Math. J. Okayama Univ. 59 (2017), 41–70

ON A NON-ABELIAN GENERALIZATION OF THE
BLOCH–KATO EXPONENTIAL MAP
Kenji Sakugawa
Abstract. The present paper establishes a non-abelian generalizationof the Bloch–Kato exponential map. Then, we relate p-adic polylogarithms introduced by Coleman to `-adic polylogarithms introduced by
Wojtkowiak. This formula is another analog of the Coleman–Ihara formula obtained by Nakamura, Wojtkowiak, and the author.

1. Introduction
Let p be a rational prime and let F be a finite extension of Qp with the+absolute Galois group GF := Gal(F /F ). Let BdR, BdR , Bcrys be the periodrings attached to F (cf. [8]).
For each de Rham representation V of GF , Spencer Bloch and Kazuya
Kato constructed a natural isomorphism called the Bloch–Kato exponential
map∼
0(V )\ DdR (V ) −→ He1 (F, V ) ⊂ H 1 (F, V ),expV : DdR
where He1 (F, V ) is the exponential part of the continuous Galois cohomol+ GF
0 (V ), D (V ) are defined to be (V ⊗andogy H 1 (F, V ) and DdR
Qp BdR )dR
GF(V ⊗Qp BdR ) , respectively (cf. [2, Definition 3.10]). The map expV , which
generalizes the p-adic exponential map for abelian varieties over F , is one of
powerful tools for studying cohomologies of algebraic varieties over F . Blochand Kato related de Rham cohomology to étale cohomology of “motives” byusing expV and formulated their Tamagawa number conjecture. The Bloch–
Kato logarithmic map, denoted by logV , is defined to be the inverse of expV .
The aim of the present paper is to give non-abelian generalizations of expV
and logV .
Let Repcrys
Qp (GF ) be the category of crystalline representations of GF and
let G be an affine group scheme in Repcrys
Qp (GF ) in the sense of Deligne (cf.
[6, Paragraphe 5]). The ring of regular functions O(G) of G is a Hopf
algebra object in the ind-category of Repcrys
Qp (GF ) by definition and the setof Qp -valued points G(Qp ) of G has a natural continuous action of GF . In
Definition 12, we will define the finite part Hf1 (F, G(Qp )) of the continuous
Galois cohomology H 1 (F, G(Qp )) which generalizes the finite part Hf1 (F, V )Mathematics Subject Classification. Primary 11R34; Secondary 14C30.
Key words and phrases. Bloch–Kato exponential map, Non-abelian p-adic Hodge theory, Coleman–Ihara formula.
41

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K. SAKUGAWA

of H 1 (F, V ). We say that G is an algebraic group in Repcrys
Qp (GF ) if O(G)is of finite type as a Qp -algebra. Let MFad
F (ϕ) be the category of weakly
admissible filtered ϕ-modules over F . Then, the Fontaine functorcrys
ϕ=1
Vcrys : MFad
∩ Fil0 (D ⊗F0 BdR )F (ϕ) −→ RepQp (GF ); D 7→ Vcrys (D) := (D ⊗F0 Bcrys )defines an equivalence of Tannakian categories over Qp (cf. [4, ThèorémeA]). Here, F0 is the maximal subfield of F unramified over Qp . Therefore,
Vcrys induces an equivalence of the category of group schemes in MFad
F (ϕ)
crys
and the category of group schemes in RepQp (GF ). We also denote by Vcrys
this equivalence by abuse of notation. The following theorem is our main
result.
Theorem 1.1 (Theorem 3.9). Let H be an algebraic group in MFadF (ϕ)
and let G be the algebraic group Vcrys (H) in Repcrys(G).
Supposethat
F
Qp
H satisfies the conditions (Non-neg) and (Bij) (see Definition 8 for thedefinitions of (Non-neg) and (Bij)). Then, there exists a canonical closed
algebraic subgroup F 0 HF of HF := H ×Spec(F0 ) Spec(F ) where F0 = F ∩Qurp .
Furthermore, we suppose the following two conditions:(a) For any right G-torsor X in RepQp (GF ), the set of Qp -rational points
X(Qp ) is non-empty.
(b) The algebraic group H is either a unipotent algebraic group over F0
in the usual sense or an algebraic group in the category of ordinary
filtered ϕ-modules in the sense of Perrin-Riou (cf. [18]).
Then, there exists a canonical injection
expG : F 0 HF (F )\H(F ) ,→ Hf1 (F, G(Qp ))satisfying the following properties:(i) If H is ωF -trivial (cf. Definition 5 and Subsection 3.1), then expG
is a bijection.
(ii) If G is a vector group scheme attached to V ∈ Obj(Repcrys
Qp (GF )),then expG coincides with expV under the canonical identifications0
Hf1 (F, G(Qp ))=Hf1 (F, V )=He1 (F, V ), F 0 HF (F )\H(F )=DdR(V )\ DdR (V ).

We call expG the generalized Bloch–Kato exponential map and define
the generalized Bloch–Kato logarithmic map logG to be its inverse when it
exists.
As an application of Theorem 1.1, we give an analog of the Coleman–
Ihara formula which is another version of [17, Theorem 1.1]. For each z ∈P1 \{0, 1, ∞}(OF ) and for each positive integer N greater than 1, we willdefine an extension Fe(z, N ) of F to be the kernel field of a certain 1-cocycle

NON-ABELIAN BLOCH–KATO EXPONENTIAL MAP

43

associated with z in Definition 18 (2). Our analog of the Coleman–Ihara
formula is then:Theorem 1.2 (Corollary 2). Let N be a positive integer greater than 1.
Suppose that p is odd and that F is unramified over Qp . Let σF be the arithmetic Frobenius automorphism of F . Then, we have the following formulafor all σ ∈ GFe(z,N ) :




−1
σF
p-adicCW(1.1) `iN (z)(σ) =Tr
1 − N LiN (z) φF,N (σ) .
(N − 1)! F/QppHere, `iN (z) : GF → Qp is the `-adic polylogarithm (cf. [16]), φCWF,N is the
Coates–Wiles homomorphism (cf. [2, Section 2]), and Lip-adic(z) is ColeN
man’s p-adic polylogarithm (cf. [3]).

Our study is motivated by the construction of the fundamental commutative diagram of Minhyong Kim (the diagram (1.3) below). We recall his
work briefly. Let K be a number field, v a finite place of K dividing p, andKv the v-adic completion of K with the ring OKv of integers. Let C be asmooth curve over OK with a good compactification over OKv . Let x bea K-valued point of C with good reduction at v and let x̄ be a geometricpoint of C over x. We denote by π1un-ét (CK , x̄) the unipotent completionover Qp of the étale fundamental group of CK := C ×Spec(OK ) Spec(K) withthe base point x̄1. Kim studied a Qp -scheme Hf1 (K, π1un-ét (CK , x̄)) whose setof R-valued points is defined to be the finite part Hf1 (K, π1un-ét (CK , x̄)(R))
of H 1 (K, π1un-ét (CK , x̄)(R)) for each Qp -algebra R (cf. [13], [14]). The Qp scheme Hf1 (K, π1un-ét (CK , x̄)) is called the Selmer variety attached to C. A
key ingredient in [14] is the morphism of Qp -schemes(1.2)


logC,v : Hf1 (Kv , π1un-ét (CK , x̄)) → RKv /Qp F 0 π1dR (CKv , x)\π1dR (CKv , x) ,where RKv /Qp is the Weil restriction functor, π1dR (CKv , x) is the de Rham
fundamental group of CKv := C×Spec(OK ) Spec(Kv ), and F 0 = F 0 π1dR (CKv , x)is the “Hodge subgroup” of π1dR (CKv , x). He constructed logC,v by rewriting
both sides of (1.2) as (subspaces of) classifying spaces of torsors of fundamental groups. The morphism logC,v fits into a commutative diagram(1.3)
C(OK )/ C(OKv )logC,v



Hf1 (K, π1un-ét (CK , x̄))(Qp )/ F 0 \π1dR (CKv , x)(Kv )4

/ Hf1 (Kv , π1un-ét (CK , x̄))(Qp )1This group is referred as the unipotent fundamental group of C

K in [14].

