Values of certain $L$-series in positive characteristic

Abstract

We introduce a class of deformations of the values of the Goss zeta function. We prove, with the use of the theory of deformations of vectorial modular forms as well as with other techniques, a formula for their value at $1$, and some arithmetic properties of values at other positive integers. Our formulas involve Anderson and Thakur’s function $\omega$. We discuss how our formulas may be used to investigate the existence of a kind of functional equation for the Goss zeta function.

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      ISBN = {0-387-96508-4},
      MRCLASS = {11F03 (11G05 11G07 11G15 14G25)},
      MRNUMBER = {0890960},
      ZBLNUMBER = {0615.14018},
      }
  • [Lutes] B. A. Lutes and M. A. Papanikolas, Algebraic independence of Goss ${L}$-functions at $s=1$, 2011.
    @misc{Lutes,
      author={Lutes, B. A. and Papanikolas, M. A.},
      TITLE={Algebraic independence of {G}oss \hbox{${L}$-functions} at $s=1$},
      ARXIV={1105.6341},
      YEAR={2011},
     }
  • [Mas] Go to document G. Mason, "Vector-valued modular forms and linear differential operators," Int. J. Number Theory, vol. 3, iss. 3, pp. 377-390, 2007.
    @article {Mas, MRKEY = {2352826},
      AUTHOR = {Mason, Geoffrey},
      TITLE = {Vector-valued modular forms and linear differential operators},
      JOURNAL = {Int. J. Number Theory},
      FJOURNAL = {International Journal of Number Theory},
      VOLUME = {3},
      YEAR = {2007},
      NUMBER = {3},
      PAGES = {377--390},
      ISSN = {1793-0421},
      MRCLASS = {11F25 (11F11)},
      MRNUMBER = {2352826},
      MRREVIEWER = {Thomas R. Shemanske},
      DOI = {10.1142/S1793042107000973},
      ZBLNUMBER = {1197.11054},
      }
  • [Pa] Go to document M. A. Papanikolas, "Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms," Invent. Math., vol. 171, iss. 1, pp. 123-174, 2008.
    @article {Pa, MRKEY = {2358057},
      AUTHOR = {Papanikolas, Matthew A.},
      TITLE = {Tannakian duality for {A}nderson-{D}rinfeld motives and algebraic independence of {C}arlitz logarithms},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {171},
      YEAR = {2008},
      NUMBER = {1},
      PAGES = {123--174},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11J93 (11G09 12H10 14L17)},
      MRNUMBER = {2358057},
      MRREVIEWER = {Liang-Chung Hsia},
      DOI = {10.1007/s00222-007-0073-y},
      ZBLNUMBER = {1235.11074},
      }
  • [Bourbaki] F. Pellarin, "Aspects de l’indépendance algébrique en caractéristique non nulle (d’après Anderson, Brownawell, Denis, Papanikolas, Thakur, Yu, et al.)," in Séminaire Bourbaki. Vol. 2006/2007, , 2008, vol. 317, p. exp. no. 973, viii, 205-242.
    @incollection {Bourbaki, MRKEY = {2487735},
      AUTHOR = {Pellarin, Federico},
      TITLE = {Aspects de l'indépendance algébrique en caractéristique non nulle (d'après {A}nderson, {B}rownawell, {D}enis, {P}apanikolas, {T}hakur, {Y}u, et al.)},
      BOOKTITLE = {S{é}minaire Bourbaki. Vol. 2006/2007},
      SERIES = {Astérisque},
      FJOURNAL = {Astérisque},
      VOLUME = {317},
      YEAR = {2008},
      PAGES = {Exp. No. 973, viii, 205--242},
      ISSN = {0303-1179},
      ISBN = {978-2-85629-253-2},
      MRCLASS = {11J93},
      MRNUMBER = {2487735},
      MRREVIEWER = {Alain Lasjaunias},
      ZBLNUMBER = {1185.11048},
     }
  • [archiv] Go to document F. Pellarin, Estimating the order of vanishing at infinity of Drinfeld quasi-modular forms.
    @misc{archiv, SORTYEAR={2012},
      AUTHOR={Pellarin, Federico},
      TITLE={Estimating the order of vanishing at infinity of {D}rinfeld quasi-modular forms},
      NOTE={to appear in {\it J. Reine Angew. Math.}; published online 02-06-2012},
      DOI = {10.1515/crelle-2012-0048},
      }
  • [mahler] F. Pellarin, An introduction to Mahler’s method for transcendence and algebraic independence, 2010.
    @misc{mahler,
      author={Pellarin, Federico},
      TITLE={An introduction to {M}ahler's method for transcendence and algebraic independence},
      NOTE={to appear in the proceedings of the conference ``Hodge Structures, Transcendence and Other Motivic Aspects" Sept. 2009, Banff, Alberta, Canada (G.~Böckle, D. Goss, U. Hartl, and M. Papanikolas, eds.)},
      ARXIV={1005.1216},
      YEAR={2010},
     }
  • [Seg] Go to document G. Segal and G. Wilson, "Loop groups and equations of KdV type," Inst. Hautes Études Sci. Publ. Math., vol. 61, pp. 5-65, 1985.
    @article {Seg, MRKEY = {0783348},
      AUTHOR = {Segal, Graeme and Wilson, George},
      TITLE = {Loop groups and equations of {K}d{V} type},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      VOLUME = {61},
      YEAR = {1985},
      PAGES = {5--65},
      ISSN = {0073-8301},
      CODEN = {PMIHA6},
      MRCLASS = {58F07 (14K99 35Q20 58G35)},
      MRNUMBER = {0783348},
      MRREVIEWER = {A. M. Vinogradov},
      DOI = {10.1007/BF02698802},
      ZBLNUMBER = {0592.35112},
      }
  • [Tha] Go to document D. S. Thakur, Function Field Arithmetic, River Edge, NJ: World Scientific Publishing Co., 2004.
    @book {Tha, MRKEY = {2091265},
      AUTHOR = {Thakur, Dinesh S.},
      TITLE = {Function Field Arithmetic},
      PUBLISHER = {World Scientific Publishing Co.},
      ADDRESS = {River Edge, NJ},
      YEAR = {2004},
      PAGES = {xvi+388},
      ISBN = {981-238-839-7},
      MRCLASS = {11G09 (11J93 11M38 11R58)},
      MRNUMBER = {2091265},
      MRREVIEWER = {Mihran Papikian},
      DOI = {10.1142/9789812562388},
      ZBLNUMBER = {1061.11001},
      }

Authors

Federico Pellarin

Université de Lyon, Lyon, France

Current address:

Université Jean Monnet
Institut Camille Jordan
42023 Saint-Etienne
France