Special L-values and shtuka functions for Drinfeld modules on elliptic curves

  • Nathan Green
  • Matthew A. Papanikolas


We make a detailed account of sign-normalized rank 1 Drinfeld \(\mathbf {A}\)-modules, for \(\mathbf {A}\) the coordinate ring of an elliptic curve over a finite field, in order to provide a parallel theory to the Carlitz module for \(\mathbb {F}_q[t]\). Using precise formulas for the shtuka function for \(\mathbf {A}\), we obtain a product formula for the fundamental period of the Drinfeld module. Using the shtuka function we find identities for deformations of reciprocal sums and as a result prove special value formulas for Pellarin L-series in terms of an Anderson–Thakur function. We also give a new proof of a log-algebraicity theorem of Anderson.


Drinfeld modules Pellarin L-series Shtuka functions Reciprocal sums Anderson generating functions Log-algebraicity 

Mathematics Subject Classification

Primary 11G09 Secondary 11R58 11M38 12H10 


Author’s contributions

All contributions to this article were made jointly by NG and MAP. Both authors read and approved the final manuscript.


  1. 1.
    Anderson, G.W.: \(t\)-motives. Duke Math. J. 53(2), 457–502 (1986)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Anderson, G.W.: Rank one elliptic \(A\)-modules and \(A\)-harmonic series. Duke Math. J. 73(3), 491–542 (1994)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Anderson, G.W.: Log-algebraicity of twisted \(A\)-harmonic series and special values of \(L\)-series in characteristic \(p\). J. Number Theory 60(1), 165–209 (1996)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    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)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Anderson, G.W., Thakur, D.S.: Tensor powers of the Carlitz module and zeta values. Ann. Math. (2) 132(1), 159–191 (1990)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Anglès, B., Ngo Dac, T., Tavares Ribeiro, F.: Twisted characteristic \(p\) zeta functions. J. Number Theory 168, 180–214 (2016)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Anglès, B., Ngo Dac, T., Tavares Ribeiro, F.: Special functions and twisted \(L\)-series. J. Théor. Nombres Bordeaux 29(3), 931–961 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Anglès, B., Ngo Dac, T., Tavares Ribeiro, F.: Stark units in positive characteristic. Proc. Lond. Math. Soc. (3) 115(4), 763–812 (2017)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Anglès, B., Pellarin, F.: Functional identities for \(L\)-series in positive characteristic. J. Number Theory 142, 223–251 (2014)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Anglès, B., Pellarin, F.: Universal Gauss–Thakur sums and \(L\)-series. Invent. Math. 200(2), 653–669 (2015)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Anglès, B., Pellarin, F., Tavares Ribeiro, F.: Anderson–Stark units for \({\mathbb{F}}_q[\theta ]\). Trans. Am. Math. Soc. (2018) (to appear). arXiv:1501.06804
  12. 12.
    Anglès, B., Pellarin, F., Tavares Ribeiro, F.: Arithmetic of positive characteristic \(L\)-series values in Tate algebras. With an appendix by F. Demeslay. Compos. Math. 152(1), 1–61 (2016)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Anglès, B., Simon, D.: Power sums of polynomials over finite fields (2013) (in preparation) Google Scholar
  14. 14.
    Brownawell, W.D., Papanikolas, M.A.: Linear independence of Gamma values in positive characteristic. J. Reine Angew. Math. 549, 91–148 (2002)MathSciNetMATHGoogle Scholar
  15. 15.
    Carlitz, L.: On certain functions connected with polynomials in a Galois field. Duke Math. J. 1(2), 137–168 (1935)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Chang, C.-Y., Papanikolas, M.A.: Algebraic relations among periods and logarithms of rank 2 Drinfeld modules. Am. J. Math. 133(2), 359–391 (2011)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Chang, C.-Y., Papanikolas, M.A.: Algebraic independence of periods and logarithms of Drinfeld modules. With an appendix by B. Conrad. J. Am. Math. Soc. 25(1), 123–150 (2012)CrossRefMATHGoogle Scholar
  18. 18.
    Dummit, D.S., Hayes, D.: Rank-one Drinfeld modules on elliptic curves. Math. Comput. 62(206), 875–883 (1994)MathSciNetMATHGoogle Scholar
  19. 19.