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K. SAKUGAWA

which plays a central role in his study of the finiteness of C(OK ) which
is a non-abelian generalization of the Coleman–Chabauty method (cf. [14,Introduction]). The proof of Theorem 1.1 is executed in a similar way as
Kim’s construction of logC,v , namely, we show that the classifying spaces ofadtorsors of G and H in Repcrys
Qp (GF ) and MFF (ϕ) satisfying certain conditions

are isomorphic to Hf1 (F, G(Qp )) and F 0 HF (F )\H(F ), respectively.
The outline of this article is as follows. In Section 2, we introduce conceptsof algebraic group and its torsor in a neutral rigid exact ⊗-category. In Subsection 3.1 and Subsection 3.2, we consider five rigid exact ⊗-categories andstudy classifying spaces of torsors in those categories. We define the generalized Bloch–Kato exponential map assuming Proposition 3.3 and define
the generalized Bloch–Kato logarithmic map in Subsection 3.3. In Section
4, we recall the exponential map introduced in [15], [20] and study a relation between this exponential map and the logarithmic map introduced in
Subsection 3.3. Then, we give a proof of Proposition 3.3. In Section 5, we
prove Theorem 1.2.
Notation. In this article, p denotes a rational prime. For each field k,
we fix its separable closure k and denote by Gk the absolute Galois group
Gal(k̄/k) of k. Let A be a topological group with a continuous action of Gk .
We denote by σ a the result of the action of σ ∈ Gk on a ∈ A. For i = 0, 1,i
H i (k, A) denotes the i-th continuous Galois cohomology Hcont(Gk , A).
Let X be a scheme over a field k and let R be a k-algebra. We denote
by XR or by X ⊗k R the base change X ×Spec(k) Spec(R) of X to Spec(R)and O(X) denotes the ring of regular functions on X. Denote by Veck thecategory of finite dimensional k-vector spaces.
For an object X of a category C, [X] denotes the isomorphism class of X
in C.
2. Algebraic groups in a rigid exact ⊗-category
Let us fix a field k in this section. Recall that a rigid exact ⊗-category overk is a k-linear exact category T equipped with an associative, unitary, andcommutative k-bilinear ⊗-structure ⊗ : T × T → T satisfying the following
three conditions (cf. [7, Definition 1.1, Definition 1.7]):(a) For each M, N ∈ Obj(T ), the internal hom Hom(M, N ) exists.
(b) For all four objects M1 , M2 , N1 , N2 ∈ Obj(T ), the canonical morphism
Hom(M1 , N1 ) ⊗ Hom(M2 , N2 ) → Hom(M1 ⊗ M2 , N1 ⊗ N2 )is an isomorphism.
(c) Any object in T is reflexive.

NON-ABELIAN BLOCH–KATO EXPONENTIAL MAP

45

The unit object in T is denoted by 1T . We say that T is neutral if thenatural map k → EndT (1T ) is an isomorphism and if there exists a faithfulexact k-linear ⊗-functor ω : T → Veck . Such an ω is called a fiber functor
on T . We denote by Ind(T ) the ind-category of T . Now on, we fix T a rigid
exact ⊗-category over k.
Definition 1 (cf. [6, Paragraphe 5.3]).
(1) A commutative algebra in T is an object A of Ind(T ) equipped withthe unit 1T → A and the multiplication A ⊗ A → A which is associative, unitary, and commutative. Denote by Alg(T ) the categoryof commutative algebras in T .
(2) An affine scheme in T is an object in the opposite category Alg(T )opof Alg(T ). The category of affine schemes in T is denoted by Aff(T ).
We denote by Sp(A) the image of A ∈ Obj(Alg(T )) in Aff(T ) under
the natural equivalence Alg(T ) ∼= Aff(T ).
0(3) Let T be a rigid exact ⊗-category over a field k 0 and let D : T → T 0
be a ⊗-functor in the sense of [7, Definition 1.8]. Then, we also
denote by
D : Aff(T ) → Aff(T 0 )the functor defined by Sp(A) 7→ Sp(D(A)).
Example 1. A commutative algebra in Veck is nothing but a commutative
k-algebra in the usual sense. Hence, the category of affine schemes in Veckcan be identified with the category of affine k-schemes.
Definition 2. Suppose that T is neutral and let ω be a fiber functor on T .
Let f : X → Y be a morphism of affine schemes in T . For each property
P of morphisms of k-schemes (resp. property P of k-schemes), we say thatf is P (resp. X is P ) if ω(f ) : ω(X) → ω(Y ) (resp. ω(X)) is P . Here, we
identify the category of affine schemes in Veck with the category of affinek-schemes.
Remark 1. Recall that if T is a Tannakian category over k, then any two
fiber functors are locally isomorphic for the fpqc topology (cf. [19, Theorem
3.2.3]). Therefore, in Definition 2, if the property P is local on the basefield k in the fpqc topology (e. g. of finite type, closed immersion, open
immersion, unipotent etc.), then the concepts defined in Definition 2 does
not depend on the choice of the fiber functor ω.
For any two affine schemes X = Sp(A) and Y = Sp(B) in T , we put
X × Y := Sp(A ⊗ B). We define the empty scheme in T to be Sp(0). Thereexists the following standard functor from T to Aff(T ).

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K. SAKUGAWA

Definition 3. We define the functor A : T → Aff(T ) by the equality A(M ):= Sp(Sym(M ∨ )). Here, Sym(M ∨ ) is the symmetric algebra associated withM ∨ . We call such A(M ) an affine space in T .

Example 2. For any object V of Veck , A(V ) is the affine scheme representing the functor (k-algebras) → (Sets); R 7→ V ⊗k R.

Lemma 2.1. Suppose that T is neutral. Let ω be a fiber functor on T and
let X = Sp(A) be an affine scheme in T . If X is of finite type, then there
exists a closed immersion X ,→ A(V ) for some V ∈ Obj(T ).

Proof. We write A as a direct limit limi Ai with Ai ∈ Obj(T ). Since X is of−→finite type, for sufficiently large i, the image of ω(Ai ) in ω(A) contains a setof generators of the k-algebra ω(A). Because the algebra homomorphism
Sym(ω(Ai )) → ω(A) is surjective, the induced morphism X ,→ A(A∨i ) is a
closed immersion by definition.
Definition 4. Let T be a neutral rigid exact ⊗-category over k.
(1) An affine group scheme in T is a group object in Aff(T ). Further,
we say that an affine group scheme G in T is an algebraic group in
T if G is of finite type.
(2) Let X be an affine scheme in T and let G be an affine group scheme
in T . A left action of G on X is a functorial left action of G on X. In
other words, it is a collection of actions of G(Y ) := HomAff(T ) (Y, G)
on X(Y ) := HomAff(T ) (Y, X) from the left for each object Y of
Aff(T ) which are functorial in Y . We define a right action of G on
X in a similar way.
(3) Let G be an affine group scheme in T . A left G-torsor in T is a nonempty object X in Aff(T ) equipped with a left action a : G×X → X
such that a×pr2 : G×X → X×X is an isomorphism of affine schemes
in T . We define a right G-torsor in T by replacing a left action by aright action. We denote by H 1 (T , G) the set of isomorphism classesof right G-torsors in T .
(4) Let G and G0 be affine group schemes in T . A (G, G0 )-bitorsor in T
is an affine scheme X in T equipped with a left (resp. right) actionof G (resp. G0 ) such that X is a left G-torsor and a right G0 -torsor.
We regard H 1 (T , G) as a pointed set with the point [G] consisting oftrivial torsors. The following well-known fact is useful for later discussions.
Lemma 2.2 (cf. [22, Proposition 2.3.6 (iii)]). Let k be a field and let G
be an algebraic group over k. Let V be a k-vector space equipped with analgebraic action of G. Then, for each finite dimensional subspace W of V ,there exists a finite dimensional subspace W 0 of V which contains W and isstable under the action of G.

NON-ABELIAN BLOCH–KATO EXPONENTIAL MAP

47

Proposition 2.3. Let G be an algebraic group in T and let X = Sp(A) be
an affine scheme in T equipped with a left action of G. If T is a neutral
Tannakian category over k, then for each object M in T contained in A,
there exists a G-stable object N ⊂ A in T containing M . In particular, A
is a direct limit of G-stable sub-objects of A in T .

Proof. By Tannakian duality, we may assume that T is the category Repk (π)
of algebraic representations of a pro-algebraic group π over k on finite dimensional k-vector spaces. Then, A (resp. M ) is an object in Ind(Repk (π))(resp. Repk (π)) and G is an algebraic group over k equipped with an algebraic action of π. We define a subspace N of A to be the intersectionof subspaces of A which contain M and are stable under the action of G.
Then, according to Lemma 2.2, N is a finite dimensional k-vector space. By
definition, N contains M and is stable under the action of G. Moreover,this space is also stable under the action of π because M is stable under theaction of π and N is characterized as the minimal k-vector subspace of A
containing M and stable under the action of G. Hence, N is an object of
Repk (π) and this completes the proof of the proposition.

Let T 0 be a rigid exact ⊗-category over a field k 0 and let D : T → T 0
be a ⊗-functor. Then, for any affine group scheme G in T , D(G) is anaffine group scheme in T 0 . If D is faithful, then D sends each G-torsor to a
D(G)-torsor. We define a notion named D-trivial as follows.
Definition 5. Let Di : T → Ti , 1 ≤ i ≤ n, be faithful ⊗-functors between
rigid exact ⊗-categories and let G be an affine group scheme in T . Let X
be a right G-torsor in T . We say that X is (D1 , . . . , Dn )-trivial if Di (X)is a trivial Di (G)-torsor for each i. When n = 1, we mean D1 -trivial for1(D1 )-trivial. Define a subset HD(T , G) of H 1 (T , G) by
1 ,...,Dn
!
n
Y
1
HD
(T , G) := Ker H 1 (T , G) →H 1 (Ti , Di (G)) .
1 ,...,Dn
i=1
11
We say that G is (D1 , . . . , Dn )-trivial if H (T , G) = HD
(T , G).
1 ,...,Dn

1
In other words, the pointed set HD(T , G) is canonically identified
1 ,...,Dn
with the set of isomorphism classes of (D1 , . . . , Dn )-trivial G-torsors.

