] the p-adic completion of O [t−1]. Over O \[t−1], we know that f
Xˆ,x0 Xˆ,x0 Xˆ,x0