Arithmetic of positive characteristic -series values in Tate algebras

The second author has recently introduced a new class of$L$-series in the arithmetic theory of function fields over finite fields. We show that the values at one of these$L$-series encode arithmetic information of a generalization of Drinfeld modules defined over Tate algebras that we introduce (the coefficients can be chosen in a Tate algebra). This enables us to generalize Anderson’s log-algebraicity theorem and an analogue of the Herbrand–Ribet theorem recently obtained by Taelman.