Abstract
We state and prove a formula for a certain value of the Goss $L$-function of a Drinfeld module. This gives characteristic-$p$-valued function field analogues of the class number formula and of the Birch and Swinnerton-Dyer conjecture. The formula and its proof are presented in an entirely self-contained fashion.
-
[Anderson00]
G. W. Anderson, "An elementary approach to $L$-functions mod $p$," J. Number Theory, vol. 80, iss. 2, pp. 291-303, 2000.
@article {Anderson00, MRKEY = {1740516},
AUTHOR = {Anderson, Greg W.},
TITLE = {An elementary approach to {$L$}-functions mod {$p$}},
JOURNAL = {J. Number Theory},
FJOURNAL = {Journal of Number Theory},
VOLUME = {80},
YEAR = {2000},
NUMBER = {2},
PAGES = {291--303},
ISSN = {0022-314X},
CODEN = {JNUTA9},
MRCLASS = {11M38},
MRNUMBER = {1740516},
MRREVIEWER = {Yoshinori Hamahata},
DOI = {10.1006/jnth.1999.2452},
ZBLNUMBER = {0977.11036},
} -
@article {Anderson86, MRKEY = {0850546},
AUTHOR = {Anderson, Greg W.},
TITLE = {{$t$}-motives},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {53},
YEAR = {1986},
NUMBER = {2},
PAGES = {457--502},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {11F67 (11G05 11R58 14K05)},
MRNUMBER = {0850546},
MRREVIEWER = {David Goss},
DOI = {10.1215/S0012-7094-86-05328-7},
ZBLNUMBER = {0679.14001},
} -
[Anderson96]
G. W. Anderson, "Log-algebraicity of twisted $A$-harmonic series and special values of $L$-series in characteristic $p$," J. Number Theory, vol. 60, iss. 1, pp. 165-209, 1996.
@article {Anderson96, MRKEY = {1405732},
AUTHOR = {Anderson, Greg W.},
TITLE = {Log-algebraicity of twisted {$A$}-harmonic series and special values of {$L$}-series in characteristic {$p$}},
JOURNAL = {J. Number Theory},
FJOURNAL = {Journal of Number Theory},
VOLUME = {60},
YEAR = {1996},
NUMBER = {1},
PAGES = {165--209},
ISSN = {0022-314X},
CODEN = {JNUTA9},
MRCLASS = {11G09 (11G40 11R18 11R58)},
MRNUMBER = {1405732},
MRREVIEWER = {David Goss},
DOI = {10.1006/jnth.1996.0119},
ZBLNUMBER = {0868.11031},
} -
[Anderson90]
G. W. Anderson and D. S. Thakur, "Tensor powers of the Carlitz module and zeta values," Ann. of Math., vol. 132, iss. 1, pp. 159-191, 1990.
@article {Anderson90, MRKEY = {1059938},
AUTHOR = {Anderson, Greg W. and Thakur, Dinesh S.},
TITLE = {Tensor powers of the {C}arlitz module and zeta values},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {132},
YEAR = {1990},
NUMBER = {1},
PAGES = {159--191},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {11G09 (11F67 11J91 11R58)},
MRNUMBER = {1059938},
MRREVIEWER = {Jing Yu},
DOI = {10.2307/1971503},
ZBLNUMBER = {0713.11082},
} -
[Boeckle09]
G. Böckle and R. Pink, Cohomological Theory of Crystals over Function Fields, European Mathematical Society (EMS), Zürich, 2009, vol. 9.
@book {Boeckle09, MRKEY = {2561048},
AUTHOR = {B{ö}ckle, Gebhard and Pink, Richard},
TITLE = {Cohomological Theory of Crystals over Function Fields},
SERIES = {EMS Tracts in Math.},
VOLUME = {9},
PUBLISHER = {European Mathematical Society (EMS), Zürich},
YEAR = {2009},
PAGES = {viii+187},
ISBN = {978-3-03719-074-6},
MRCLASS = {14F05 (11G09 14F43 14G10 14G15)},
MRNUMBER = {2561048},
MRREVIEWER = {Martin C. Olsson},
DOI = {10.4171/074},
ZBLNUMBER = {1186.14002},
} -
[Carlitz35]
L. Carlitz, "On certain functions connected with polynomials in a Galois field," Duke Math. J., vol. 1, iss. 2, pp. 137-168, 1935.
