Multiple zeta value cycles in low weight

Abstract

Abstract. In a recent work, the author has constructed two families of algebraic cycles in Bloch's cycle algebra over P 1 \ {0, 1, ∞} that are expected to correspond to multiple polylogarithms in one variable and have a good specialization at 1 related to multiple zeta values. This is a short presentation, by the way of toy examples in low weight ( 5), of this construction and could serve as an introduction to the general setting. Working in low weight also makes it possible to push ("by hand") the construction further. In particular, we will not only detail the construction of the cycles but we will also associate to these cycles explicit elements in the bar construction over the cycle algebra and make as explicit as possible the "bottom-left" coefficient of the Hodge realization period matrix. That is, in a few relevant cases we will associated to each cycle an integral showing how the specialization at 1 is related to multiple zeta values. We will be particularly interested in a new weight 3 example corresponding to −ζ(2, 1).