A <i>P</i> ‐adic class formula for Anderson <i>t</i> ‐modules

Abstract In 2012, Taelman proved a class formula for ‐series associated to Drinfeld ‐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 ‐modules. Let be a monic irreducible polynomial of , we define the ‐adic ‐series associated with Anderson ‐modules and prove a ‐adic class formula à la Taelman linking a ‐adic regulator, the class module and a local factor at . Next, we study the vanishing of the ‐adic ‐series and give some applications to Drinfeld modules defined over itself. Finally, we extend this result to the multi‐variable setting à la Pellarin.