@article {Carlitz35, MRKEY = {1545872},
AUTHOR = {Carlitz, Leonard},
TITLE = {On certain functions connected with polynomials in a {G}alois field},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {1},
YEAR = {1935},
NUMBER = {2},
PAGES = {137--168},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {Contributed Item},
MRNUMBER = {1545872},
DOI = {10.1215/S0012-7094-35-00114-4},
ZBLNUMBER = {0012.04904},
} -
[Chang10b] C. Chang and M. A. Papanikolas, Algebraic independence of periods and logarithms of Drinfeld modules, 2010.
@misc{Chang10b,
author={Chang, Chieh-Yu and Papanikolas, Matthew A.},
TITLE = {Algebraic independence of periods and logarithms of {D}rinfeld modules},
NOTE={preprint},
YEAR={2010},
ARXIV={1005.5120},
} -
[Chang10]
C. Chang and M. A. Papanikolas, "Algebraic relations among periods and logarithms of rank 2 Drinfeld modules," Amer. J. Math., vol. 133, iss. 2, pp. 359-391, 2011.
@article {Chang10, MRKEY = {2797350},
AUTHOR = {Chang, Chieh-Yu and Papanikolas, Matthew A.},
TITLE = {Algebraic relations among periods and logarithms of rank 2 {D}rinfeld modules},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {133},
YEAR = {2011},
NUMBER = {2},
PAGES = {359--391},
ISSN = {0002-9327},
CODEN = {AJMAAN},
MRCLASS = {11J93 (11G09 14G25)},
MRNUMBER = {2797350},
DOI = {10.1353/ajm.2011.0009},
ZBLNUMBER = {1216.11065},
} -
[Drinfeld74E] V. G. Drinfelcprimed, "Elliptic modules," Mat. Sb., vol. 94(136), pp. 594-627, 656, 1974.
@article {Drinfeld74E, MRKEY = {0384707},
AUTHOR = {Drinfel{\cprime}d, V. G.},
TITLE = {Elliptic modules},
JOURNAL = {Mat. Sb.},
VOLUME = {94(136)},
YEAR = {1974},
PAGES = {594--627, 656},
MRCLASS = {10D99 (14K15)},
MRNUMBER = {0384707},
MRREVIEWER = {Pierre Deligne},
ZBLNUMBER={0321.14014},
} -
[Gardeyn02]
F. Gardeyn, "A Galois criterion for good reduction of $\tau$-sheaves," J. Number Theory, vol. 97, iss. 2, pp. 447-471, 2002.
@article {Gardeyn02, MRKEY = {1942970},
AUTHOR = {Gardeyn, Francis},
TITLE = {A {G}alois criterion for good reduction of {$\tau$}-sheaves},
JOURNAL = {J. Number Theory},
FJOURNAL = {Journal of Number Theory},
VOLUME = {97},
YEAR = {2002},
NUMBER = {2},
PAGES = {447--471},
ISSN = {0022-314X},
CODEN = {JNUTA9},
MRCLASS = {11G09 (11M38)},
MRNUMBER = {1942970},
MRREVIEWER = {David Goss},
DOI = {10.1016/S0022-314X(02)00014-8},
ZBLNUMBER = {1053.11054},
} -
[Goss92] D. Goss, "$L$-series of $t$-motives and Drinfel\cprime d modules," in The Arithmetic of Function Fields, Berlin: de Gruyter, 1992, vol. 2, pp. 313-402.
