A P-adic class formula for Anderson t-modules

Let $P$ be a monic prime of ${\mathbb{F}}_q[\theta ]$, 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 ${\mathbb{F}}_q[\theta ]$ itself.