Additive polylogarithms and their functional equations

Abstract

Let k[ε] 2 := k[ε]/(ε 2 ). The single valued real analytic n-polylogarithm L n : C → R is fundamental in the study of weight n motivic cohomology over a field k, of characteristic 0. In this paper, we extend the construction in Ünver (Algebra Number Theory 3:1-34, 2009) to define additive n-polylogarithms li n :k[ε] 2 → k and prove that they satisfy functional equations analogous to those of L n . Under a mild hypothesis, we show that these functions descend to an analog of the nth Bloch group B n (k[ε] 2 ) defined by Goncharov (Adv Math 114:197-318, 1995). We hope that these functions will be useful in the study of weight n motivic cohomology over k[ε] 2 .

Tangent to Bloch-Suslin and Grassmannian Complexes over the dual numbers

“…After calculating all these values. Expand the sums (18) and (19) and put all values what we have calculated above. Let us talk about (18).…”

“…Expand the sums (18) and (19) and put all values what we have calculated above. Let us talk about (18). In this sum we have huge amount of terms, so we group them in a suitable way.…”

Motivic cohomology of fat points in Milnor range

“…The general integral cycle case is reduced to the geometrically integral case by constructing a Nisnevich cover. Some follow-up works will treat the cases of off-Milnor range of the relative Chow group of (k m , (t)), with a cycle-theoretic version of the regulator maps on the additive polylogarithmic complex constructed and studied in [30] and [31]. cf.…”