@incollection {Goss92, MRKEY = {1196527},
AUTHOR = {Goss, David},
TITLE = {{$L$}-series of {$t$}-motives and {D}rinfel\cprime d modules},
BOOKTITLE = {The Arithmetic of Function Fields},
VENUE={{C}olumbus, {OH},
1991},
SERIES = {Ohio State Univ. Math. Res. Inst. Publ.},
VOLUME = {2},
PAGES = {313--402},
PUBLISHER = {de Gruyter},
ADDRESS = {Berlin},
YEAR = {1992},
MRCLASS = {11G09 (11G40)},
MRNUMBER = {1196527},
MRREVIEWER = {Yuichiro Taguchi},
ZBLNUMBER={0806.11028},
} -
[Igusa00] J. Igusa, An Introduction to the Theory of Local Zeta Functions, Providence, RI: Amer. Math. Soc., 2000, vol. 14.
@book {Igusa00, MRKEY = {1743467},
AUTHOR = {Igusa, Jun-ichi},
TITLE = {An Introduction to the Theory of Local Zeta Functions},
SERIES = {AMS/IP Stud. Adv. Mat.},
VOLUME = {14},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2000},
PAGES = {xii+232},
ISBN = {0-8218-2015-X},
MRCLASS = {11S40 (11G25 11M99)},
MRNUMBER = {1743467},
MRREVIEWER = {Willem Veys},
ZBLNUMBER = {0959.11047},
} -
[Lafforgue09]
V. Lafforgue, "Valeurs spéciales des fonctions $L$ en caractéristique $p$," J. Number Theory, vol. 129, iss. 10, pp. 2600-2634, 2009.
@article {Lafforgue09, MRKEY = {2541033},
AUTHOR = {Lafforgue, Vincent},
TITLE = {Valeurs spéciales des fonctions {$L$} en caractéristique {$p$}},
JOURNAL = {J. Number Theory},
FJOURNAL = {Journal of Number Theory},
VOLUME = {129},
YEAR = {2009},
NUMBER = {10},
PAGES = {2600--2634},
ISSN = {0022-314X},
CODEN = {JNUTA9},
MRCLASS = {11M38},
MRNUMBER = {2541033},
DOI = {10.1016/j.jnt.2009.04.002},
ZBLNUMBER = {1194.11089},
} -
[Papanikolas08]
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 {Papanikolas08, 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 = {05236753},
} -
[Poonen97]
B. Poonen, "Local height functions and the Mordell-Weil theorem for Drinfel\cprime d modules," Compositio Math., vol. 97, iss. 3, pp. 349-368, 1995.
@article {Poonen97, MRKEY = {1353279},
AUTHOR = {Poonen, Bjorn},
TITLE = {Local height functions and the {M}ordell-{W}eil theorem for {D}rinfel\cprime d modules},
JOURNAL = {Compositio Math.},
FJOURNAL = {Compositio Mathematica},
VOLUME = {97},
YEAR = {1995},
NUMBER = {3},
PAGES = {349--368},
ISSN = {0010-437X},
CODEN = {CMPMAF},
MRCLASS = {11G09},
MRNUMBER = {1353279},
MRREVIEWER = {David Goss},
URL = {http://www.numdam.org/item?id=CM_1995__97_3_349_0},
ZBLNUMBER={0839.11024},
} -
[SwinnertonDyer67] H. P. F. Swinnerton-Dyer, "An application of computing to class field theory," in Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 280-291.
@incollection {SwinnertonDyer67, MRKEY = {0219514},
AUTHOR = {Swinnerton-Dyer, H. P. F.},
TITLE = {An application of computing to class field theory},
BOOKTITLE = {Algebraic {N}umber {T}heory ({P}roc. {I}nstructional {C}onf., {B}righton, 1965)},
PAGES = {280--291},
PUBLISHER = {Thompson, Washington, D.C.},
YEAR = {1967},
MRCLASS = {10.68},
MRNUMBER = {0219514},
MRREVIEWER = {K. Masuda},
ZBLNUMBER={0153.07403},
} -
[Taelman10]
