## Arithmetic of positive characteristic $$L$$-series values in Tate algebras.(English)Zbl 1336.11042

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 $${\mathbb C}_{ \infty}\to{\mathbb C}_{\infty}$$, $$x\mapsto x^q$$.
Among many other results, the authors prove the class number formula for the $$L$$-series value $$L(1,T)$$ (Theorem 5.11). Then they generalize 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 L. Taelman [Invent. Math. 188, No. 2, 253–275 (2012; Zbl 1278.11102)].

### MSC:

 11F52 Modular forms associated to Drinfel’d modules 14L05 Formal groups, $$p$$-divisible groups 11R60 Cyclotomic function fields (class groups, Bernoulli objects, etc.) 14G25 Global ground fields in algebraic geometry

### Citations:

Zbl 0868.11031; Zbl 1278.11102; Zbl 1336.11064
Full Text:

### References:

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