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Recent Developments in the Theory of Anderson Modules

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Abstract

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.

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Acknowledgments

We would like to thank both referees for carefully reading our manuscript and for giving helpful comments which helped improving the quality of the paper. Part of this work was done during the authors’ visit to Vietnam Institute for Advanced Study in Mathematics (VIASM) in June–August 2018. We are grateful to VIASM for its hospitality and great working conditions.

Funding

The second author (T. ND.) was partially supported by ANR Grant PerCoLaTor ANR-14-CE25-0002.

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Authors and Affiliations

  1. Laboratoire de Mathématiques Nicolas Oresme, CNRS UMR 6139, Université de Caen Normandie, Campus II, Boulevard Maréchal Juin, B.P. 5186, 14032, Caen Cedex, France

    Bruno Anglès & Floric Tavares Ribeiro

  2. Laboratoire de Mathématiques Nicolas Oresme, CNRS UMR 6139, CNRS and Université de Caen Normandie, Campus II, Boulevard Maréchal Juin, B.P. 5186, 14032, Caen Cedex, France

    Tuan Ngo Dac

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  1. Bruno Anglès
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Correspondence to Tuan Ngo Dac.

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Anglès, B., Dac, T.N. & Ribeiro, F.T. Recent Developments in the Theory of Anderson Modules. Acta Math Vietnam 45, 199–216 (2020). https://doi.org/10.1007/s40306-019-00348-z

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