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Taylor coefficients of Anderson–Thakur series and explicit formulae

Abstract

For each positive characteristic multiple zeta value (defined by Thakur in Function field arithmetic, World Scientific Publishing, River Edge, 2004), the first and third authors in Chang and Mishiba (Invent Math, 2020, https://doi.org/10.1007/s00222-020-00988-1) constructed a t-module together with an algebraic point such that a specified coordinate of the logarithmic vector of the algebraic point is a rational multiple of that multiple zeta value. The objective of this paper is to use the Taylor coefficients of Anderson–Thakur series and t-motivic Carlitz multiple star polylogarithms to give explicit formulae for all of the coordinates of this logarithmic vector.

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Acknowledgements

We are grateful to M. Papanikolas for sharing his manuscript [26] with us, and to J. Yu for his many helpful suggestions and comments. The first and third authors thank National Center for Theoretical Sciences and Kyushu University for their financial support and hospitality. Finally, we thank the referees for their helpful comments that improve the exposition of this paper.

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Correspondence to Chieh-Yu Chang.

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Communicated by Wei Zhang.

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Chieh-Yu Chang was partially supported by MOST Grant 107-2628-M-007-002-MY4 . Yoshinori Mishiba was supported by JSPS KAKENHI Grant Number JP18K13398. This project was partially supported by JSPS Bilateral Open Partnership Joint Research Projects.

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Chang, CY., Green, N. & Mishiba, Y. Taylor coefficients of Anderson–Thakur series and explicit formulae. Math. Ann. 379, 1425–1474 (2021). https://doi.org/10.1007/s00208-020-02103-4

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  • DOI: https://doi.org/10.1007/s00208-020-02103-4

Mathematics Subject Classification

  • Primary 11M38
  • Secondary 11G09