Math. J. Okayama Univ. 66 (2024), 159–169

DUALITY-REFLECTION FORMULAS
OF MULTIPLE POLYLOGARITHMS
AND THEIR `-ADIC GALOIS ANALOGUES
Densuke Shiraishi
Abstract. In this paper, we derive formulas of complex and ℓ-adic
multiple polylogarithms, which have two aspects: a duality in terms ofindexes and a reﬂection in terms of variables. We provide an algebraic
proof of these formulas by using algebraic relations between associators
arising from the S3 -symmetry of the projective line minus three points.

1. Introduction and main results
The purpose of the present paper is to derive a series of functional equations that generalizes Oi-Ueno’s reflection formulas between complex multiple polylogarithms at z and 1 − z. This specializes to the duality formulafor multiple zeta values when z → 1. We also show the `-adic Galois analogof these equations by tracing the same argument in a parallel way to thecomplex case.
For= (k1 . . . , kd ) ∈ Nd and a topological path γ ∈ a multi-index k−→−→π1top P1 (C)\{0, 1, ∞}; 01, z from the standard tangential base point 01 toa C-rational base point z, the complex multiple polylogarithm Lik (z; γ) is
defined as an iterated integral along γ (see §2.1 for details). As is well known,Lik (z; γ) coincides with a certain signed coefficient of the KZ solution
G0 (X, Y )(z; γ) ∈ C⟨⟨X, Y ⟩⟩.
The multiple zeta value ζ(k) appears as its special value at the tangential−→ −→
−→base point 10 with the straight path δ ∈ π1top P1 (C)\{0, 1, ∞}; 01, 10 along
the unit interval (0, 1) ⊂ P1 (R)\{0, 1, ∞}. Our main result of the complex
case is then as follows.
Theorem 1.1 (The duality-reflection formula of complex multiple polylogarithms). Givena (possibly, tangential base) point z of P1 (C)\{0, 1, ∞} and−→a path γ ∈ π1top P1 (C)\{0, 1, ∞}; 01, z , deﬁne the path γ ′ associated to γby
−→(1.1)γ ′ := δ · φ(γ) ∈ π1top P1 (C)\{0, 1, ∞}; 01, 1 − z ,Mathematics Subject Classiﬁcation. Primary 11G55; Secondary 11F80, 14H30.
Key words and phrases. multiple polylogarithm, ℓ-adic Galois multiple polylogarithm,duality-reﬂection formula.
159

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D. SHIRAISHI

where φ ∈ Aut P1 (C)\{0, 1, ∞} is given by φ(t) = 1 − t and paths arecomposed from left to right. For any n, m ∈ Z≥2 , the following holds:
(1.2)
m−1
X (− log (z; γ))jj=0

j!

n−2X (− log (1 − z; γ ′ ))j
Li1,...,1,m−j (z; γ) +Li 1,...,1 ,n−j (1 − z; γ ′ )
j!
|{z}
|{z}
j=0n−2 times

m−2 times= ζ( 1, . . . , 1 , m),| {z }

where log(z; γ) :=Rn−2 timesdtδ −1 ·γ t is the logarithm function with respect to γ.