Remark 2. Suppose that T is neutral and let ω : T → Veck be a fiber functor
on T . Then, a G-torsor X is ω-trivial if and only if ω(X) has a k-rational
point.
We recall the contraction and the pushforward of torsors.
Definition 6. ([11, Definition 1.3.1]). Let X (resp. X 0 ) be an affine schemewith a right (resp. left) action of an affine group scheme G in T . Then,

48

K. SAKUGAWA

we define the contraction X ∧G X 0 to be the quotient of X × X 0 by thenatural G-action defined by (x, x0 ) 7→ (xg, g −1 x0 ). If X 0 is an affine groupscheme in T and the action of G on X 0 is induced by a group homomorphismu : G → X 0 , we denote X ∧G X 0 by u∗ (X) and call it the pushforward of X
by u.
Remark 3. Let X (resp X 0 ) be a right G-torsor (resp. (G, G0 )-bitorsor).
(1) The contraction X ∧G X 0 is a right G0 -torsor. We call such a contraction the composition of torsors and call X ∧G X 0 the composite
X and X 0 .
(2) The composition is functorial in the following sense. Let D : T → T 0
be a faithful ⊗-functor between rigid exact ⊗-categories. Then, thequotient morphism X × X 0 → X ∧G X 0 induces D(G0 )-equivariant
morphismc : D(X) ∧D(G) D(X 0 ) → D(X ∧G X 0 ).

Since both hand sides are right D(G0 )-torsors, c is an isomorphism.
Finally, we introduce the condition (Unip).
Definition 7. Suppose that T is neutral and let ω be a fiber functor on T .
For each algebraic group G in T , we say that G satisfies (Unip) if G satisfies
the following condition: (Unip) ω(G) is a unipotent algebraic group over k.

3. Generalized Bloch–Kato exponential map and logarithmicmap
In this section, we define the generalized Bloch–Kato exponential map
and logarithmic map as maps between two classifying spaces of torsors incertain rigid exact ⊗-categories. We fix the following system of notation for
the rest of this paper. Let F be a finite extension of Qp and let F0 be themaximal subfield of F unramified over Qp . Let f := [F0 , Qp ] be the residual
degree of F over Qp .
3.1. Classifying spaces of torsors in MFF (ϕ) and in MFad
F (ϕ). In thisand the next subsections, we give examples of classifying spaces of torsors
in rigid exact ⊗-categories. Let MFF (ϕ) be the category of finite dimensional filtered ϕ-modules over F . Recall that a filtered ϕ-module is an F0 vector space M equipped with a Frobenius semi-linear endomorphism ϕM
and a separated and exhaustive decreasing filtration F i (M ⊗F0 F ) of the F vector space M ⊗F0 F . A Hodge-Tate weight of M is an integer n satisfying
F n (M ⊗F0 F ) 6= F n+1 (M ⊗F0 F ). This category has a natural ⊗-structure

NON-ABELIAN BLOCH–KATO EXPONENTIAL MAP

49

and internal hom (cf. [9, 4.3.4]) and one can check that MFF (ϕ) is a neutral
rigid exact ⊗-category over Qp . We let
ωF0 : MFF (ϕ) −→ VecF0 ; M 7→ M
be the forgetful functor. Note that ωF0 is a ⊗-functor. We denote by
MFF the category of finite dimensional filtered F -vector spaces. Then, thiscategory also has a natural ⊗-structure and internal hom (cf. loc. cit.) and isa neutral rigid exact ⊗-category over F . We also define a Hodge-Tate weightof M for each object of MFF as we did. The correspondence M 7→ M ⊗F0 F
defines a faithful exact ⊗-functor of rigid exact ⊗-categories
ωF : MFF (ϕ) −→ MFF .
We fix an algebraic group H in MFF (ϕ). We denote by ϕH : H → H
the Qp -scheme endomorphism induced by the Frobenius endomorphism of
the filtered ϕ-module O(H). Note that ϕfH is an endomorphism of an F0 scheme H because f is the residual degree of F0 over Qp . Let HF :=H ×Spec(F0 ) Spec(F ). By definition, the following two inclusion relations are
satisfied for all integers i and j:
(3.1)
(3.2)

F i O(HF )F j O(HF ) ⊂ F i+j O(HF ),X
cmH (F i O(HF )) ⊂F j O(HF ) ⊗F F k O(HF ).
j+k=i

Here, cmH is the comultiplication of O(H). We sometimes identify ωF (H)with HF .
Definition 8. Let Y be an affine scheme in MFF (ϕ).
(1) We say that Y satisfies the condition (Non-neg) if the following holds:(Non-neg) All Hodge-Tate weights of O(Y ) are non-negative, thatis, we have F 0 (O(YF )) = O(YF ).
(2) Suppose that Y is an affine group scheme in MFF (ϕ). Then, we
define the condition (Bij) as follows:(Bij) The morphism of F0 -schemes ϕf − 1 : Y → Y ; x 7→ ϕf (y)y −1
is an isomorphism.
Remark 4. Let Y be an affine scheme in MFF (ϕ) and let YF := Y ×Spec(F0 )Spec(F ). If Y satisfies the condition (Non-neg), then F i O(YF ) is an idealof O(YF ).
Suppose that H satisfies the condition (Non-neg). Then, F 1 O(HF ) isa Hopf-ideal of O(HF ) by the inclusion relation (3.1) in the beginning of
this subsection. We define a closed subgroup scheme F 0 HF of HF to be

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K. SAKUGAWA

Spec(O(HF )/F 1 O(HF )). We say that an H-torsor Y in MFF (ϕ) is nonnegative if Y satisfies the condition (Non-neg). For each non-negative Htorsor Y , we put F 0 YF := Spec(O(YF )/F 1 O(YF )). It can be checked thatF 0 YF is a torsor under F 0 HF in the usual sense. By definition, any ωF trivial torsor satisfies the condition (Non-neg).
Lemma 3.1. Let Y be a right H-torsor in MFF (ϕ) and let YF := Y ×Spec(F0 )Spec(F ). If the underlying algebraic group of H is unipotent, then Y satisfies
the condition (Non-neg).
Proof. Let UN be the closed subgroup scheme of GLN,F0 consisting of upper
triangle unipotent matrices and suppose that H is a closed subgroup schemeof UN . Then, there exists xi,j ∈ O(H), 1 ≤ i < j ≤ N such that F0 [xi,j |1 ≤i < j ≤ N ] = O(H) andX
cmH (xi,j ) = xi,j ⊗ 1 + 1 ⊗ xi,j +
xi,l ⊗ xl,ji<l<jfor all 1 ≤ i < j ≤ N . Since the algebraic group H over F0 is unipotent,
there exists a trivialization H ∼= Y of an H-torsor in VecF0 . We fix a∼trivialization t : Y −→ H and denote by yi,j ∈ O(Y ) the image of xi,j under∼]t : O(H) −→ O(Y ). Then, we have the equality
X(3.3)
caH,Y (yi,j ) = yi,j ⊗ 1 + 1 ⊗ xi,j +
yi,l ⊗ xl,j ,i<l<jwhere caH,Y is the coaction of O(H) on O(Y ). Let us denote by ca0H,Y∼
the isomorphism of filtered ϕ-modules O(Y ) ⊗F0 O(Y ) −→ O(Y ) ⊗F0 O(H)defined by y ⊗ y 0 7→ (y ⊗ 1)caH,Y (y 0 ). We show that yi,j ∈ F 0 O(YF ) by
induction on d := j − i.
First, we suppose that d = 1. Then, by the equality (3.3), we have
0
caH,Y (1 ⊗ yi,j − yi,j ⊗ 1) = 1 ⊗ xi,j . Since 1 ⊗ xi,j ∈ F 0 (O(YF ) ⊗F O(HF )) by
the condition (Non-neg), 1 ⊗ yi,j − yi,j ⊗ 1 is also contained in F 0 (O(YF ) ⊗F
O(YF )). This implies that yi,j ∈ F 0 O(YF ).
Next, we suppose that d > 1 and that yi0 ,j 0 ∈ F 0 O(YF ) if i0 − j 0 < d.
Then, by the induction hypothesis, ca0H,Y (1 ⊗ yi,j − yi,j ⊗ 1) = 1 ⊗ xi,j +P0
i<l<j yi,l ⊗ xl,j is contained in F (O(YF ) ⊗F O(HF )). Hence, we haveyi,j ∈ F 0 O(YF ) and this completes the proof of the lemma.
Remark 5. In general, an H-torsor Y is not always to be non-negative. Let
F = Qp and let H = Gm,Qp = Spec(Qp [t, t−1 ]) equipped with the trivial
Frobenius action and the filtration defined by F i O(H) = O(H) if i ≤ 0 and
by F i O(H) = 0 if i > 0. Let Y = Gm,Qp equipped with the trivial Frobenius
action, the filtration defined by F i O(Y ) = ti Qp [t], and the right action of

NON-ABELIAN BLOCH–KATO EXPONENTIAL MAP

51

H defined by the group structure of H. Then, Y is an H-torsor in MFQp (ϕ)
which is not non-negative.
Definition 9. Let M be a filtered ϕ-module over F and let f : M ⊗F0
F → M ⊗F0 F be an F -linear automorphism. Then, we define the filtered ϕ-module Mf to be M equipped with the filtration F i (Mf ⊗F0 F ) :=f (F i (M ⊗F0 F )) and with the same Frobenius endomorphism.

Proposition 3.2. Let H be an algebraic group in MFF (ϕ) satisfying theconditions (Non-neg) and (Bij). Then, there exists a canonical isomorphismof pointed sets∼

Ψ : Hω1F ,ωF (MFF (ϕ), H) −→ F 0 HF (F )\H(F ).
0

Proof. We repeat the same argument as in [14, Section 1]. First, we construct a map
0Ψ : {(ωF0 , ωF )-trivial H-torsors in MFad
F (ϕ)} → F HF (F )\H(F ).

