On Multiple Polylogarithms in Characteristic $p$: $v$-Adic Vanishing Versus $\infty$-Adic Eulerianness

Abstract

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 ∞-adic counterpart being Eulerian. This reveals a nontrivial connection between the v-adic and ∞-adic worlds in positive characteristic.

On multi-poly-Bernoulli–Carlitz numbers

“…Also, Chang-Mishiba and D. S. Thakur concerned v-adic variant ( [10], [19]) and finite variant ( [9], [20]). In this paper, we consider Chang and Mishiba's finite variant ( [9]).…”

Taylor coefficients of Anderson-Thakur series and explicit formulae

“…Note that Φ ′ is the square matrix of size r cut from the upper left square of Φ. Following [CM19a], for each 1 ≤ i ≤ r we put (2.3.6)…”

“…2.3.2 when replacing Q i by u i can be explicitly written down. We will use this explicit description of G heavily going forward, so we take a moment to recall it from [CM19a]. Let d i be given in (2.3.6) for 1 ≤ i ≤ r, and put…”

“…, u r ) ∈ K r with |u i | ∞ ≤ q s i q q−1 for each 1 ≤ i ≤ r − 1 and |u r | ∞ < q sr q q−1 . Following [CM19a], we explicitly construct a t-module G s,u and an algebraic point v s,u ∈ G s,u (K) in (3.3.3) and (3.3.4) using s and u, and note that the logarithm of the t-module G s,u , denoted by Log Gs,u , converges ∞-adically at v s,u . We then establish the following explicit formula for this logarithmic vector Log Gs,u (v s,u ) (stated as Theorem 3.3.7 via Proposition 3.2.3 using hyperderivatives).…”

Multizeta values for function fields: A survey

“…[7] formulas, he showed that the multizeta are K-linear combination of their values at tuples with integral co-ordinates. We refer to [12] for some general transcendence results on their values, and to [14], where he and Mishiba study a v-adic analog and show correspondence between the v-adic vanishing and the ∞-adic Eulerian nature, connecting to the torsion phenomena.…”