A P-adic class formula for Anderson t-modules

TL;DR

The paper develops a $P$-adic analogue of Taelman’s class formula for Anderson $t$-modules by constructing $P$-adic $L$-series that converge in a Tate algebra $\mathbb{T}_z(K_P)$ and by defining a $P$-adic regulator on unit modules. It extends the framework to a multi-variable Pellarin-type setting via $z$-deformations and evaluations at $z=\zeta$ for $\zeta\in\overline{\mathbb{F}}_q$, establishing $P$-adic class formulas that relate $L_P$ to regulators and class modules, and it analyzes non-vanishing/vanishing criteria at $z=1$. The results are generalized to the multi-variable context with $A_s$ and include integral-level considerations, together with a detailed analysis of the vanishing behavior and Leopoldt-type phenomena. The work yields new tools for studying $P$-adic special values of function-field $L$-series attached to Drinfeld and Anderson modules, with implications for period lattices, Stark-type units, and potential Leopoldt-type conjectures in positive characteristic.

Abstract

Let $P$ be a monic prime of $\mathbb F_q[θ]$, we define the $P$-adic $L$-series associated with Anderson $t$-modules and prove a $P$-adic class formula à la Taelman linking a $P$-adic regulator, the class module and a local factor at $P$. Next, we extend this result to the multi-variable setting à la Pellarin. Finally, we give some applications to Drinfeld modules defined over $\mathbb F_q[θ]$ itself.

Theorems & Definitions (103)

• Theorem A: Theorem \ref{['th:paspole']}
• Theorem B: Theorem \ref{['maincf']}
• Theorem C: Proposition \ref{['main3']} and Conjecture \ref{['leopoldt']}
• Theorem D: Theorem \ref{['main2']} and Theorem \ref{['main0']}
• Theorem E: Proposition \ref{['5']}
• Proposition 2.1
• Definition 2.2
• Proposition 2.3
• proof
• Proposition 3.1