Special relations between multizeta values and parity results

TL;DR

This paper investigates algebraic relations among multizeta values over function fields, proving several conjectures including the parity conjecture, and elucidates how products of these values can be expressed as linear combinations within the algebra.

Contribution

The paper proves multiple conjectures related to multizeta values over function fields, notably establishing the parity conjecture and describing product relations as linear combinations.

Findings

Proved the parity conjecture for multizeta values.

Established that the F_p-span of multizeta values forms an algebra.

Demonstrated how products  can be expressed as linear combinations of multizeta values.

Abstract

We study relations between multizeta values for function fields introduced by D. Thakur. The F_p-span of Thakur's multizeta values is an algebra (Thakur. Shuffle relations for function field multizeta values). In particular, the product \zeta(a)\zeta(b) is a linear combination of multizeta values. In this paper, several of the conjectures formulated by the author and by D. Thakur for small values or for special families of a about how to write \zeta(a)\zeta(b) as an F_p-linear combination of multizeta values, are proved. Also, the parity conjecture formulated by Thakur is proved.