Multiple zeta value cycles in low weight

TL;DR

This paper explores algebraic cycles related to multiple polylogarithms and zeta values in low weight, providing explicit constructions and period computations to deepen understanding of their algebraic and analytic properties.

Contribution

It introduces explicit low-weight examples of algebraic cycles linked to multiple zeta values, detailing their construction and period computations.

Findings

Explicit low-weight algebraic cycles constructed

Associated integrals relate cycles to multiple zeta values

Detailed period matrix coefficients computed

Abstract

In a recent work, the author has constructed two families of algebraic cycles in Bloch cycle algebra over the prjective line minus 3 points 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 contruc- tion 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 cycle 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 "bottow-left" coefficient of the Hodge realization periods matrix. That is, in a few relevant cases we will associated to each cycles 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 .