Let Y be an (ωF0 , ωF )-trivial right H-torsor in MFad
F (ϕ). Then, we can0take an F -rational point pY,d of F YF because Y is ωF -trivial. Further, by
the same argument of [1, Corollary 3.2], we see that there exists a uniqueF0 -valued point pY,c of Y invariant under ϕf . Here, for applying Besser’s
argument, we need the conditions that Y is ωF0 -trivial and that H satisfies(Bij). We define Ψ(Y ) to be the class of h ∈ H(F ) in F 0 HF (F )\H(F )satisfying pY,c = pY,d h. One can check that the class of h in F 0 HF (F )\H(F )depends only on the isomorphism class of Y and that Ψ induces a welldefined injective map Hω1F ,ωF (MFF (ϕ), H) ,→ F 0 HF (F )\H(F ). By abuse
0of notation, we also denote this injection by Ψ.
Finally, we show the surjectivity of Ψ. Let h be an element of H(F )and denote by h] the ring automorphism of O(HF ) induced by the left
multiplication of h. Then, the filtered ϕ-module O(H)h] is a commutative
algebra in MFF (ϕ). We define Hh to be Sp(O(H)h] ). The affine scheme Hh
is ωF0 -trivial and the unit of H is the unique Frobenius invariant point of
Hh . On the other hand, the closed subscheme F 0 Hh,F of Hh,F coincides with
h−1 F 0 HF by construction. Therefore, we may take pHh ,d as h−1 . Hence,
Ψ(Hh ) = h modulo F 0 HF (F ) and this implies the surjectivity of Ψ.
Now, we assume that H is an algebraic group in the category MFadF (ϕ)
of weakly admissible filtered ϕ-modules (cf. [9]). According to [4, ThéorèmeA], the category MFad
F (ϕ) is a neutral Tannakian category over Qp . Let
ord
MFF (ϕ) be the category of ordinary filtered ϕ-modules in the sense of
Perrin-Riou (cf. [18]). Recall that M ∈ Obj(MFF (ϕ)) is said to be ordinary if its Hodge numbers coincide with Newton numbers counted with

52

K. SAKUGAWA

multiplicity and that each ordinary filtered ϕ-module is weakly admissible.
Perrin-Riou proved that MFord
F (ϕ) is a neutral Tannakian category over Qp(cf. loc. cit. Théorème 1.5). The following proposition is a key of this paper.
Proposition 3.3. Let H be an algebraic group in MFad
F (ϕ) satisfying theconditions (Non-neg) and (Bij). Furthermore, we suppose one of the following conditions:(a) The algebraic group H is an algebraic group in MFord
F (ϕ).
(b) The algebraic group H satisfies the condition (Unip).
Then, the canonical injection
1
Hω1F ,ωF (MFad
F (ϕ), H) ,→ HωF ,ωF (MFF (ϕ), H)00

is a bijection.
Remark 6. If an algebraic group H in MFad
F (ϕ) satisfies the condition (Unip),then H is ωF -trivial by Lemma 3.1. Furthermore, it is clear that H is ωF0 trivial in this case (cf. Remark 2). Hence, Proposition 3.3 states that wehave∼→ H 1 (MFF (ϕ), H) ∼(3.4) H 1 (MFad (ϕ), H) −= F 0 HF (F )\H(F )F

if H satisfies three conditions (Non-neg), (Bij), and (Unip).

The proofs of Proposition 3.3 for the ordinary case (a) and the unipotent
case (b) are given separately. The proof of the former case is simper thanthe latter case and we give a proof of that case in the last of this subsection.
The proof of the latter case will be given in Subsection 4.3 depending on a
totally different idea.
Before starting the proof of the ordinary case of Proposition 3.3, we recall
the slope filtration of ϕ-modules. Let σ be the arithmetic Frobenius automorphism of F0 . Let M be a dualizable finite dimensional ϕ-module overF0 , that is, the F0 -linear homomorphism M → σ∗ M induced by ϕM is an
isomorphism. Then, by the Dieudonné–Manin classification, there exists a
unique decomposition M = ⊕α M [α] such that M [α] is pure of slope α (cf.
[12, Corollary 14.6.4], [18, Subsection 1.2]). We define the slope filtration{Mα }α∈R of M by Mα := ⊕β≤α M [β] . By definition, this filtration is anincreasing, separated, and saturated filtration of M . If M is ordinary, thenM is dualizable and each slope of M is an integer. Thus, we regard theslope filtration of M as a filtration indexed by integers when M is ordinary.
Lemma 3.4. Let C be a sub-Tannakian category of MFord
F (ϕ) closed undersubquotients and extensions. Let M be an object in C and let f : M ⊗F0
F → M ⊗F0 F be an F -linear automorphism. If f stabilizes the filtration{Mα ⊗F0 F }α∈Z , then the filtered ϕ-module Mf is also an object in C. In
particular, Mf is weakly admissible.

NON-ABELIAN BLOCH–KATO EXPONENTIAL MAP

53

Proof. We prove this lemma by induction on n := ]{Hodge numbers of M }.
If n = 1, then M is pure of slope α for some integer α. The Hodge filtrationof M is give by the following equalities:(
M ⊗F0 F if i ≤ α,
F i (M ⊗F0 F ) =0
if i > α.
Hence, we have Mf = M and the assertion holds for M .
Next, we show the general case. We assume that the assertion of thelemma holds for each N ∈ Obj(C) such that ]{Hodge numbers of N } < n.
Let α be the maximum integer such thatF α (M ⊗F0 F ) = M ⊗F0 F.

Then, according to [18, Lemma 2.5], Mα is pure of slope α and Mα is a
sub-object of M in C. By the assumption of the induction, the assertion of
the lemma for M 0 := (M/Mα ) ∈ Obj(C) holds. Consider the following exact
sequence of filtered ϕ-modules:
0 → Mα,f = Mα → Mf → Mf0¯ → 0

where f¯ : M 0 ⊗F0 F → M 0 ⊗F0 F is the F -linear automorphism induced by
f . Since Mα and Mf0¯ are ordinary, Mf is also ordinary (cf. [18, Proposition2.4]). Furthermore, since Mα and Mf0¯ are objects in C, we see that Mf isalso an object in C.

Proof of Proposition 3.3 when H satisfies the condition (b). By the proof of
Proposition 3.2, it is sufficient to show that O(H)h] is admissible for anyh ∈ H(F ). According to Lemma 3.4, it is sufficient to show that h] preserves the slope filtration of O(HF ). Let α be a slope of O(H). Since thecomultiplication cmH of O(H) is compatible with its slope filtration, h] isfactored as follows:cm

O(HF )α := O(H)α ⊗F0 F −−−H
→

X

β+γ=α

1⊗h

O(HF )β ⊗F O(HF )γ −−−→

X

β+γ=α

O(HF )γ .

Note thatPeach slope of O(H) P
is non-negative by the condition (Non-neg). Hence,α
we have β+γ=α O(HF )γ = γ=0 O(HF )γ = O(HF )α . Thus, h] preserves thefiltration {O(HF )α }α∈Z≥0 and this completes the proof of Proposition 3.3 when
the condition (a) holds.


By Lemma 3.4 and the above proof, we obtain the following corollary.
Corollary 1. Let C be a sub-Tannakian category of MFord
F (ϕ) stable undersubquotients and extensions. Let H be an algebraic group in C satisfying
the conditions (Non-neg) and (Bij). Then, the canonical injections
1
Hω1F ,ωF (C, H) ,→ Hω1F ,ωF (MFad
F (ϕ), H) ,→ HωF ,ωF (MFF (ϕ), H)
0

0

0

54

K. SAKUGAWA

are bijections.
3.2. Classifying spaces of torsors in RepQp (GF ) and in Repcrys
Qp (GF ).
The next example of a rigid exact ⊗-category is the category RepQp (GF )of continuous representations of GF on finite dimensional Qp -vector spaces.
We study fundamental properties of affine schemes and classifying spaces of
torsors in this category.
Definition 10. Let X be an affine scheme in RepQp (GF ) of finite type.
Choose V ∈ Obj(RepQp (GF )) and a closed immersion X ,→ A(V ) (cf.
Lemma 2.1). Then, for any topological Qp -algebra R, we define the topology on X(R) to be the relative topology defined by the inclusion X(R) ,→A(V )(R) = V ⊗Qp R. Here, we equip V ⊗Qp R with the product topology
on R.
Lemma 3.5. The topology on X(R) does not depend on the choice of aclosed immersion i : X ,→ A(V ).
Proof. First, we consider the case when X is an affine space A(W ) with W ∈Obj(RepQp (GF )) and i is induced by an injective Qp [GF ]-homomorphism
W ,→ V . In this case, the product topology on W ⊗Qp R coincides withthe topology induced by W ⊗Qp R ,→ V ⊗Qp R because W ⊗Qp R is a direct
factor of V ⊗Qp R as an R-module.
Next, we consider the general case. Take another object V 0 of RepQp (GF )and a closed immersion i : X ,→ A(V 0 ). By replacing V 0 by V ⊕ V 0 , we
may assume that V is contained in V 0 and that there exists a commutative
diagram of closed immersions
i

X
i0

"

{

/ A(V )f

A(V 0 )
such that f is induced by the inclusion V ,→ V 0 . Since the topology on A(V )coincides with the relative topology defined by f , the relative topology on
X(R) induced by i coincides with its relative topology defined by i0 .
Definition 11. Let X be an affine scheme in RepQp (GF ) of finite type.
Then, for each topological Qp -algebra R equipped with a continuous actionof GF , we define theaction ofσ ∈ GF on x ∈ X(R) = Hom(O(X), R) by the−1equality σ x(a) := σ x(σ a) for each a ∈ O(X).
Remark 7. We use the same notation as in Definition 11.