This functional equation has two aspects: a duality n ↔ m with respect toindexes and a reflection z ↔ 1 − z with respect to variables. We derive thefunctional equation from an algebraic relation (chain rule)−→
G0 (X, Y )(z; γ) = G0 (Y, X)(1 − z; γ ′ ) · G0 (X, Y )(10; δ),−→
where G0 (X, Y )(10; δ) is the so-called Drinfeld associator.
We also deal with the `-adic Galois case for any prime number `. Let K
be a subfield of C and GK := Gal(K/K) the absolute Galois group of K
with respect to its algebraic closure K. For a K-rational base point z of−→P1 \{0, 1, ∞}, consider each topologicalpath γ ∈ π1top
P1 (C)\{0, 1, ∞}; 01, z−→
as a pro-` étale path γ ∈ π1ℓ-ét P1K \{0, 1, ∞}; 01, z by the comparison map.
Then, for σ ∈ GK , the `-adic Galois multiple polylogarithm Liℓk (z; γ, σ) isdefined as a certain signed coefficient of the `-adic Galois associatorfz,γσ (X, Y ) ∈ Qℓ ⟨⟨X, Y ⟩⟩(see §2.2 for details). This `-adic multiple polylogarithm is an `-adic étaleavatar of Lik (z; γ) introduced by Wojtkowiak. The `-adic Galois multipleℓzeta value (or called the `-adic
 multiple Soulé element) ζ k (σ) is defined as−→its special value Liℓk 10; δ, σ . Our another main result is then as follows.
Theorem 1.2 (The duality-reflection formula of `-adic Galois multiple
polylogarithms). Given a K-rational (possibly, tangentialbase) point z of−→ 
top
11P \{0, 1, ∞} and γ ∈ π1P (C)\{0, 1, ∞}; 01, z , deﬁne the path γ ′ associated to γ as in (1.1). For any σ ∈ GK , the following holds:
(1.3)
m−1
X (ρz,γ (σ))jj=0j!

ℓLi1,...,1,m−j (z; γ, σ) +|{z}n−2 timesn−2X

j(ρ1−z,γ ′ (σ))
Liℓ1,...,1 ,n−j (1 − z; γ ′ , σ)
j!
|{z}
j=0
m−2 timesℓ= ζ1,...,1,m (σ),|{z}n−2 times

DUALITY-REFLECTION FORMULA

161kwhere ρz,γ : GK → Zℓ is the Kummer 1-cocycle deﬁned by σ(z 1/ℓ ) =kk
ρ (σ)ζℓkz,γ z 1/ℓ with respect to the `-th power roots {z 1/ℓ }k determined by γ.
By reinterpreting the proof of the complex functional equation (1.2) afterreplacing G0 (X, Y )(z; γ) by fz,γσ (X, Y ), we derive the `-adic functional equation (1.3) from the following chain rule between `-adic Galois associators′−→

1−z,γfz,γ(Y, X) · f10,δ
σ (X, Y ) = fσσ (X, Y )along the path composition (1.1) (cf. [N21, p.701, the key identity (∗)]).
Remark 1.3. The formula (1.2) is a generalization of the following functional equation (1.4) due to Oi and Ueno in [Oi09],[OU13]. The formula(1.3) is a generalization of the following functional equation (1.5) due toNakamura in [NS22],[N21].
(1.4)
m−1X (− log (z; γ))jj!

j=0

m−2 times(1.5)
m−1X (ρz,γ (σ))jj=0

Lim−j (z; γ) + Li 1,...,1 ,2 (1 − z; γ ′ ) = ζ(m),|{z}j!

Liℓm−j (z; γ, σ) + Liℓ1,...,1 ,2 (1 − z; γ ′ , σ) = ζ ℓm (σ) (σ ∈ GK ).
|{z}
m−2 times−→
Remark 1.4. By setting z = 10 (i.e. z → 1 along the real interval) in (1.2)
and (1.3), we obtain the well-known duality formula of multiple zeta values
and its `-adic Galois analog.
(1.6)

ζ( 1, . . . , 1 , n) = ζ( 1, . . . , 1 , m),| {z }| {z }
m−2 times(1.7)

n−2 timesζ ℓ1,...,1 ,n (σ) = ζ ℓ1,...,1 ,m (σ) (σ ∈ GK ).
|{z}|{z}m−2 times

n−2 times

Remark 1.5. In [F04], Furusho constructed the theory of the p-adic KZ
equation and studied the p-adic multiple polylogarithm, which is a p-adic
crystalline avatar of Lik (z; γ). Using the results in [F04], it is possible toobtain a p-adic analog of (1.2) in the same way as in the proof of (1.2).

Acknowledgement. The author would like to express deep gratitude to Professor Hiroaki Nakamura for his helpful advice and warm encouragement.
This work was supported by JSPS KAKENHI Grant Numbers JP20J11018.

