On multiple polylogarithms in characteristic $p$: $v$-adic vanishing versus $\infty$-adic Eulerianness

TL;DR

This paper establishes a deep connection between the vanishing of $v$-adic Carlitz multiple polylogarithms at algebraic points and their $ abla$-adic Eulerian property, revealing interplay between different arithmetic worlds in positive characteristic.

Contribution

It proves a simultaneous vanishing principle linking $v$-adic and $ abla$-adic properties of CMPLs at algebraic points in positive characteristic.

Findings

$v$-adic vanishing of CMPLs implies $ abla$-adic Eulerian property.

The equivalence between $v$-adic vanishing and $ abla$-adic Eulerian status.

Reveals a nontrivial connection between $v$-adic and $ abla$-adic worlds.

Abstract

In this paper, we give a simultaneous vanishing principle for the $v$-adic Carlitz multiple polylogarithms (abbreviated as CMPLs) at algebraic points, where $v$ is a finite place of the rational function field over a finite field. This principle establishes the fact that the $v$-adic vanishing of CMPLs at algebraic points is equivalent to its $\infty$-adic counterpart being Eulerian. This reveals a nontrivial connection between the $v$-adic and $\infty$-adic worlds in positive characteristic.