NON-ABELIAN BLOCH–KATO EXPONENTIAL MAP

55(1) The action of GF on X(R) is functorial in X. In other words, forany morphism X → Y between affine schemes in RepQp (GF ), themap X(R) → Y (R) is GF -equivariant.
(2) If X = A(V ) with V ∈ Obj(RepQp (GF )), then the action of GF on
X(R) = V ⊗Qp R coincides with the diagonal action. In particular,
this action is continuous.
Lemma 3.6. Let X and R be the same as in Definition 11. Then, theaction of GF on X(R) is continuous.
Proof. Choose V ∈ Obj(RepQp (GF )) and a closed immersion X ,→ A(V ).
Recall that the inclusion X(R) ,→ A(V )(R) is GF -equivariant (cf. Remark 7(1)). Furthermore, the action of GF on A(V )(R) is continuous by Remark 7(2). Since the topology on X(R) is defined by this inclusion, we obtain the
conclusion of the lemma.
Let ωQp : RepQp (GF ) → VecQp be the forgetful functor. Then, we candescribe the classifying space of ωQp -trivial torsors as follows:
Proposition 3.7. Let G be an algebraic group in RepQp (GF ). Then, there
exists a natural isomorphism of pointed sets∼

Φ : Hω1Q (RepQp (GF ), G) −→ H 1 (F, G(Qp )).
p

Proof. Let X be an ωQp -trivial right G-torsor in RepQp (GF ). Since X is ωQp trivial, X(Qp ) is non-empty (cf. Remark 2) and is a right torsor under G(Qp )in the usual sense. We fix an element x of the non-empty set X(Qp ). Then,for each σ ∈ GF , there exists a unique element cx (σ) ∈ G(Qp ) satisfying σ x =xcx (σ). By construction, cx is a continuous 1-cocycle of GF with coefficientsin G(Qp ). By the standard argument, one can prove that the cohomology
class [cx ] ∈ H 1 (F, G(Qp )) of cx depends only on the isomorphism class of
X and [cx ] determines the isomorphism class of X. We put Φ([X]) :=[cx ]. Then, Φ is injective because the isomorphism class of X is uniquelydetermined by the cohomology class Φ([X]).
We shall check the surjectivity of Φ. Let c : GF → G(Qp ) be a continuous
1-cocycle. For each σ ∈ GF , we define ac (σ) : G → G by ac (σ)(g) := σ g c(σ).
Since c is a 1-cocycle, ac defines a left action of GF on G. We define Gc to
be G equipped with the action of GF defined by ac . Further, we equip Gc
with the natural right action of G defined by translations. We shall showthat Gc is a right torsor under G in RepQp (GF ). It is sufficient to show that
Gc is an affine scheme in RepQp (GF ). According to Proposition 2.3, there
exists a collection of finite dimensional sub-Qp -vector spaces {Vλ }λ of O(G)56

K. SAKUGAWA

stable under the actions of GF and G such that O(G) = ∪λ Vλ . Then, Vλ isalso stable under the new action of GF defined by ac . Therefore, Gc is anaffine scheme in RepQp (GF ). By the construction of Gc and the definitionof Φ, we have Φ([Gc ]) = [c]. Thus, Φ is surjective.

For the rest of this subsection, we consider torsors under algebraic groups
in the category Repcrys
Qp (GF ) of crystalline representations of GF .
Definition 12. We define the finite part Hf1 (F, G(Qp )) of H 1 (F, G(Qp )) by
Hf1 (F, G(Qp )) := Ker H 1 (F, G(Qp )) → H 1 (F, G(Bcrys )) .
Recall that there exists an equivalence of Tannakian categories∼

0
Dcrys : Repcrys→ MFadF (ϕ); V 7→ Dcrys (V ) := H (GF , V ⊗Qp Bcrys )Qp (GF ) −which is a quasi-inverse of the functor Vcrys (cf. [4]).
Proposition 3.8. Let G be an algebraic group in Repcrys
Qp (GF ). Then, themap Φ in Proposition 3.7 induces the canonical bijection∼(Repcrys→ Hf1 (F, G(Qp )),Φ : Hω1Q ,Dcrys
0
Qp (GF ), G) −p0
is the composite ωF0 ◦ Dcrys : Repcrys
where Dcrys
Qp (GF ) → VecF0 .

Proof. Let X be a right G-torsor in RepQp (GF ) which is ωQp -trivial. Notethat Φ([X]) is contained in the kernel of H 1 (F, G(Qp )) → H 1 (F, G(Bcrys ))if and only if the GF -invariant part of X(Bcrys ) is non-empty. In particular,
we have O(X) ⊗Qp Bcrys ∼= O(G) ⊗Qp Bcrys as GF -modules if Φ([X]) ∈1
Hf (F, G(Qp )). This implies that O(X) is ind-crystalline, namely, X is anaffine scheme in Repcrys
Qp (GF ).
Now, we suppose that X is a G-torsor in Repcrys
Qp (GF ). By definition, we
have the following equalities:
X(Bcrys )GF= HomQp -alg. (O(X), Bcrys )GF= HomBcrys -alg. (O(X) ⊗Qp Bcrys , Bcrys )GF= HomBcrys -alg. (Dcrys (O(X)) ⊗F0 Bcrys , Bcrys )GF= HomF0 -alg. (Dcrys (O(X)), Bcrys )GF= HomF0 -alg. (Dcrys (O(X)), F0 )= Dcrys (X)(F0 ).

0Therefore, Φ(X) is contained in Hf1 (F, G(Qp )) if and only if X is Dcrys
trivial. This completes the proof of the proposition.
NON-ABELIAN BLOCH–KATO EXPONENTIAL MAP

57

3.3. Generalized exponential map and logarithmic map. In this subsection, we define the generalized Bloch–Kato exponential map and logarithmic map assuming Proposition 3.3. We fix an algebraic group G in
RepcrysQp (GF ) and denote Dcrys (G) by H.
Theorem 3.9. We suppose that G is ωQp -trivial and that H satisfies theconditions (Non-neg) and (Bij).
(1) If H satisfies one of the conditions (a) and (b) of Proposition 3.3,∼
then the equivalence Vcrys : MFad
→ Repcrys
F (ϕ) −Qp (GF ) of Tannakian
categories induces a natural injection
F 0 HF (F )\H(F ) ,→ Hf1 (F, G(Qp )).
(2) If H is ωF -trivial, then the equivalence of Tannakian categories∼
Dcrys : Repcrys→ MFadF (ϕ) induces an injection between pointed
Qp (GF ) −sets
Hf1 (F, G(Qp )) ,→ F 0 HF (F )\H(F ).
(3) If H satisfies both conditions of (1) and (2), then the injections in(1) and (2) are bijections and one of them is the inverse of the otherone.
Proof. By the assumption of the triviality, we have
(3.5)

1
1
(Repcrys(RepcrysHD00
Qp (GF ), G).
Qp (GF ), G) = HωQ ,Dcrys
cryspThen, by Proposition 3.8, we have the following morphisms of pointed sets:
(3.6)
Vcrys

1
1
Hω1F0 , ωF (MFad−−→ HD(Repcrys0
F (ϕ), H) −Qp (GF ), G) ,→ Hf (F, G(Qp )).
crys , ωF ◦Dcrys∼

First, we show the assertion (1). If H satisfies one of the conditions (a) and(b) of Proposition 3.3, then the natural injection Hω1F , ωF (MFad
F (ϕ), H) ,→00
F HF (F )\H(F ) is bijective by Proposition 3.3. By composing the inverseof that bijection and the injection (3.6), we have the desired injection.
Next, we show the assertion (2). If H is ωF -trivial, then G is ωF ◦ Dcrys trivial. Therefore, the injection (3.6) is bijective by Proposition 3.8 and theequality (3.5). Since Dcrys is a quasi-inverse of Vcrys , the inverse of the bijection (3.6) is induced by the functor Vcrys . Then, by composing the inverse of0(3.6) and the natural injection Hω1F , ωF (MFad
F (ϕ), H) ,→ F HF (F )\H(F ),0
we obtain the second inclusion.
The assertion (3) is easily proved by the constructions of two injections.
58

K. SAKUGAWA

Definition 13. We call the injection in Theorem 3.9 (1) the generalized
Bloch–Kato exponential map for G and denote it by expG . We also define thegeneralized Bloch–Kato logarithmic map logG to be the injection in Theorem
3.9 (2).
The following theorem is deduced from the definition of the exponential
maps and Remark 3.
Theorem 3.10. Let 1 → G0 → G → G00 → 1 be an exact sequence of
0
0
00
00
algebraic groups in Repcrys
Qp (GF ) and set H := Dcrys (G ), H := Dcrys (G ).
We assume that whole the groups satisfy whole the conditions in Theorem
3.9 (1), (2). Then, we have the following commutative diagram of pointed
sets with exact rows:(3.7)

F 0 HF0 (F )\H 0 (F )/ F 0 HF (F )\H(F )Hf1 (F, G0 (Qp ))F∼= expG∼= expG0

/ F 0 H 00 (F )\H 00 (F )