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D. SHIRAISHI

2. Preliminaries
In this section, we review the basic properties of complex multiple polylogarithms and `-adic Galois multiple polylogarithms in preparation for proving
the main theorems (1.2) and (1.3).
For a (possibly, tangential base) point z on P1 \{0, 1, ∞}, we shall write−→ 
π1top P1 (C)\{0, 1, ∞}; 01, z
for the set of homotopy classes of piece-wise smooth topological paths on−→P1 (C)\{0, 1, ∞} from the tangential base point 01 to z, and−→−→ −→
π1top P1 (C)\{0, 1, ∞}, 01 := π1top P1 (C)\{0, 1, ∞}; 01, 01
for the topological fundamental group of P1 (C)\{0, 1, ∞} at the base point−→
01 with respect to the path composition γ1 · γ2 := γ1 γ2 , i.e. paths arecomposed from left to right. Let−→l0 , l1 ∈ π1top P1 (C)\{0, 1, ∞}, 01
be smooth loops circling counterclockwise around 0, 1 respectively, as FIGURE 1 shows. In FIGURE 1, the dashed line represents P1 (R)\{0, 1, ∞}and the upper 
half-plane is located 
above. Then, {l0 , l1 } is a free generating−→
top1P (C)\{0, 1, ∞}, 01 . Fix a smooth pathsystem of π1−→ (2.1)γ ∈ π1top P1 (C)\{0, 1, ∞}; 01, z .
Moreover, we denote by−→ −→
δ ∈ π1top P1 (C)\{0, 1, ∞}; 01, 10−→−→
the straight path on P1 (C)\{0, 1, ∞} from 01 to 10along the real interval
as FIGURE 1 shows. Let φ ∈ Aut P1 (C)\{0, 1, ∞} be the automorphismof P1 (C)\{0, 1, ∞} defined by φ(t) = 1 − t. Then, we shall define−→(2.2)γ ′ := δ · φ(γ) ∈ π1top P1 (C)\{0, 1, ∞}; 01, 1 − z .
2.1. Complex multiple polylogarithms. Let z be a C-rational (possibly,dtangential base) point on P1 \{0,k= 1, ∞}. For a pair (k1 . . . , kd ) ∈ N and−→a fixed path γ(= γz ) ∈ π1top P1 (C)\{0, 1, ∞}; 01, z , we shall define thecomplex logarithmZdtlog(z; γ) :=(2.3)
δ −1 ·γ t

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163

Figure 1. Topological paths on P1 (C)\{0, 1, ∞}
z•γδ

•

l00

•l11

•φ(γ)

1−z

and the complex multiple polylogarithm Lik (z; γ) as the iterated integral
along γ below:(R1(kd ̸= 1),
γ t Lik1 ,··· ,kd −1 (t; γt ) dt
R(2.4)
Lik (z; γ) :=1
γ 1−t Lik1 ,··· ,kd−1 (t; γt ) dt (kd = 1),
Zdt′(2.5),Li1 (z; γ) := − log(1 − z; γ ) =γ 1−twhichcontinued to the pointed universal covering space can be analytically−→
of P1 (C)\{0, 1, ∞}, 01 . In particular, we define the multiple zeta value
−→ (2.6)ζ(k) := Lik 10; δ ∈ R.
The complex multiple polylogarithm Lik (z; γ) is closely related to the KZ(Knizhnik-Zamolodchikov) equation. The KZ equation on P1 (C)\{0, 1, ∞}is the differential equationdX
YG(X, Y )(z) =+G(X, Y )(z)dzz
z−1where G(X, Y )(z) is an analytic (i.e. each of whose coefficient is analytic)
function with values in the non-commutative formal power series algebraC⟨⟨X, Y ⟩⟩. There exists a unique solution
 G0 (X, Y )(z; γ) ∈ C⟨⟨X, Y ⟩⟩ at−→toptached to γ ∈ π1
P1 (C)\{0, 1, ∞}; 01, z characterized by the asymptotic
behavior∞X1(2.7)
G0 (X, Y )(z; γ) ≈(X · log (z; γ))m (z → 0).
m!
m=0

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D. SHIRAISHI

Moreover, we define the Drinfeld associator
−→ Φ(X, Y ) := G0 (X, Y ) 10; δ ∈ C⟨⟨X, Y ⟩⟩.
Then the following chain rule holds (cf. [D90],[F04],[F14]):(2.8)

G0 (X, Y )(z; γ) = G0 (Y, X)(1 − z; γ ′ ) · Φ(X, Y ).

