Chapter On adic iterated integrals V: linear independence properties of adic polylogarithms adic sheaves

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Chapter 14 On -adic iterated integrals V: linear independence, properties of -adic polylogarithms, -adic sheaves Zdzis?aw Wojtkowiak Abstract In a series of papers we have introduced and studied -adic polylogarithms and -adic iterated integrals which are analogues of the classical complex polyloga- rithms and iterated integrals in -adic Galois realizations. In this note we shall show that in the generic case -adic iterated integrals are linearly independent over Q. In particular they are non trivial. This result can be viewed as analogous of the statement that the classical iterated integrals from 0 to z of se- quences of one forms dzz and dz z?1 are linearly independent over Q. We also study ramification properties of -adic polylogarithms and the minimal quo- tient subgroup of the absolute Galois group GK of a number field K on which -adic polylogarithms are defined. In the final sections of the paper we study -adic sheaves and their relations with -adic polylogarithms. We show that if an -adic sheaf has the same monodromy representation as the classical complex polylogarithms then the action of GK in stalks is given by -adic polylogarithms. Key words: Galois group, polylogarithms, fundamental group 14.1 Introduction In this paper we study properties of -adic iterated integrals and -adic polyloga- rithms introduced in [Wo04] and [Wo05a]. We describe briefly the main results of the paper, though in the introduction we do not present them in full generality.

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Chapter 14 On`-adic iterated integrals V: linear independence, properties of`-adic polylogarithms,`-adic sheaves

Zdzisław Wojtkowiak

AbstractIn a series of papers we have introduced and studied`-adic polylogarithms and`-adic iterated integrals which are analogues of the classical complex polyloga-rithms and iterated integrals in`-adic Galois realizations. In this note we shall show that in the generic case`-adic iterated integrals are linearly independent overQ`. In particular they are non trivial. This result can be viewed as analogous of the statement that the classical iterated integrals from 0 tozof se-formsdz z quences of onezandzd−1are linearly independent overQ. We also study ramiﬁcation properties of`-adic polylogarithms and the minimal quo-tient subgroup of the absolute Galois groupGKof a number ﬁeldKon which`-adic polylogarithms are deﬁned. In the ﬁnal sections of the paper we study`-adic sheaves and their relations with`-adic polylogarithms. We show that if an`-adic sheaf has the same monodromy representation as the classical complex polylogarithms then the action ofGKin stalks is given by`-adic polylogarithms.

Key words:Galois group, polylogarithms, fundamental group

14.1 Introduction

In this paper we study properties of`-adic iterated integrals and`-adic polyloga-rithms introduced in [Wo04] and [Wo05a]. We describe brieﬂy the main results of the paper, though in the introduction we do not present them in full generality. ¯ LetKbe a number ﬁeld with algebraic closureK. Throughout this paper we ﬁx an embeddingK¯⊂C. Letz∈K\ {01}or letzbe a tangential point of

Zdzisław Wojtkowiak Universite´ de Nice-Sophia Antipolis, De´partement de Mathe´matiques, Laboratoire Jean Alexandre Dieudonne´, U.R.A. au C.N.R.S., No 168, Parc Valrose - B.P.N◦71, 06108 Nice Cedex 2, France e-mail: wojtkow@unice.fr

Zdzisław Wojtkowiak

P1K¯\ {01∞} deﬁned overK, and letγbe an`-adic path from 0−→1 tozonP1K¯\ {01∞}.For any σ∈GK=Gal(K¯/K)we set fγ(σ):=γ−1∙σ(γ)∈π´et1(P1K¯\ {01∞}0−→1)pro−`. Here and later our convention of composing a pathαfromytozwith a pathβfrom xtoywill be thatα∙βis deﬁned as a path fromxtoz. LetVbe an algebraic variety deﬁned overKand letvbe aK-point or a tangential point deﬁned overK. By the comparison homomorphism ´t(VK¯v)pro−` π1(V(C)v)→π1e

any element ofπ1(V(C)v)determines canonically an element ofπ´et1(VK¯v)pro−`, and we shall use the same notation for an element ofπ1(V(C)v)and its image. In particular, we have the comparison homomorphism π1(U0−→1)→π1´et(SpecK¯((z)0−→1)pro−` ) 

whereU⊂C\ {0}is a punctured inﬁnitesimal neighbourhood of 0 and SpecK¯((z)) is an algebraic inﬁnitesimal punctured neighbourhood of 0 inP1K¯. Hence a loop − around 0 inC\ {0}determines canonically an element ofπ1´te(SpecK¯((z))0→1). Sim-ilarly we have the comparison map from the torsor of paths fromvtozonV(C)to the torsor of`-adic paths fromvtozonVK¯. −→ Informally, we deﬁne`-adic iterated integrals from 01 tozas functions

lb(z) =lb(z)γ:GK→Q` given by coefﬁcients offγ( )indexed by elementsbin a Hall basisBof the free Lie algebra Lie(XY)on two generatorsXandY. LetBnbe the set of elements of degreeninB. Let Hn⊂GK(µ`∞) be the subgroup ofGK(µ`∞)deﬁned by the condition that alllb(z)andlb(1−→0)vanish onHnfor allb∈Si<nBi. Our ﬁrst result concerns linear independence of`-adic iterated integrals.

Theorem 1.Assume that z∈K\ {01}root of any equation of the formis not a zp∙(1−z)q=1, where p and q are integers such that p2+q2>0.Then the functions

lb(z):Hn→Q` for b∈Bnare linearly independent overQ`.

Our next results concerns`-adic polylogarithms. Hence we recall here their def-inition (see [Wo05a, Deﬁnition 11.0.1.]). Letxandybe the standard generators

14 On` 3-adic iterated integrals V π´te1(P1K¯\ {01−→)pro−`(see for example [Wo05a, Picture 1 on page 126]). of∞}01 LetQ`{{XY}}be theQ`-algebra of non-commutative formal power series in non-commutative variablesXandY. Let K¯\ {01−→) E:´te1(P1∞}01 πpro−`→Q`{{XY}} be a continuous multiplicative embedding ofπt1´e(P1K¯\ {01∞}0−→1)pro−`into the Q`-algebra of non-commutative formal power seriesQ`{{XY}}given by E(x) =exp(X)

E(y) =exp(Y). The`-adic polylogarithmsln(z)and the`-adic logarithml(z)are deﬁned as func-tions onσ∈GKby the coefﬁcients of the following expansion ∞ n− logE(fγ(σ)) =l(z)(σ)X+∑ln(z)(σ)Y X1+. . .  n=1

where only relevent terms on the right hand side are written. The`-adic polylog-−→ arithmsln(z)andl(z)depend on a choice of a pathγfrom 01 toz. If we want to indicate the dependence on a pathγwe shall writeln(z)γandl(z)γ. The function l(z):GK→Q` takes its values inZ`and agrees with the Kummer characterκ(z)associated toz (see [Wo05b, Proposition 14.1.0.]). Our second result concerns the minimal quotient ofGK, on which the`-adic polylogarithmsln(z)deﬁned and their ramiﬁcation properties. Forare z∈K\ {01} we consider the ﬁeldsK(µ`∞)andK(µ`∞z1/`∞).Let M(K(µ`∞z1/`∞))a`b1−z be the maximal pro-`abelian extension ofK(µ`∞z1/`∞)that is unramiﬁed outside` and 1−z.