/ H 1 (F, G(Q ))fp∼= expG00

/ H 1 (F, G00 (Q )).
fpHere, the horizontal maps are defined by the pushforward of torsors. Further, if G0 is contained in the center of G, then Hf1 (F, G0 (Qp )) (resp.
F 0 HF0 (F )\H 0 (F )) acts on Hf1 (F, G(Qp )) (resp. F 0 HF (F )\H(F )) and the
diagram (3.7) is compatible with those actions.
The last assertion of Theorem 3.10 follows from well-known facts about
long exact sequence of non-abelian group cohomology for central extensions
and from the construction of expG .
4. Unipotent caseIn this section, we recall an exponential map exp0G defined in the previous
works [15] and [20] when G satisfies (Unip). The main result Proposition 4.9of this section states that exp0G is the inverse map of the generalized Bloch–
Kato logarithm logG defined in Subsection 3.3. Then, we complete the proofof Proposition 3.3 when H satisfies the condition (b) of Proposition 3.3 asa consequence of Proposition 4.9.
4.1. Remarks on nilpotent Lie algebras and unipotent algebraic
groups. Let k be a field of characteristic 0 and g a finite dimensional nilpotent Lie algebra over k. Then, for any k-algebra R, the Campbell–Hausdorff
product (cf. Campbell–Hausdorff’s formula [21, Chapter IV, Section 8, p.
27 line 30]) defines a canonical group structure on g ⊗k R. We denote thisgroup by (g ⊗k R)CH and can regard the functor
gCH : (k-algebras) → (groups); R 7→ (g ⊗k R)CH

NON-ABELIAN BLOCH–KATO EXPONENTIAL MAP

59

as a unipotent algebraic group over k because this functor is represented by
the scheme Spec(Sym(g∨ )) where g∨ is the dual k-vector space of g. Let
NilLiek (resp. Unipk ) be the category of finite dimensional nilpotent Liealgebras over k (resp. unipotent algebraic groups over k). Then, according
to [5, Chapter IV, Section 2, Proposition 4.1, Corollaire 4.5 (b)], the functor
NilLiek → Unipk ; g 7→ gCH
is an equivalence of categories with a quasi-inverse
Lie : Unipk → NilLiek ; G 7→ Lie(G).
The following proposition is easily checked by gCH = Spec(Sym(g∨ )).
Proposition 4.1. The functor Lie induces an equivalence between the category of algebraic groups in Repcrys
Qp (GF ) satisfying (Unip) and the categoryof nilpotent Lie algebra objects in Repcrys
Qp (GF ). In particular, for an algebraic group G in RepQp (GF ) satisfying (Unip), G is an algebraic group in
RepcrysQp (GF ) if and only if Lie(G) is a crystalline representation of GF .
Let g be a nilpotent Lie algebra object in Repcrys
Qp (GF ) and let h :=Dcrys (g).
Lemma 4.2. The algebraic group hCH in MFad
F (ϕ) satisfies the condition(Non-neg) if and only if any Hodge-Tate weight of h∨ is non-negative.
Proof. Since hCH = Spec(Sym(h∨ )), h∨ is a sub-admissible filtered ϕ-moduleof O(hCH ). Thus, if hCH satisfies (Non-neg), then any Hodge-Tate weightof h∨ is non-negative. On the other hand, if any Hodge-Tate weight ofh∨ is non-negative, then any Hodge-Tate weight of Symn (h∨ ) is also nonnegative for each n ∈ Z≥0 . Hence hCH satisfies (Non-neg) and we have theconclusion.

We define the central descending series Γi g by Γ1 g := g and Γi+1 g :=[Γi g, g].
Lemma 4.3. We put Vi := Γi g/Γi+1 g. If ϕf -invariant parts of Dcrys (Vi )are zero for all i, then the algebraic group hCH in MFad
F (ϕ) satisfies thecondition (Bij).
Proof. We prove this lemma by induction on the nilpotency n of g. If n = 1,
then the assertion is easily checked. We assume that n > 1 and the assertion
holds when we replace n by n − 1. Let H be hCH , Z the center of H, and
H 0 = H/Z. Then, we obtain the following commutative diagram with exact

60

K. SAKUGAWA

rows:
1

/Z
f
∼
= ϕZ −1

1


/Z

/H

/ H0

/1

f
ϕfH −1 ∼= ϕH 0 −1


/H


/ H0

/ 1.

We note that ϕfH − 1 is not a group homomorphism except the case n =1. However, the diagram above is compatible with the action of Z on H
defined by multiplications. Indeed, for any z ∈ Z and h ∈ H, the equality(ϕfZ (z)z −1 )(ϕfH (h)h−1 ) = ϕfH (zh)(zh)−1 holds. Hence, by the snake lemma,
we have the conclusion of the lemma.

4.2. The fundamental exact sequence. Let us fix an algebraic group
G in RepQp (GF ) satisfying the condition (Unip). We denote by g the Liealgebra of G. The key of the construction of the exponential map in [15]and [20] is the following lemma:
Lemma 4.4 (The fundamental exact sequence, [20, Lemma 2.1], [15, Proofof Proposition 1.4]). There exists the following GF -equivariant exact sequence of topological pointed sets:
(4.1)

α

β

+
1 → G(Qp ) −→ G(Be ) −→ G(BdR)\G(BdR ) → 1.

ϕ=1Here, Be is the ϕ-invariant part Bcrys
of Bcrys . Moreover, the map β has a
set theoretical continuous section.

Recall that g is the Lie algebra of G. Then, DdR (g) := H 0 (F, g ⊗Qp BdR )+0 (g) := H 0 (F, g ⊗and DdR
Qp BdR ) have natural structures of nilpotent Liealgebras over F . We define the unipotent algebraic group DdR (G) (resp.
0 (G)) over F to be D (g)CH (resp. D 0 (g)CH ).
DdR
dR
dR
Lemma 4.5 ([20, Lemma 5.3]). Assume that g is a de Rham representationof GF . Then, we have a canonical isomorphism of pointed sets
∼

+
0
DdR
(G)(F )\ DdR (G)(F ) −→ H 0 (F, G(BdR)\G(BdR )).

Definition 14.
(1) We define the exponential part He1 (F, G(Qp )) of H 1 (F, G(Qp )) to be
the kernel of the canonical map of pointed sets H 1 (F, G(Qp )) →H 1 (F, G(Be )) induced by the homomorphism α in (4.1).
(2) Assume that Lie(G) = g is a de Rham representation of GF . Then,
0 (G)(F )\ D (G)(F ) →we define the exponential map exp0g : DdRdR
1
He (F, G(Qp )) to be the composite of the isomorphism of LemmaNON-ABELIAN BLOCH–KATO EXPONENTIAL MAP

61

4.5 and the connecting homomorphism of the fundamental exact sequence (4.1) in Lemma 4.4. We also denote exp0g by exp0G .
We can prove the following proposition by the induction on the nilpotencyof g.
Proposition 4.6 (cf. [20, Proposition 5.7]). Assume that g satisfies thefollowing conditions:(a) The GF -representation g = Lie(G) is a de Rham representation.
(b) The algebraic group Dcrys (G) := Dcrys (g)CH in MFad
F (ϕ) satisfies the1condition (Bij) .
Then, the following assertions hold.
(1) The exponential map exp0G is bijective.
(2) If g is crystalline, then He1 (F, G(Qp )) coincides with Hf1 (F, G(Qp )).
Now, we fix a Zp -basis ζp∞ = (ζpn )n≥0 of Zp (1) := limn µpn and let←−
Qp (r) := Zp (1)⊗r ⊗Zp Qp . We regard Qp (r) as an abelian Lie algebra object
0in Repcrys
Qp (GF ). Although it is hard to calculate expG in general, there
exists an explicit formula for exp0Q (r) due to Bloch and Kato. Recall thatpr
the one-dimensional F0 -vector space Dcrys (Qp (r)) is generated by ζp⊗r
∞ /t√where t ∈ Bcrys is the p-adic analog of “2π −1” attached to ζp∞ (cf. [8]).

Theorem 4.7 (The explicit reciprocity law, [2, Theorem 2.1 (4.8.2)]). Assume that F is unramified over Qp . Then, for each a ∈ F and for eachpositive integer r greater than 1, the following formula holds:
!
 


⊗r
ζ
∞
1
σ
p
F
exp0Qp (r) a r=
Tr.
1 − r a φCWF,rt(r − 1)! F/QppHere, σF ∈ Gal(F/Qp ) is the arithmetic Frobenius automorphism of F
∼ 1
and φCW
F,r ∈ HomGal(F (µp∞ )/F ) (Gal(F /F (µp∞ )), F (r)) = H (F, F (r)) is the
Coates–Wiles homomorphism.
We will use this theorem in Subsection 5.2.
4.3. End of the proof of Proposition 3.3. In this subsection, we assume
that G is an algebraic group in Repcrys
Qp (GF ) satisfying (Unip) and that H :=Dcrys (G) satisfies the conditions (Non-neg) and (Bij). We denote Dcrys (g)by h. Note that H is ωF -trivial by Lemma 3.1. Therefore, the generalized
Bloch–Kato logarithm logG for G is defined by Theorem 3.9 (2).
1This condition is slightly different from the condition used in [20]. However, the same

proof works.

62

K. SAKUGAWA

0 (G) over F is canonically
Lemma 4.8. The unipotent algebraic group DdR
isomorphic to F 0 HF .

Proof. We have the following canonical isomorphism:
H = Dcrys (G) ∼= Spec Sym(h∨ ) .

