Arithmetic of positive characteristic \(L\)-series values in Tate algebras (Q2787619)

\textit{F. Pellarin} introduced in [Ann. Math. (2) 176, No. 3, 2055--2093 (2012; Zbl 1336.11064)] a new class of \(L\)-series over global function fields. In the paper under review, the authors show that the values at one of these \(L\)-series encode arithmetic information of a generalization of Drinfeld modules. The basic object is the following generalization of Drinfeld modules. Let \(k\) be the finite field of \(q\) elements and let \(A=k[T]\) be the ring of polynomials of one variable over \(k\). The Tate algebra \({\mathbb T}_s\) of dimension \(s\) is the completion of the polynomial algebra \({\mathbb C}_{\infty}[t_1,\ldots,t_s]\). A Drinfeld \(A[t_1,\ldots,t_s]\)-module \(\phi\) of rank \(r\) over \({\mathbb T}_s\) is an injective \(k[t_1,\ldots,t_s]\)-algebra homomorphism \(\phi: A[t_1,\ldots,t_s]\to \roman{End}_{k[t_1,\ldots,t_s]-{\text{lin}}}( {\mathbb T}_s)\) given by \(\phi_T=T+\alpha_1\tau+\cdots+\alpha_r\tau^r\) with \(\alpha_1,\ldots,\alpha_r\in{\mathbb T}_s\), \(\alpha_r\neq 0\) and \(\tau\) is the continuous extension of \(k[t_1,\ldots,t_s]\)--algebras \(\tau:{\mathbb T}_s\to{\mathbb T}_s\) of the homomorphism \({\mathbb C}_{ \infty}\to{\mathbb C}_{\infty}\), \(x\mapsto x^q\).NEWLINENEWLINEAmong many other results, the authors prove the class number formula for the \(L\)-series value \(L(1,T)\) (Theorem 5.11). Then they generalize \textit{G. W. Anderson}'s log algebraicity theorem [J. Number Theory 60, No. 1, 165--209 (1996; Zbl 0868.11031)] and show that the class formula implies Anderson's theorem for the Carlitz module. As another main result, the authors generalize an analogue of the Herbrand-Ribet theorem recently obtained by \textit{L. Taelman} [Invent. Math. 188, No. 2, 253--275 (2012; Zbl 1278.11102)].