A P-adic class formula for Anderson t-modules

In 2012, Taelman proved a class formula for $L$-series associated to Drinfeld $F_q\left[\theta \right]$-modules and considered it as a function field analogue of the Birch and Swinnerton-Dyer conjecture. Since then, Taelman's class formula has been generalized to the setting of Anderson $t$-modules. Let $P$ be a monic irreducible polynomial of $F_q\left[\theta \right]$, 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 study the vanishing of the $P$-adic $L$-series and give some applications to Drinfeld modules defined over $F_q\left[\theta \right]$ itself. Finally, we extend this result to the multi-variable setting à la Pellarin.