By definition, the ideal of Sym(h∨ ) ⊗F0 F = Sym(DdR (g∨ )) corresponding
to the closed subgroup scheme F 0 HF of HF is generated by F 1 DdR (g∨ ).
Thus, the ring of regular functions
on F 0 HF is canonically isomorphic to∨1∨
Sym DdR (g )/F (DdR (g )) . Hence, for each F -algebra R, we have canonical isomorphisms of sets
∨
CH
0
0
F 0 HF (R) ∼(g) ⊗F R = DdR(g) ⊗F R= DdR (g∨ )/F 1 (DdR (g∨ )) ⊗F R ∼= DdRfunctorial in R. Furthermore, the group structures of both hand sides coincide under the isomorphism above because they are subgroups of HF (R) =(DdR (g) ⊗F R)CH . This completes the proof of the lemma.
Proposition 4.9. Let G be an algebraic group in Repcrys
Qp (GF ) satisfying(Unip). Put H := Dcrys (G). If H satisfies the conditions (Non-neg) and(Bij), then logG defined in Definition 13 is a bijection and
0(G)(F )\ DdR (G)(F ) = F 0 HF (F )\H(F ) → Hf1 (F, G(Qp ))exp0G : DdR

is the inverse map of logG .

Proof. Since logG is injective, it is sufficient to show that the composite
logG ◦ exp0G is the identity map. Let us take h ∈ H(F ) and b ∈ G(Be ) whose+)\G(BdR ) coincides with the image of h. Then,
image under G(Be ) → G(BdR
0
by the definition of expG , the continuous 1-cocycle cb : GF → G(Qp ); σ 7→b σ (b−1 ) represents the cohomology class exp0G (h mod F 0 HF (F )). We put
X := Gcb (cf. Proof of Proposition 3.7). By definition, the action of GF on
X = Gcb is defined by g 7→ cb (σ) σ g = b σ (b−1 g). Therefore, the left multiplication b−1 on G ×Spec(Qp ) Spec(Be ) defines a GF -equivariant isomorphism∼

X ×Spec(Qp ) Spec(Be ) −→ G ×Spec(Qp ) Spec(Be ) of right G ×Spec(Qp ) Spec(Be )torsors. In other words, the left multiplication by b−1 induces the following
GF -equivariant Bcrys -algebra isomorphism:
(4.2)

∼

b−1,] : O(G) ⊗Qp Bcrys −→ O(X) ⊗Qp Bcrys .

Since b is Frobenius invariant, the isomorphism (4.2) is compatible with
Frobenius endomorphisms of both hand sides. Furthermore, because b co+
incides with h modulo G(BdR), the ring homomorphism b−1,] induces thenatural isomorphism∼

F i (O(Hh ) ⊗F BdR ) = h] F i (O(G) ⊗Qp BdR ) −→ F i (O(X) ⊗Qp BdR )NON-ABELIAN BLOCH–KATO EXPONENTIAL MAP

63

for each i. Therefore, by taking GF -invariant parts of (4.2), we have an
isomorphism∼
O(Hh ) −→ O(Dcrys (X))
in Ind(MFF (ϕ)). Hence, logG is the left inverse of exp0G that we want toprove.

Proof of Proposition 3.3 when H satisfies the condition (b). Let us supposethat H is an algebraic group in MFad
F (ϕ) satisfying (Non-neg), (Bij), and(Unip). Let Y be a right torsor under H in MFF (ϕ) which is (ωF0 , ωF )trivial. We will show that Y is an H-torsor in MFad
F (ϕ).
crys
Let G be an algebraic group in RepQp (GF ) such that Dcrys (G) = H. Since
G also satisfies (Unip), logG is a bijection (cf. Proposition 4.9). Therefore, by0
Proposition 3.8 and the definition of logG , there exists an (ωQp , Dcrys)-trivial
crys∼
right torsor X under G in RepQp (GF ) such that Dcrys (X) = Y as right Htorsors in MFF (ϕ). Since Dcrys is an equivalence between Repcrys
Qp (GF ) andadMFad
F (ϕ), Dcrys (X) is an affine scheme in MFF (ϕ). This implies that Y
is also an affine scheme in MFad
F (ϕ) and this completes the proof of the
proposition.
Proposition 4.9 states that expG defined in Definition 13 coincides with
exp0G when G satisfies (Unip). Hence we always identify these two exponential maps expG and exp0G for the rest of the present paper.
5. Path torsors under fundamental groups of P1 \{0, 1, ∞}In this section, we give an analog of the Coleman–Ihara formula which isa different version of [17, Theorem 1.1]. We fix a positive integer N greaterthan 1.
5.1. Polylogarithmic quotients. We denote by π1 the maximal pro-p→−
quotient of the étale fundamental group π1ét (P1F \{0, 1, ∞}, 01) of the algebraic curve P1F \{0, 1, ∞}. The topological group π1 has a natural continuous→−
action of GF defined by the base point 01. Let x, y ∈ π1 be the standard
generators of π1 (cf. [23, Sectoion 8, Picture 4]). It is a well-known fact thatcyc
the equality σ x = xχ (σ) holds for each σ ∈ GF where χcyc : GF → Z×p is
the p-adic cyclotomic character. Let p : π1 → Zp (1) be the GF -equivariant
group homomorphism defined by p(x) = ζp∞ , p(y) = 1. Then, we have acanonical GF -equivariant section s : Zp (1) → π1 ; ζp∞ 7→ x of p.

64

K. SAKUGAWA

Definition 15.
(1) We define the pro-p group π1pol to be the quotient π1 /[Ker p, Ker p]
where [Ker p, Ker p] is the closed subgroup of π1 generated by commutators of Ker p. We also denote by p the homomorphism π1pol →Zp (1) induced by p by abuse of notation.
−ét (→(2) We define the N -th p-adic étale polylogarithmic quotient PN01) topolpolN+1be the unipotent completion of π1 /Γ
π1 over Qp , where Γn π1pol
is the central descending series of π1pol defined by Γ1 π1pol := π1pol andby Γi+1 π1pol := [Γi π1pol , π1pol ].
→−−ét (→
Remark 8. By construction, PN01) is a quotient of π1un-ét (PF1 \{0, 1, ∞}, 01)→−
and the p-adic étale realization of the motivic fundamental group U (01)(N )in the sense of Deligne (cf. [6, (16.11.2)]).
We recall some fundamental properties of the p-adic étale polylogarithmic
quotient.
Lemma 5.1 ([6, Paragraphe 16.11, 16.12]).
(1) The section s induces a GF -equivariant isomorphism
π1pol ∼
= Zp (1) n Zp [[ Zp (1) ]](1).
Here, the action of Zp (1) on Zp [[ Zp (1) ]](1) is induced by translationsof Zp (1) on itself.
−ét (→(2) The Lie algebra of PN01) is canonically isomorphic to the Lie algeQN
bra Qp (1)n n=1 Qp (n). Here, the abelian Lie algebra Qp (1) acts on
Q
the abelian Lie algebra N
n=1 Qp (n) via the canonical homomorphism∼→ Qp (n + 1) for 1 ≤ n ≤ N − 1 and annihilates
Qp (1) ⊗Qp Qp (n) −Qp (N ).
−ét (→
According to Proposition 4.1, PN01) is an algebraic group in the catecrys
gory RepQp (GF ). We introduce the N -th crystalline polylogarithmic quotient.
Definition 16. For any positive integer N and for any finite extension F−
crys →of Qp , we define the crystalline N -th polylogarithmic quotient PN(01) to−ét (→
be Dcrys (PN01)).
−
crys →
By definition, PN(01) is an algebraic group in MFord
F (ϕ) satisfying(Unip). Furthermore, by Proposition 4.1, Lemma 4.2, and Lemma 4.3, we−crys →
see that PN(01) satisfies the conditions (Non-neg) and (Bij). Let k be
the residue field of F and let z be an element of P1 \{0, 1, ∞}(OF ). We

NON-ABELIAN BLOCH–KATO EXPONENTIAL MAP

65

→
−
→
−
denote by π1crys (P1k \{0, 1, ∞}; 01) and by π1crys (P1k \{0, 1, ∞}; z, 01) the crystalline fundamental group and the crystalline path torsor, respectively (cf.
[10, Definition 1.6]). It is well-known that there exist comparison isomorphisms of F -schemes→−→−
π1crys (P1k \{0, 1, ∞}; 01) ⊗F0 F ∼= π1dR (P1F \{0, 1, ∞}; 01)
and
→
−
→
−
π1crys (P1k \{0, 1, ∞}; z, 01) ⊗F0 F ∼= π1dR (P1F \{0, 1, ∞}; z, 01)→−(cf. [10, Lemma 2.2]). Therefore, π1crys (P1k \{0, 1, ∞}; 01) has a natural struc→−ture of an affine group scheme in MFF (ϕ) and π1crys (P1k \{0, 1, ∞}; z, 01)→−
has a natural structure of a right π1crys (P1k \{0, 1, ∞}; 01)-torsor in MFF (ϕ).
−
crys →(01) is a quotient
Meanwhile, the crystalline polylogarithmic quotient PN→−of π1crys (P1k \{0, 1, ∞}; 01) by the explicit descriptions of the Frobenius action (cf. [10, Lemma 2.9]) and the Hodge filtration (cf. [6, Paragraphe13.7]). For the symbol ? = un-ét or crys, we denote by u?N the canoni→−−? (→cal surjective homomorphism π1? (P1 \{0, 1, ∞}, 01)  PN
01). We define
→
−
→
−
?
?
the right PN (01)-torsor PN (z; 01) to be the pushforward of the path tor→−sor π1? (P1 \{0, 1, ∞}; z, 01) by u?N . According to Corollary 1, we have thefollowing proposition.
Proposition 5.2. Let M T ad
F (ϕ) be the minimal Tannakian subcategory ofad
MFF (ϕ) containing F0 hni for all n ∈ Z closed under extensions. Then, for−→−crys →
crys(01)-torsor
each point z ∈ P1 \{0, 1, ∞}(OF ), PN(z; 01) is a right PN
in M T ad
F (ϕ).
In the last of this subsection, we recall two unipotent Albanese maps
defined by Minhyong Kim.
Definition 17 (cf. [14, Introduction]).
(1) We define the étale Albanese map
− ∼ 1
−
1
1
ét →
ét →
AlbétF,N : P \{0, 1, ∞}(F ) → H (RepQp (GF ), PN (01)) = H (F, PN (01)(Qp ))→−étby Albét
F,N (z) := [PN (z; 01)].
(2) We define the crystalline-de Rham Albanese map− ∼ crys →−crys →
ad
1
1
Albcr-dR
F,N : P \{0, 1, ∞}(OF ) → H (M T F (ϕ), PN (01)) = PN (01)(F )

→
−
crys
by Albcr-dR
F,N (z) := [PN (z; 01)].