    El-Guindy, A., Papanikolas, M.A.: Identities for Anderson generating functions for Drinfeld modules. Monatsh. Math. 173(4), 471–493 (2014)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Fresnel, J., van der Put, M.: Rigid Analytic Geometry and its Applications. Birkhäuser, Boston (2004)CrossRefMATHGoogle Scholar
  21. 21.
    Gekeler, E.-U.: Drinfeld Modular Curves, Lecture Notes in Math., vol. 1231. Springer, Berlin (1986)Google Scholar
  22. 22.
    Goss, D.: On a new type of \(L\)-function for algebraic curves over finite fields. Pac. J. Math. 105(1), 143–181 (1983)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Goss, D.: \(L\)-series of \(t\)-motives and Drinfeld modules. In: The Arithmetic of Function Fields (Columbus, OH, 1991), de Gruyter, Berlin, pp. 313–402 (1992)Google Scholar
  24. 24.
    Goss, D.: Basic Structures of Function Field Arithmetic. Springer, Berlin (1996)CrossRefMATHGoogle Scholar
  25. 25.
    Goss, D.: On the \(L\)-series of F. Pellarin. J. Number Theory 133(3), 955–962 (2013)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Hartl, U., Juschka, A.-K.: Pink’s theory of Hodge structures and the Hodge conjecture over function fields (2016). arXiv:1607.01412
  27. 27.
    Hayes, D.R.: Explicit class field theory in global function fields. In: Studies in algebra and number theory. Adv. Math. Suppl. Stud. vol. 6. Academic Press, New York, pp. 173–217 (1979)Google Scholar
  28. 28.
    Hayes, D.R.: A brief introduction to Drinfeld modules. In: The Arithmetic of Function Fields (Columbus, OH, 1991), W. de Gruyter, Berlin, pp. 1–32 (1992)Google Scholar
  29. 29.
    Lutes, B.A.: Special values of the Goss \(L\)-function and special polynomials. Ph.D. thesis, Texas A&M University (2010)Google Scholar
  30. 30.
    Lutes, B.A., Papanikolas, M.A.: Algebraic independence of values of Goss \(L\)-functions at \(s=1\). J. Number Theory 133(3), 1000–1011 (2013)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Mumford, D.: An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg deVries equation and related nonlinear equations. In: Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), Kinokuniya Book Store, Tokyo, pp. 115–153 (1978)Google Scholar
  32. 32.
    Papanikolas, M.A.: Tannakian duality for Anderson–Drinfeld motives and algebraic independence of Carlitz logarithms. Invent. Math. 171(1), 123–174 (2008)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Papanikolas, M.A.: Log-algebraicity on tensor powers of the Carlitz module and special values of Goss \(L\)-functions (in preparation) Google Scholar
  34. 34.
    Pellarin, F.: Aspects de l’indépendance algébrique en caractéristique non nulle, Sém. Bourbaki, vol. 2006/2007, Astérisque 317, no. 973, viii, pp. 205–242 (2008)Google Scholar
  35. 35.
    Pellarin, F.: Values of certain \(L\)-series in positive characteristic. Ann. Math. (2) 176(3), 2055–2093 (2012)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Pellarin, F., Perkins, R.B.: On certain generating functions in positive characteristic. Monatsh. Math. 180(1), 123–144 (2016)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Perkins, R.B.: Explicit formulae for \(L\)-values in finite characteristic. Math. Z. 278(1–2), 279–299 (2014)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Perkins, R.B.: On Pellarin’s \(L\)-series. Proc. Am. Math. Soc. 142(10), 3355–3368 (2014)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Silverman, J.H.: The Arithmetic of Elliptic Curves, 2nd edn. Springer, Dordrecht (2009)CrossRefMATHGoogle Scholar
  40. 40.
    Sinha, S.K.: Periods of \(t\)-motives and transcendence. Duke Math. J. 88(3), 465–535 (1997)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Taelman, L.: Special \(L\)-values of Drinfeld modules. Ann. Math. (2) 175(1), 369–391 (2012)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Thakur, D.S.: Gamma functions for function fields and Drinfeld modules. Ann. Math. (2) 134(1), 25–64 (1991)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Thakur, D.S.: Drinfeld modules and arithmetic in the function fields. Int. Math. Res. Not. 1992(9), 185–197 (1992)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Thakur, D.S.: Shtukas and Jacobi sums. Invent. Math. 111(3), 557–570 (1993)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Thakur, D.S.: Function Field Arithmetic. World Scientific Publishing, River Edge (2004)CrossRefMATHGoogle Scholar

Copyright information

© SpringerNature 2018

Authors and Affiliations

  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA

Personalised recommendations