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

Sophia Antipolis

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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.

• let


• arbitrary -adic iterated


• infinitesimal


• k¯ ?


• result concerns


• main results


• adic polylogarithms


• galois group


• independent over

Sujets

• Antipolis
• Infinitesimal
• Galois group

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• Publié par: pefav
• Nombre de lectures: 18
• Langue: English

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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- forms dz z quences of one z and zd −1 are linearly independent overQ. We also study ramification properties of`-adic polylogarithms and the minimal quo- tient subgroup of the absolute Galois groupG K of 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 ofG K 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. ¯ 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: [e-mail]

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Zdzisław Wojtkowiak

P 1 K ¯ \ {01∞} defined overK, and letγbe an`-adic path from 0 −→ 1 tozonP 1 K ¯ \ {01∞}.For any σ∈G K =Gal(K ¯ /K)we set f γ (σ):=γ −1 ∙σ(γ)∈π ´et 1 (P 1 K ¯ \ {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 (V K ¯ v) pro−` π 1 (V(C)v)→π 1e

any element ofπ 1 (V(C)v)determines canonically an element ofπ ´et 1 (V K ¯ v) pro−` , and we shall use the same notation for an element ofπ 1 (V(C)v)and its image. In particular, we have the comparison homomorphism π 1 (U0 −→ 1)→π 1 ´ 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 inP 1 K ¯ . Hence a loop − around 0 inC\ {0}determines canonically an element ofπ 1 ´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 fromvtozonV K ¯ . −→ Informally, we define`-adic iterated integrals from 01 tozas functions

l b (z) =l b (z) γ :G K →Q ` given by coefficients off γ ( )indexed by elementsbin a Hall basisBof the free Lie algebra Lie(XY)on two generatorsXandY. LetB n be the set of elements of degreeninB. Let H n ⊂G K(µ ` ∞ ) be the subgroup ofG K(µ ` ∞ ) defined by the condition that alll b (z)andl b (1 −→ 0)vanish onH n for allb∈ S i<n B i . 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 z p ∙(1−z) q =1, where p and q are integers such that p 2 +q 2 >0.Then the functions

l b (z):H n →Q ` for b∈B n are 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 ´te 1 (P 1 K ¯ \ {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: ´te 1 (P 1 ∞}01 π pro−` →Q ` {{XY}} be a continuous multiplicative embedding ofπ t 1 ´e (P 1 K ¯ \ {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 polylogarithmsl n (z)and the`-adic logarithml(z)are defined as func- tions onσ∈G K by the coefficients of the following expansion ∞ n− logE(f γ (σ)) =l(z)(σ)X+ ∑ l n (z)(σ)Y X 1 +. . .  n=1

where only relevent terms on the right hand side are written. The`-adic polylog- −→ arithmsl n (z)andl(z)depend on a choice of a pathγfrom 01 toz. If we want to indicate the dependence on a pathγwe shall writel n (z) γ andl(z) γ . The function l(z):G K →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 ofG K , on which the`-adic polylogarithmsl n (z)defined and their ramification properties. Forare z∈K\ {01} we consider the fieldsK(µ ` ∞ )andK(µ ` ∞ z 1/` ∞ ).Let M(K(µ ` ∞ z 1/` ∞ )) a ` b 1−z be the maximal pro-`abelian extension ofK(µ ` ∞ z 1/` ∞ )that is unramified outside` and 1−z.

Theorem 2.Assume that z∈K\ {01}root of any equation of the formis not a z p ∙(1−z) q =1, where p and q are integers such that p 2 +q 2 >0.Then we have:

(1) The`-adic polylogarithm l n (z):G K →Q ` factors through the group

Gal(M(K(µ ` ∞ z 1/` ∞ )) ` a  b 1−z /K). (2) The`-adic polylogarithm l n (z)ramifies only at prime factors of the fractional ideals