66

K. SAKUGAWA

5.2. Computations of Galois L-functions. Let K be a subfield2 of C.
In the paper [16], Nakamura and Wojtkowiak studied a measure valued
functionκz : GK → Meas(Zp , Zp (1))(1) = Zp [[ Zp (1) ]](1) ⊂ π1pol

for each z ∈ K (cf. [24, Proposition 2.3])3. In this subsection, we calculate
integrations of this measure when K is a finite extension F of Qp . By theconstruction of κz , we have the following equality:
Lemma 5.3. Let z be an element of P1 \{0, 1, ∞}(OF ). Then, for each−ét (→σ ∈ GF (ζp∞ ) , we have the following equation in PN
01)(Qp ) ∼= Qp (1) n
QN
→
−
ét
n=1 Qp (n) up to inner automorphisms of PN (01)(Qp ) :
(5.1)

(Z
)
!N 
1
κ0,z (σ);
Albétxn−1 dκz (σ)(x) ζp⊗n∞
F,N (z)(σ) =(n − 1)!
Zp
n=1

where κ0,z is the Kummer character associated to z.

Next, we recall that Albcr-dR
is calculated by using Coleman’s p-adic
F,Np-adicpolylogarithms Lin(z).
Theorem 5.4 (A special case of [10, Theorem 2.3]). For each OF -valued−
crys →
point z of P1 \{0, 1, ∞}, we have the following equality in PN(01)(F ) ∼
=
QN
ζp⊗n
∞
ζp∞
F t n n=1 F tn :


!
⊗n N
ζp ∞
∞
logp (z) ζp ; −Lip-adic
.
(z) n
(5.2)
Albcr-dRnF,N (z) =ttn=1Here, logp is the Iwasawa p-adic logarithm.

The following lemma is a direct consequence of Theorem 3.10.
Lemma 5.5. We have the following exact sequence of pointed sets:
(5.3)
pr
−
→
−
ét →
ét
1 → Hf1 (F, Qp (N )) → Hf1 (F, PN(01)(Qp )) −−N→ Hf1 (F, PN−1 (01)(Qp )) → 1.

−ét (→
The abelian group Hf1 (F, Qp (N )) acts on Hf1 (F, PN
01)(Qp )) whose action
is compatible with the generalized exponential maps. We denote the ac−ét (→
tion of c ∈ Hf1 (F, Qp (N )) on c0 ∈ Hf1 (F, PN
01)(Qp )) by c ∗ c0 . For all
2In the paper [16], they assumed that K is a number field. However, we can constructκz for an arbitrary z ∈ K ⊂ C by exactly the same way.
→
3More precisely, we need to fix an étale path γ from −01 to z and we should write κz,γ .

NON-ABELIAN BLOCH–KATO EXPONENTIAL MAP

67−ét (→
c1 , c2 ∈ Hf1 (F, PN
01)(Qp )) satisfying prN (c1 ) = prN (c2 ), there exists a
unique element c3 ∈ Hf1 (F, Qp (N )) such that c1 = c3 ∗ c2 .
0 (Q (n)) = 0 for
Proof. Note that F 0 (Pncrys ×Spec(F0 ) Spec(F )) = 1 and DdRp
any n ∈ Z>0 . Therefore, by Theorem 3.10, we have the following commutative diagram with exact rows:
(5.4)
−
−
/ P crys (→/ P crys (→01)(F )
DdR (Qp (N ))N
N −1 01)(F )→expP ét (−
∼
01)
=
N −1

→
expP ét (−
∼
01)
=
N

∼
= expQp (N )−
/ H 1 (F, P ét (→01)(Q ))



Hf1 (F, Qp (N )))

f

N

p



/ H 1 (F, P étf→−

N −1 (01)(Qp )).

It is clear that the first (resp. the last) map in the upper sequence is injective(resp. surjective). Thus, we obtain the short exact sequence (5.3). The restof assertions follow from the commutative diagram (5.4) and the compatibility of the actions of the abelian groups DdR (Qp (N )) and Hf1 (F, Qp (N ))) on−−
crys →ét (→(01)(F ) and Hf1 (F, PN
01)(Qp )), respectively (cf. Theorem 3.10). PN
The following proposition is one of the key propositions of [17].
Proposition 5.6 ([17, Proposition 5.1 (2)]). For each positive integer N
and for each unramified extension F of Qp , the composite of two maps−−∼
crys →
cr-dRét →− ◦AlbF,N : P1 \{0, 1, ∞}(OF ) → P
expP ét (→→ Hf1 (F, PN(01)(Qp ))N (01)(F ) −01)
Ncoincides with AlbétF,N .
For describing the integrations of the measure κz , we introduce a concept−ét (→of standard lifting of an element of Hf1 (F, PN−1 01)(Qp )).
Definition 18.

−ét (→(1) Let c be an element of the finite part Hf1 (F, PN−1 01)(Qp )) and−crys →→ (c) = (d0 , d1 , . . . , dN −1 ) ∈ P
write logP ét (−N −1 (01)(F ). Then, we
01)
N −1

→ ((d0 , d1 , . . . , dN −1 , 0))∈define the standard lifting of c to be expP ét (−
N 01)
→
−
ét (01)(Q )) and denote it by c̃.
Hf1 (F, PN
p1(2) For each z ∈ P \{0, 1, ∞}(OF ), we define the field Fe(z, N ) ⊃ F to
be the kernel field of the standard lifting of prN Albét
F,N (z) .

−ét (→→ (c) byLemma 5.7. Let c ∈ Hf1 (F, PN
01)(Qp )) and we denote logP ét (−01)^(d0 , . . . , dN ). Then, we have c = expQp (N ) (dN ) ∗ pr
N (c).

N

68

K. SAKUGAWA

Proof. It follows from Lemma 5.5 and the definition of the standard lifting.

Theorem 5.8. Let F be a finite unramified extension of Qp and let z be
an element of P1 \{0, 1, ∞}(OF ). Let N be a positive integer greater than 1.
Then, for each σ ∈ GFe(z,N ) , the following formula holds:
1(N − 1)!

Z

Zp

x

N −1

−1
dκz (σ)(x) =TrF/Qp(N − 1)!





σF
p-adicCW
1 − N LiN(z) φF,N (σ) .
p

Proof. According to Theorem 5.4 and Lemma 5.7, we have
!
⊗N
ζ
∞
p
p-adic^étAlbét(z) N∗ prN (AlbF,N (z) = expQp (N ) −LiN
F,N (z)).
t^étSince Fe(z, N ) is the kernel filed of prN (AlbF,N (z)), the formula
!
⊗N
ζ
∞
p
p-adic(σ)(z) NAlbétF,N (z)(σ) = expQp (N ) −LiNtholds for any σ ∈ GFe(z,N ) . Therefore, we see the conclusion of Theorem 5.8
by Theorem 4.7 and Lemma 5.3.
Corollary 2. Under the same setting as in Theorem 5.8, we have the following formula for each σ ∈ GFe(z,N ) :




σF
−1
p-adicTr(5.5) `iN (z)(σ) =1 − N LiN (z) φCWF,N (σ) .
(N − 1)! F/QppHere, `iN (z) is the N -th `-adic polylogarithm (cf. [16]).
Proof. In the paper [16], the following formula was proved:
Z
1
(5.6)
`iN (z)(σ) =xN −1 κz (σ)(x), for any σ ∈ GFe(z,N ) .
(N − 1)! Zp
Hence, by Theorem 5.8, we obtain the formula (5.5).

Acknowledgement. The author would like to thank Professor Hiroaki
Nakamura for valuable discussions and encouragement. He would also liketo thank Professor Seidai Yasuda for discussions about classifying spaces of
torsors in MFad
F (ϕ). He thank Doctor Sho Ogaki for careful reading of an
earlier version of the present manuscript. Finally, he would like to express
his gratitude to the referee for a careful reading and many helpful comments.

NON-ABELIAN BLOCH–KATO EXPONENTIAL MAP

69

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Kenji Sakugawa
Department of Mathematics
Graduate School of Science
Osaka University
Toyonaka, Osaka 560-0043 Japane-mail address: k-sakugawa@cr.math.sci.osaka-u.ac.jp(Received March 18, 2015 )(Accepted June 29, 2016 )