L. Taelman, "A Dirichlet unit theorem for Drinfeld modules," Math. Ann., vol. 348, iss. 4, pp. 899-907, 2010.
@article {Taelman10, MRKEY = {2721645},
AUTHOR = {Taelman, Lenny},
TITLE = {A {D}irichlet unit theorem for {D}rinfeld modules},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {348},
YEAR = {2010},
NUMBER = {4},
PAGES = {899--907},
ISSN = {0025-5831},
CODEN = {MAANA},
MRCLASS = {11G09},
MRNUMBER = {2721645},
MRREVIEWER = {Mihran Papikian},
DOI = {10.1007/s00208-010-0506-6},
ZBLNUMBER = {1217.11062},
} -
[Taelman11]
L. Taelman, "The Carlitz shtuka," J. Number Theory, vol. 131, iss. 3, pp. 410-418, 2011.
@article {Taelman11, MRKEY = {2739043},
AUTHOR = {Taelman, Lenny},
TITLE = {The {C}arlitz shtuka},
JOURNAL = {J. Number Theory},
FJOURNAL = {Journal of Number Theory},
VOLUME = {131},
YEAR = {2011},
NUMBER = {3},
PAGES = {410--418},
ISSN = {0022-314X},
CODEN = {JNUTA9},
MRCLASS = {11G09 (11M38 11R58 14F42)},
MRNUMBER = {2739043},
MRREVIEWER = {David Goss},
DOI = {10.1016/j.jnt.2010.09.004},
ZBLNUMBER = {1221.11137},
} -
[Taguchi97]
Y. Taguchi and D. Wan, "Entireness of $L$-functions of $\phi$-sheaves on affine complete intersections," J. Number Theory, vol. 63, iss. 1, pp. 170-179, 1997.
@article {Taguchi97, MRKEY = {1438656},
AUTHOR = {Taguchi, Yuichiro and Wan, Daqing},
TITLE = {Entireness of {$L$}-functions of {$\phi$}-sheaves on affine complete intersections},
JOURNAL = {J. Number Theory},
FJOURNAL = {Journal of Number Theory},
VOLUME = {63},
YEAR = {1997},
NUMBER = {1},
PAGES = {170--179},
ISSN = {0022-314X},
CODEN = {JNUTA9},
MRCLASS = {11G09 (14F30 14G20)},
MRNUMBER = {1438656},
MRREVIEWER = {Yoshinori Hamahata},
DOI = {10.1006/jnth.1997.2055},
ZBLNUMBER = {0908.11026},
} -
[Tate68]
J. Tate, "Residues of differentials on curves," Ann. Sci. École Norm. Sup., vol. 1, pp. 149-159, 1968.
@article {Tate68, MRKEY = {0227171},
AUTHOR = {Tate, John},
TITLE = {Residues of differentials on curves},
JOURNAL = {Ann. Sci. École Norm. Sup.},
FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
VOLUME = {1},
YEAR = {1968},
PAGES = {149--159},
ISSN = {0012-9593},
MRCLASS = {14.20},
MRNUMBER = {0227171},
MRREVIEWER = {R. L. Knighten},
URL = {http://www.numdam.org/item?id=ASENS_1968_4_1_1_149_0},
ZBLNUMBER = {0159.22702},
} -
[Tate84] J. Tate, Les Conjectures de Stark sur les Fonctions $L$ d’Artin en $s=0$, Boston, MA: Birkhäuser, 1984, vol. 47.
@book {Tate84, MRKEY = {0782485},
AUTHOR = {Tate, John},
TITLE = {Les Conjectures de {S}tark sur les Fonctions {$L$} d'{A}rtin en {$s=0$}},
SERIES = {Progr. Math.},
VOLUME = {47},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston, MA},
YEAR = {1984},
PAGES = {143},
ISBN = {0-8176-3188-7},
MRCLASS = {11R42},
MRNUMBER = {0782485},
MRREVIEWER = {Leslie Jane Federer},
ZBLNUMBER = {0545.12009},
}