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Recent developments in the theory of Anderson modules. (English) Zbl 1461.11089

Summary: Let \(K\) be a global function field over a finite field of characteristic \(p\) and let \(A\) be the ring of elements of \(K\) which are regular outside a fixed place of \(K\). This report presents recent developments in the arithmetic of special \(L\)-values of Anderson \(A\)-modules. Provided that \(p\) does not divide the class number of \(K\), we prove an “analytic class number formula” for Anderson \(A\)-modules with the help of a recent work of Debry. For tensor powers of the Carlitz module, we explain how to derive several log-algebraicity results from the class number formula for these Anderson modules.

MSC:

11G09 Drinfel’d modules; higher-dimensional motives, etc.
11M38 Zeta and \(L\)-functions in characteristic \(p\)
11R58 Arithmetic theory of algebraic function fields
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