On André periods of mixed Tate motives

TL;DR

This work shows that André periods, as defined by Ancona–Frăţilă, coincide with classical $p$-adic periods in the mixed Tate setting and then connects them to Coleman integration via Frobenius-fixed de Rham paths, which are shown to be motivic for MTM. By comparing the André-period torsor $ ext{O}H(Z)$ with the classical de Rham torsor $ ext{O}U_{ ext{dR}}(Z)$ in a Tannakian framework, the authors produce an isomorphism of $K$-algebras that preserves the two period maps. They further establish that Frobenius-fixed paths arising in the unipotent fundamental group are motivic in the mixed Tate world, and demonstrate explicitly that $p$-adic iterated integrals, including $p$-adic multiple polylogarithms, arise as André periods of motivic path torsors. This provides a concrete bridge between André periods and established $p$-adic period theories, with implications for evaluating special values via motivic methods beyond the mixed Tate setting.

Abstract

In this note, we show that the $p$-adic periods of motives introduced recently by Ancona and Frăţilă (``André periods'') reduce to the classically studied notion in the case of Mixed Tate motives. We also connect André 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 André periods in a concrete way.

Theorems & Definitions (15)

• Theorem 4.1
• Proposition 4.2
• proof
• Proposition 4.4
• proof
• Proposition 4.5
• proof
• Proposition 4.6
• proof
• proof : Proof of Theorem \ref{['A7']}