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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
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References:

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