We prove the strong Suslin reciprocity law conjectured by A. Goncharov. The Suslin reciprocity law is a generalization of the Weil reciprocity law to higher Milnor $K$-theory. The Milnor $K$-groups can be identified with the top cohomology groups of the polylogarithmic motivic complexes; Goncharov's conjecture predicts the existence of a contracting homotopy underlying Suslin reciprocity. The main ingredient of the proof is a homotopy invariance theorem for the cohomology of the polylogarithmic motivic complexes in the ‘next to Milnor’ degree. We apply these results to the theory of scissors congruences of hyperbolic polytopes. For every triple of rational functions on a compact projective curve over $\mathbb {C}$ we construct a hyperbolic polytope (defined up to scissors congruence). The hyperbolic volume and the Dehn invariant of this polytope can be computed directly from the triple of rational functions on the curve.

In this paper, we consider certain double series analogous to Tornheim’s double series and real analytic Eisenstein series. By computing double integrals in two ways, we express the double series as a sum of products of polylogarithms. The technique generalises one given by Kanemitsu, Tanigawa and Yoshimoto. Evaluating the double series at particular points gives new evaluations for certain double series in terms of values of the Riemann zeta function and the dilogarithm which are analogues of formulas of Mordell and Goncharov.

In this article we show that the polylogarithmic currents introduced by Andrey Levin describe the polylogarithm of an abelian scheme at the topological level. From this result, which had been conjectured by Andrey Levin himself, we deduce a method to explicit the Eisenstein classes of abelian schemes at the topological level. These classes are of special interest since they have a motivic origin by a result of Guido Kings. In a forthcoming work entitled ‘Les classes d'Eisenstein des variétés de Hilbert–Blumenthal’, the results of the present article are used to prove that the Eisenstein classes of Hilbert–Blumenthal varieties degenerate at the boundary of the Baily–Borel compactification of the base in a special value of an L-function associated to the underlying totally real number field, and we get as a corollary a non-vanishing result for some of these Eisenstein classes in this geometric situation.

We study $l$-adic analogues of the classical polylogarithms. Using an explicit arithmetic formula we show that they satisfy the same distribution equations as classical polylogarithms. We also identify $l$-adic polylogarithms evaluated at roots of unity with Soulé classes.

AMS 2000 Mathematics subject classification: Primary 11G55; 11D88; 17B70

For an arbitrary field $F$ we study the double scissors congruence groups $A_n(F)$ generated by admissible pairs of simplices in the projective $n$-space ${\mathbb P}_F^n$ over $F$. Let $(L, M)$ be an admissible pair of simplices whose faces are defined as hyperplanes in ${\mathbb P}_F^n$ such that they do not have common faces of the same dimension. When $L$ and $M$ are in general positions we define a linearly constructible motivic perverse sheaf producing a motivic cohomology whose Betti realization is the relative cohomology $H^{\ast}({\mathbb P}_F^n\setminus L, M\setminus L; {\mathbb Q})$. The motivic cohomology provides a simple explanation of why the generic part of $A_\bullet$ forms a Hopf algebra with well-defined coproduct. When $n = 2$ we define explicitly the coproduct on $A_2$ which simplifies the approach of Beilinson et al. We also complete the same task for $A_3$ and $A_4$ which enables us to develop further results in another paper connecting Aomoto trilogarithms to the classical ones.

In this note we are studying the Lie algebras associated to non-abelian unipotent periods on $P^1_{\mathbb{Q}(\mu_n)}\setminus\{0,\mu_n,\infty\}$. Let $n$ be a prime number. We assume that for any $m\geq 1$ the numbers $Li_{m+1}(\xi_n^k)$ for $1\leq k\leq (n-1)/2$ are linearly independent over $\mathbb{Q}$ in $\mathbb{C}/(2\pi\ri)^{m+1}\mathbb{Q}$. Let $S=\{k_1,\cdots,k_q\}$ be a subset of $\{1,\dots,p-1\}$ such that if $k\in S$, then $p-k\in S$ and $(S+S)\cap S=\emptyset$ (the sum of two elements of $S$ is calculated $\mathrm{Mod}p$). Then we show that in the Lie algebra associated to non-abelian unipotent periods on $P^1_{\mathbb{Q}(\mu_n)}\setminus \{0,\mu_n,\infty\}$ there are derivations $D^{k_1}_{m+1},\dots,D^{k_q}_{m+1}$ in each degree $m+1$ and these derivations are free generators of a free Lie subalgebra of this Lie algebra.

AMS 2000 Mathematics subject classification: Primary 11G55

We investigate a connection between the differential of polylogarithms (as considered by Cathelineau) and a finite variant of them. This allows to answer a question raised by Kontsevich concerning the construction of functional equations for the finite analogs, using in part the p-adic version of polylogarithms and recent work of Besser. Kontsevich's original unpublished note is supplied (with his kind permission) in an ‘Appendix’ at the end of the paper.

The finite nth polylogarithm lin(z) ∈ Z/p(z) is defined as [sum ]k=1p−1zk/kn. We state and prove the following theorem. Let Lik: $ C$p → $ C$p be the p-adic polylogarithms defined by Coleman. Then a certain linear combination Fn of products of polylogarithms and logarithms, with coefficients which are independent of p, has the property that p1−nDFn(z) reduces modulo p>n+1 to lin−1(σ(z)), where D is the Cathelineau operator z(1−z)d/dz and σ is the inverse of the p-power map. A slightly modified version of this theorem was conjectured by Kontsevich. This theorem is used by Elbaz-Vincent and Gangl to deduce functional equations of finite polylogarithms from those of complex polylogarithms.

In the previous works of the first author, two completely different constructions of single valued Grassmannian trilogarithms were given. One of the constructions, in Math. Res. Lett. 2 (1995), 99–114, is very simple and provides Grassmannian n-logarithms for all n. However its motivic nature is hidden. The other construction in Adv. in Math. 114 (1995), 197–318, is very explicit and motivic, but the generalization for n>4 is not known. In this paper we will compute explicitly the Grassmannian trilogarithm constructed in Math. Res. Lett. 2 (1995), 99–114 and prove that it differs from the motivic Grassmannian trilogarithm by an explicitly given product of logarithms. We also derive some general results about the Grassmannian polylogarithms.

In this article we introduce a natural ‘elliptic’ generalization of the classical polylogarithms, study the properties of these functions and their relations with Eisenstein series.