This algebraic relation reflects the path composition (2.2). Let M be the noncommutative free monoid generated by the non-commuting indeterminates
X, Y . Since G0 (X, Y )(z; γ) is group-like in C⟨⟨X, Y ⟩⟩, the expansion ofG0 (X, Y )(z; γ) looks likeX(2.9)
G0 (X, Y )(z; γ) = 1 +Coeffw (G0 (X, Y )(z; γ)) · ww∈M\{1}where {Coeffw (G0 (X, Y )(z; γ))}w∈M is a family of complex numbers. Forw(k) := X kd −1 Y · · · X k1 −1 Y ∈ M and j ∈ N, the following equalities hold(cf. [F04],[F14],[LM96]):(2.10)

Coeffw(k) (G0 (X, Y )(z; γ)) = (−1)d · Lik (z; γ),(2.11)

CoeffX j (G0 (X, Y )(z; γ)) =(2.12)

logj (z; γ),j!
logj (1 − z; γ ′ ).
CoeffX j (G0 (X, Y )(1 − z; γ ′ )) =j!

2.2. `-adic Galois multiple polylogarithms. Let ` be a prime number
and K a subfield of C with the algebraic closure K ⊂ C. Suppose that z
is a K-rational (possibly, tangential base) point on P1 \{0, 1, ∞}. Then the`-adic Galois (multiple) polylogarithm introdecud by Zdzislaw Wojtkowiak
in his series of papers [W0]-[W3] is defined as follows.
We shall write−→ 
π1ℓ-ét P1K \{0, 1, ∞}; 01, z
1 \{0, 1, ∞} from the Kfor the pro-`-finite set of pro-` étale paths on PK−→rational tangential base point 01 to z, and
−→−→ −→
1\{0, 1, ∞}, 01 := π1ℓ-ét P1K \{0, 1, ∞}; 01, 01
π1ℓ-ét PK
1 \{0, 1, ∞} with the base point
for the pro-` étale fundamental group of PK−→
01. By the canonical comparison map−→ −→ 
1π1top P1 (C)\{0, 1, ∞}; 01, ∗ → π1ℓ-ét PK(2.13)\{0, 1, ∞}; 01, ∗DUALITY-REFLECTION FORMULA

165

n−o→for ∗ ∈ 01, z, 1 − z , regard topological paths l0 , l1 , γ, γ ′ on P1 (C)\{0, 1, ∞}→
1 \{0, 1, ∞}, −as pro-` étale paths on P1K \{0, 1, ∞}. Then π1ℓ-ét PK
01 is thefree pro-` group of rank 2 with topologically generating system {l0 , l1 }.
We focus on the natural action of the absolute Galois group
GK := Gal(K/K)
−→on π1ℓ-ét P1K \{0, 1, ∞}; 01, z (cf. [NW99, (1.1)]). Since z is K-rational, thisGalois action is well-defined. For each σ ∈ GK , we define a pro-` étale loop−→
−1ℓ-ét1fz,γ:=γ·σ(γ)∈πP(2.14)\{0,1,∞},01.
σ1K
Consider the multiplicative Magnus embedding into the algebra of noncommutative formal power series
−→E : π1ℓ-ét P1K \{0, 1, ∞}, 01 ,→ Qℓ ⟨⟨X, Y ⟩⟩P∞ 1 ndefined by E(l0 ) = exp(X) :=n=0 n! X , E(l1 ) = exp(Y ). We get aformal power series(2.15)z,γfz,γσ (X, Y ) := E(fσ ) ∈ Qℓ ⟨⟨X, Y ⟩⟩−→
called the `-adic Galois associator associated to γ. If z = 10 and γ = δ, it
is called the `-adic Ihara associator in [F07, Definition 2.32]. By (2.2) and(2.14), the following algebraic relation (chain rule) holds:(2.16)′−→

1−z,γfz,γ(Y, X) · f10,δ
σ (X, Y ) = fσσ (X, Y ).

