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Taylor coefficients of Anderson–Thakur series and explicit formulae


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, 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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  1. Anderson, G.W.: \(t\)-motives. Duke Math. J. 53(2), 457–502 (1986)

    MathSciNet  Article  Google Scholar 

  2. Anderson, G.W., Brownawell, W.D., Papanikolas, M.A.: Determination of the algebraic relations among special \(\Gamma \)-values in positive characteristic. Ann. Math. (2) 160(1), 237–313 (2004)

    MathSciNet  Article  Google Scholar 

  3. Anderson, G.W., Thakur, D.S.: Tensor powers of the Carlitz module and zeta values. Ann. Math. (2) 132(1), 159–191 (1990)

    MathSciNet  Article  Google Scholar 

  4. Anderson, G.W., Thakur, D.S.: Multizeta values for \({\mathbb{F}}_{q}[t]\), their period interpretation, and relations between them. Int. Math. Res. Not. 11, 2038–2055 (2009)

    MATH  Google Scholar 

  5. André, Y.: Une Introduction Aux Motifs (Motifs purs, Motifs Mixtes, Périodes). Panoramas et Synthéses, vol. 17. Société Mathématique de France, Paris (2004)

    MATH  Google Scholar 

  6. Anglès, B., Ngo Dac, T., Tavares Ribeiro, F.: Special functions and twisted L-series. J. Théor. Nombres Bordeaux 29(3), 931–961 (2017)

    MathSciNet  Article  Google Scholar 

  7. Baker, A., Wüstholz, G.: Logarithmic Forms and Diophantine Geometry, New Mathematical Monographs, vol. 9. Cambridge University Press, Cambridge (2007)

    MATH  Google Scholar 

  8. Brown, F.: Single-valued periods and multiple zeta values. Forum Math. Sigma 2(e25), 37 (2014).

    MathSciNet  Article  MATH  Google Scholar 

  9. Burgos Gil, J.I., Fresán, J.: Multiple zeta values: From numbers to motives. Clay Math. Proc

  10. Bosch, S., Güntzer, U., Remmert, R.: Non-Archimedean Analysis, Grundlehren der Mathematischen Wissenschaften, vol. 261. Springer, Berlin (1984)

    MATH  Google Scholar 

  11. Carlitz, L.: On certain functions connected with polynomials in a Galois field. Duke Math. J. 1(2), 137–168 (1935)

    MathSciNet  Article  Google Scholar 

  12. Chang, C.Y.: Linear independence of monomials of multizeta values in positive characteristic. Composit. Math. 150, 1789–1808 (2014)

    MathSciNet  Article  Google Scholar 

  13. Chang, C.Y., Mishiba, Y.: On multiple polylogarithms in characteristic \(p\): \(v\)-adic vanishing versus \(\infty \)-adic Eulerianness. Int. Math. Res. Not. 3, 923–947 (2019)

    MathSciNet  Article  Google Scholar 

  14. Chang, C.Y., Mishiba, Y.: On a conjecture of Furusho over function fields. Invent. Math. (2020).

    Article  MATH  Google Scholar 

  15. Chang, C.Y., Papanikolas, M.A., Yu, J.: An effective criterion for Eulerian multizeta values in positive characteristic. J. Eur. Math. Soc. (JEMS) 21(2), 405–440 (2019)

    MathSciNet  Article  Google Scholar 

  16. Chang, C.Y., Yu, J.: Determination of algebraic relations among special zeta values in positive characteristic. Adv. Math. 216(1), 321–345 (2007)

    MathSciNet  Article  Google Scholar 

  17. Chen, H.J.: On shuffle of double zeta values for \(\mathbb{F}_q[t]\). J. Number Theory 148, 153–163 (2015)

    MathSciNet  Article  Google Scholar 

  18. Green, N., Papanikolas, M.A.: Special \(L\)-values and shtuka functions for Drinfeld modules on elliptic curves. Res. Math. Sci. 5(1), 47 (2007)

    MathSciNet  MATH  Google Scholar 

  19. Green, N.: Tensor powers of rank 1 Drinfeld modules and periods. J. Number Theory (2019).

    Article  Google Scholar 

  20. Green, N.: Special zeta values using tensor powers of Drinfeld modules. Math. Res. Lett. 26(6), 1629–1676 (2019)

    MathSciNet  Article  Google Scholar 

  21. Hartl, U., Juschka, A.K.: Pink’s theory of Hodge structures and the Hodge conjecture over function fields. In: Böckle, G., Goss, D., Hartl, U., Papanikolas, M. (eds.) \(t\)-Motives: Hodge Structures, Transcendence and Other Motivic Aspects. EMS Congress Reports, European Mathematical Society, pp. 183–260 (2020)

  22. Maurischat, A.: Prolongations of \(t\)-motives and algebraic independence of periods. Doc. Math. 23, 815–838 (2018)

    MathSciNet  MATH  Google Scholar 

  23. Mishiba, Y.: On algebraic independence of certain multizeta values in characteristic \(p\). J. Number Theory 173, 512–528 (2017)

    MathSciNet  Article  Google Scholar 

  24. Namoijam, C., Papanikolas, M.A.: Hyperderivatives of periods and quasi-periods for Anderson t-modules (2020) (preprint)

  25. Papanikolas, M.A.: Tannakian duality for Anderson–Drinfeld motives and algebraic independence of Carlitz logarithms. Invent. Math. 171(1), 123–174 (2008)

    MathSciNet  Article  Google Scholar 

  26. Papanikolas, M.A.: Log-algebraicity on tensor powers of the Carlitz module and special values of Goss \(L\)-functions (in preparation)

  27. Thakur, D.S.: Shtukas and Jacobi sums. Invent. Math. 111(3), 557–570 (1993)

    MathSciNet  Article  Google Scholar 

  28. Thakur, D.S.: Function Field Arithmetic. World Scientific Publishing, River Edge (2004)

    Book  Google Scholar 

  29. Thakur, D.S.: Shuffle relations for function field multizeta values. Int. Math. Res. Not. 11, 1973–1980 (2010)

    MathSciNet  MATH  Google Scholar 

  30. Uchino, Y., Satoh, T.: Function field modular forms and higher-derivations. Math. Ann. 311(3), 439–466 (1998)

    MathSciNet  Article  Google Scholar 

  31. Wüstholz, G.: Algebraische Punkte auf analytischen Untergruppen algebraischer Gruppen [Algebraic points on analytic subgroups of algebraic groups] (German). Ann. Math. (2) 129(3), 501–517 (1989)

    MathSciNet  Article  Google Scholar 

  32. Yu, J.: Transcendence and special zeta values in characteristic \(p\). Ann. Math. (2) 134(1), 1–23 (1991)

    MathSciNet  Article  Google Scholar 

  33. Yu, J.: Analytic homomorphisms into Drinfeld modules. Ann. Math. (2) 145(2), 215–233 (1997)

    MathSciNet  Article  Google Scholar 

  34. Zhao, J.: Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values, Series on Number Theory and its Applications, vol. 12. World Scientific Publishing Co Pte Ltd., Hackensack (2016)

    Book  Google Scholar 

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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).

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Mathematics Subject Classification

  • Primary 11M38
  • Secondary 11G09