On Andr\'e periods of mixed Tate motives

TL;DR

This paper demonstrates that André periods of motives simplify to classical notions for mixed Tate motives and links them with Coleman integration, providing concrete realizations of p-adic multiple polylogarithm values.

Contribution

It establishes the reduction of André periods to classical cases for mixed Tate motives and connects them with Coleman integration, offering explicit realizations.

Findings

André periods reduce to classical notions for mixed Tate motives.

Connection established between André periods and Coleman integration.

Special values of p-adic multiple polylogarithms are realized as André periods.

Abstract

In this note, we show that the $p$-adic periods of motives introduced recently by Ancona and Fr\u{a}\c{t}il\u{a} (``Andr\'e periods'') reduce to the classically studied notion in the case of Mixed Tate motives. We also connect Andr\'e periods with Coleman integration by observing that the Frobenius-fixed de Rham paths of Besser and Vologodsky come from motivic paths in characteristic $p$ (unconditionally in the mixed Tate setting, conditionally in general). We use this to realize special values of $p$-adic multiple polylogarithms as Andr\'e periods in a concrete way.