The power series fz,γσ (X, Y ) is an `-adic Galois analog of the KZ fundamental
solution G0 (X, Y )(z; γ) in (2.7), and the relation (2.16) is an `-adic Galois
analog of the chain rule (2.8) of KZ fundamental solutions. Since fz,γσ (X, Y )is group-like in Qℓ ⟨⟨X, Y ⟩⟩, the expansion of fz,γσ (X, Y ) looks likeX(2.17)
fz,γ(X,Y)=1+Coeffw (fz,γσσ (X, Y )) · w,w∈M\{1}where {Coeffw (fz,γσ (X, Y ))}w∈M is a family of `-adic numbers. For k =(k1 . . . , kd ) ∈ Nd and w(k) := X kd −1 Y · · · X k1 −1 Y , we shall define the `-adic
Galois multiple polylogarithm and the `-adic Galois multiple zeta value(2.18)(2.19)

Liℓk (z; γ, σ) := (−1)d · Coeffw(k) (fz,γσ (X, Y )),−→ζ ℓk (σ) := Liℓk 10; δ, σ .

As ζ ℓk (σ) is called the `-adic multiple Soulé element in [F07, Definition 2.32],
ζ ℓk (σ) is closely related to the Soulé character (cf. [F07, Examples 2.33],[NW99, REMARK 2]).

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D. SHIRAISHI
kLet ρz,γ (resp. ρ1−z,γ ′ ) : GK → Zℓ be the Kummer 1-cocycle of {z 1/ℓ }kk(resp. {(1 − z)1/ℓ }k ) determined by γ (resp. γ ′ ). For j ∈ N, the followingholds (cf. [NW99],[NW20],[NS22]):(−ρz,γ (σ))j,j!
(−ρ1−z,γ ′ (σ))j′
CoeffX j (f1−z,γ(X,Y))=(2.21).
σj!
The `-adic Galois multiple polylogarithm is similar to the complex multiple polylogarithm as the TABLE 1 shows.
CoeffX j (fz,γσ (X, Y )) =(2.20)

Table 1. Analogy between `-adic Galois side and complex sideℓ-adic Galois sidecomplex sidez : K-ratinal base point on P1 \{0, 1, ∞}z : C-rational base point on P1 \{0, 1, ∞}fz,γσ (X, Y ) ∈ Qℓ ⟨⟨X, Y ⟩⟩ (σ ∈ GK )G0 (X, Y )(z; γ) ∈ C⟨⟨X, Y ⟩⟩(−→ )Φ(X, Y ) = G0 (X, Y ) 10; δ ∈ C⟨⟨X, Y ⟩⟩−→f10,δ
(X, Y ) ∈ Qℓ ⟨⟨X, Y ⟩⟩σ→−′

1−z,γfz,γ(Y, X) · f10,δ
(X, Y )σ (X, Y ) = fσσ

G0 (X, Y )(z; γ) = G0 (Y, X)(1 − z; γ ′ ) · Φ(X, Y )Liℓk (z; γ, σ) ∈ Qℓ

Lik (z; γ) ∈ C

ζ kℓ : GK → Qℓ

ζ(k) ∈ R

Liℓ1 (z; γ, σ) = ρ1−z,γ ′ (σ)

Li1 (z; γ) = − log(1 − z, γ ′ )3. Proof of main results
In this section, we
1.1 and Theorem 1.2. We fix a topolog prove Theorem −→ top
1ical path γ ∈ π1
P (C)\{0, 1, ∞}; 01, z . All other symbols are the sameas in the previous sections.
Proof of Theorem 1.1, Theorem 1.2. Let n, m ∈ Z≥2 . The following computations are inspired by a remark given in the Appendix of Furusho’s lecturenote [F14, A.24] and an insight about the `-adic Oi-Ueno’s equation in Nakamura’s Oberwolfach Report [N21].
First, we show Theorem 1.1. Since G0 (X, Y )(z; γ) is a group-like element
in C⟨⟨X, Y ⟩⟩, the shuffle relation holds for {Coeffw (G0 (X, Y )(z; γ))}w∈M(cf. [Ree58]), i.e. for w, w′ ∈ M,
(3.1)

Coeffww′ = Coeffw · Coeffw′ .

