On ℓ-adic Iterated Integrals V: Linear Independence, Properties of ℓ-adic Polylogarithms, ℓ-adic Sheaves

Abstract

In a series of papers we have introduced and studied ℓ-adic polylogarithms and ℓ-adic iterated integrals which are analogues of the classical complex polylogarithms and iterated integrals in ℓ-adic Galois realizations. In this note we shall show that in the generic case ℓ-adic iterated integrals are linearly independent over ${\mathbb{Q}}_{\mathcal{l}}$. 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 sequences of one forms $\frac{dz} {z}$ and $\frac{dz} {z-1}$ are linearly independent over $\mathbb{Q}$. We also study ramification properties of ℓ-adic polylogarithms and the minimal quotient subgroup of the absolute Galois group G _K of a number field K on which ℓ-adic polylogarithms are defined. In the final sections of the paper we study ℓ-adic sheaves and their relations with ℓ-adic polylogarithms. We show that if an ℓ-adic sheaf has the same monodromy representation as the classical complex polylogarithms then the action of G _K in stalks is given by ℓ-adic polylogarithms.