Euler reflexion formulas for motivic multiple zeta functions

TL;DR

This paper introduces a new algebraic product for motivic multiple zeta functions, establishes its properties, and formulates an Euler reflexion formula, connecting it to existing motivic theorems via arc space theory.

Contribution

It defines a novel $oxast$-product for motivic series, proves its associativity, and formulates an Euler reflexion formula for motivic zeta functions, linking to the motivic Thom-Sebastiani theorem.

Findings

The $oxast$-product preserves integrability and commutes with limits.

The motivic Euler reflexion formula is established using arc space theory.

Limit of the motivic Euler reflexion recovers the motivic Thom-Sebastiani theorem.

Abstract

We introduce a new notion of $\boxast$-product of two integrable series with coefficients in distinct Grothendieck rings of algebraic varieties, preserving the integrability and commuting with the limit of rational series. In the same context, we define a motivic multiple zeta function with respect to an ordered family of regular functions, which is integrable and connects closely to Denef-Loeser's motivic zeta functions. We also show that the $\boxast$-product is associative in the class of motivic multiple zeta functions. Furthermore, a version of the Euler reflexion formula for motivic zeta functions is nicely formulated to deal with the $\boxast$-product and motivic multiple zeta functions, and it is proved using the theory of arc spaces. As an application, taking the limit for the motivic Euler reflexion formula we recover the well known motivic Thom-Sebastiani theorem.