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167

By the definition of the shuffle product,(3.2)
X j X m−j−1 Y n−1 = X(X j−1 X m−j−1Y n−1) + X(X j  X m−j−2Y n−1).

For w, w′ ∈ M, we set
Coeffw+w′ := Coeffw + Coeffw′ .
Then, we obtainm−1X

j(− log(z; γ))Li 1,...,1 ,m−j (z; γ)j!
|{z}j=0n−2 times

=m−1X




(−1)n+j−1 · CoeffX j G0 (Y, X)(z; γ) · CoeffX m−j−1 Y n−1 G0 (X, Y )(z; γ)

j=0(by (2.10), (2.11))
=m−1X



(−1)n+j−1 · CoeffX j X m−j−1 Y n−1 G0 (X, Y )(z; γ)(by (3.1))

j=0=(−1)n+m−2 · CoeffY (X m−1 Y n−2 ) G0 (X, Y )(z; γ)(by (3.2)).

−→ Using (2.6), (2.10), (2.12), log 10; δ = 0, (3.1) and (3.2), we have thefollowing equalities by making the same computations as above:n−2X(− log(1 − z; γ ′ ))j
Li 1,...,1 ,n−j (1 − z; γ ′ )j!
|{z}
j=0m−2 times= (−1)n+m−3 · CoeffX(Y m−1 X n−2 ) G0 (X, Y )(1 − z; γ ′ ) ,and−→ jm−1−
X − log 10; δ→−→ζ( 1, . . . , 1 , m) = Li 1,...,1 ,m 10; δ +Li 1,...,1 ,m−j 10; δ | {z }j!
|{z}|{z}j=1n−2 times

n−2 times

n−2 times

−→ 
=(−1)n+m−2 · CoeffY (X m−1 Y n−2 ) G0 (Y, X) 10; δ .
Combining these equalities and the following equalityCoeffY (X m−1 Y n−2 ) G0 (X, Y )(z; γ)= CoeffY (X m−1 Y n−2 ) G0 (Y, X)(1 − z; γ ′ )−→ + CoeffY (X m−1 Y n−2 ) G0 (X, Y ) 10; δ= CoeffX(Y m−1 X n−2 ) G0 (X, Y )(1 − z; γ ′ )(by (2.8))

168

D. SHIRAISHI

−→ + CoeffY (X m−1 Y n−2 ) G0 (X, Y ) 10; δ ,we get the desired equation (1.2). This completes the proof of Theorem 1.1.
Next, we show Theorem 1.2. Let σ ∈ GK . Since fz,γσ (X, Y ) is grouplike in Qℓ ⟨⟨X, Y ⟩⟩, the shuffle relation holds for {Coeffw (fz,γσ (X, Y ))}w∈M .
Using (2.18), (2.19), (2.20), (2.21), (3.1) and (3.2), we obtain the following
equalities by making the same computations as above:m−1X(ρz,γ (σ))j ℓ
Li 1,...,1 ,m−j (z; γ, σ)j!
|{z}
j=0n−2 times= (−1)n+m−2 · CoeffY (X m−1 Y n−2 ) fz,γσ (X, Y ) ,n−2X(ρ1−z,γ ′ (σ))j ℓ
Li 1,...,1 ,n−j (1 − z; γ ′ , σ)j!
|{z}
j=0m−2 times′= (−1)n+m−3 · CoeffX(Y m−1 X n−2 ) f1−z,γ(X, Y ) ,σ−→
ζ ℓ1,...,1 ,m (σ) = (−1)n+m−2 · CoeffY (X m−1 Y n−2 ) f10,δ(X,Y)).
σ|{z}n−2 times

Combining these equalities and the following equalityCoeffY (X m−1 Y n−2 ) fz,γσ (X, Y )−→′10,δ
m−1n−2= CoeffY (X m−1 Y n−2 ) f1−z,γ(Y,X)+Coefff(X,Y)
Y (XY ) σσ(by (2.16))
−→′= CoeffX(Y m−1 X n−2 ) f1−z,γ(X, Y ) + CoeffY (X m−1 Y n−2 ) f10,δσσ (X, Y ) ,we get the equation (1.3). This completes the proof of Theorem 1.2.

□References
V.G.Drinfeld, On quasitriangular quasi-Hopf algebras and on a group that isclosely connected with Ga(Q/Q) (Russian), Algebra i Analiz 2 (1990), 149–181;translation in Leningrad Math. J. 2 (1991), 829–860[F04] H. Furusho, p-adic multiple zeta values. I. p-adic multiple polylogarithms and thep-adic KZ equation. Invent. Math. 155 (2004), no. 2, 253–286.
[F07] H. Furusho, p-adic multiple zeta values II – tannakian interpretations.
Amer.J.Math, Vol 129, No 4, (2007),1105-1144.
[F14] H.Furusho, Knots and Grothendieck-Teichmüller group (in Japanese), Math-forindustry Lecture Note 68, 2014.
[LM96] T. T. Q. Le, J. Murakami. Kontsevich’s integral for the Kauﬀman polynomial.
Nagoya Math. J. 142 (1996), 39–65.
[NS22] H. Nakamura, D. Shiraishi, Landen’s trilogarithm functional equation and ℓadic Galois multiple polylogarithms. Preprint [31/10/2022 –] arXiv:2210.17182[math.NT].
[D90]

DUALITY-REFLECTION FORMULA

169[NW99] H. Nakamura, Z. Wojtkowiak. On explicit formulae for l-adic polylogarithms.
Arithmetic fundamental groups and noncommutative algebra (Berkeley, CA,1999), 285–294, Proc. Sympos. Pure Math., 70, Amer. Math. Soc., Providence,RI, 2002.
[NW20] H. Nakamura, Z. Wojtkowiak, On distribution formulas for complex and l-adicpolylogarithms. Periods in quantum ﬁeld theory and arithmetic, 593–619, Springer
Proc. Math. Stat., 314, Springer, 2020.
[N21] H. Nakamura, Some aspects of arithmetic functions in Grothendieck-Teichmüller
theory. Oberwolfach Rep. 18 (2021), no. 1, 700–702.
[Oi09] S. Oi, Gauss hypergeometric functions, multiple polylogarithms, and multiple zeta
values. Publ. Res. Inst. Math. Sci. 45 (2009), no. 4, 981–1009.
[OU13] S. Oi, K. Ueno, The inversion formula of polylogarithms and the Riemann-Hilbert
problem. Symmetries, integrable systems and representations, 491–496, Springer
Proc. Math. Stat., 40, Springer, Heidelberg, 2013
[Ree58] R. Ree, Lie elements and an algebra associated with shuﬄes. Ann. of Math. (2)
68 (1958), 210–220.
[W0]Z. Wojtkowiak, On l-adic polylogarithms. Prépublication n◦ 549, Université deNice-Sophia Antipolis, Juin 1999.
[W1]
Z.Wojtkowiak, On ℓ-adic iterated integrals, I – Analog of Zagier Conjecture,
Nagoya Math. J., 176 (2004), 113–158.
[W2]
Z.Wojtkowiak, On ℓ-adic iterated integrals, II – Functional equations and ℓ-adicpolylogarithms, Nagoya Math. J., 177 (2005), 117–153.
[W3]
Z.Wojtkowiak, On ℓ-adic iterated integrals, III – Galois actions on fundamentalgroups, Nagoya Math. J., 178 (2005), 1–36.
Densuke Shiraishi
Department of Mathematics
Graduate School of ScienceOsaka University
Toyonaka, Osaka 560-0043, Japane-mail address: densuke.shiraishi@gmail.com,u848765h@ecs.osaka-u.ac.jp(Received June 13, 2023 )(Accepted August 29, 2023 )