Hilbert modular forms and the Gross-Stark conjecture

Abstract

Let $F$ be a totally real field and $\chi$ an abelian totally odd character of $F$. In 1988, Gross stated a $p$-adic analogue of Stark’s conjecture that relates the value of the derivative of the $p$-adic $L$-function associated to $\chi$ and the $p$-adic logarithm of a $p$-unit in the extension of $F$ cut out by $\chi$. In this paper we prove Gross’s conjecture when $F$ is a real quadratic field and $\chi$ is a narrow ring class character. The main result also applies to general totally real fields for which Leopoldt’s conjecture holds, assuming that either there are at least two primes above $p$ in $F$, or that a certain condition relating the $\mathscr{L}$-invariants of $\chi$ and $\chi^{-1}$ holds. This condition on $\mathscr{L}$-invariants is always satisfied when $\chi$ is quadratic.

  • [coates-lichtenbaum] Go to document J. Coates and S. Lichtenbaum, "On $l$-adic zeta functions," Ann. of Math., vol. 98, pp. 498-550, 1973.
    @article {coates-lichtenbaum, MRKEY = {0330107},
      AUTHOR = {Coates, J. and Lichtenbaum, S.},
      TITLE = {On {$l$}-adic zeta functions},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {98},
      YEAR = {1973},
      PAGES = {498--550},
      ISSN = {0003-486X},
      MRCLASS = {12A70},
      MRNUMBER = {0330107},
      MRREVIEWER = {T. Kubota},
      DOI = {10.2307/1970916},
      ZBLNUMBER = {0279.12005},
      }
  • [colmez] Go to document P. Colmez, "Résidu en $s=1$ des fonctions zêta $p$-adiques," Invent. Math., vol. 91, iss. 2, pp. 371-389, 1988.
    @article {colmez, MRKEY = {0922806},
      AUTHOR = {Colmez, Pierre},
      TITLE = {Résidu en {$s=1$} des fonctions zêta {$p$}-adiques},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {91},
      YEAR = {1988},
      NUMBER = {2},
      PAGES = {371--389},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11R42 (11R80)},
      MRNUMBER = {0922806},
      MRREVIEWER = {Leslie Jane Federer},
      DOI = {10.1007/BF01389373},
      ZBLNUMBER = {0651.12010},
      }
  • [colmez-crelle] Go to document P. Colmez, "Fonctions zêta $p$-adiques en $s=0$," J. Reine Angew. Math., vol. 467, pp. 89-107, 1995.
    @article {colmez-crelle, MRKEY = {1355923},
      AUTHOR = {Colmez, Pierre},
      TITLE = {Fonctions zêta {$p$}-adiques en {$s=0$}},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {467},
      YEAR = {1995},
      PAGES = {89--107},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {11R42 (11F33 11F67 11G40 11S40 19F15)},
      MRNUMBER = {1355923},
      MRREVIEWER = {Alexey A. Panchishkin},
      DOI = {10.1515/crll.1995.467.89},
      ZBLNUMBER = {0864.11062},
      }
  • [dr] Go to document P. Deligne and K. A. Ribet, "Values of abelian $L$-functions at negative integers over totally real fields," Invent. Math., vol. 59, iss. 3, pp. 227-286, 1980.
    @article {dr, MRKEY = {0579702},
      AUTHOR = {Deligne, Pierre and Ribet, Kenneth A.},
      TITLE = {Values of abelian {$L$}-functions at negative integers over totally real fields},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {59},
      YEAR = {1980},
      NUMBER = {3},
      PAGES = {227--286},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {12A70 (10D21)},
      MRNUMBER = {0579702},
      MRREVIEWER = {Neal Koblitz},
      DOI = {10.1007/BF01453237},
      ZBLNUMBER = {0434.12009},
      }
  • [fg] Go to document B. Ferrero and R. Greenberg, "On the behavior of $p$-adic $L$-functions at $s=0$," Invent. Math., vol. 50, iss. 1, pp. 91-102, 1978/79.
    @article {fg, MRKEY = {0516606},
      AUTHOR = {Ferrero, Bruce and Greenberg, Ralph},
      TITLE = {On the behavior of {$p$}-adic {$L$}-functions at {$s=0$}},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {50},
      YEAR = {1978/79},
      NUMBER = {1},
      PAGES = {91--102},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {12B30},
      MRNUMBER = {0516606},
      MRREVIEWER = {Daniel Barsky},
      DOI = {10.1007/BF01406470},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {0441.12003},
      }
  • [greenberg-artin] Go to document R. Greenberg, "On $p$-adic Artin $L$-functions," Nagoya Math. J., vol. 89, pp. 77-87, 1983.
    @article {greenberg-artin, MRKEY = {0692344},
      AUTHOR = {Greenberg, Ralph},
      TITLE = {On {$p$}-adic {A}rtin {$L$}-functions},
      JOURNAL = {Nagoya Math. J.},
      FJOURNAL = {Nagoya Mathematical Journal},
      VOLUME = {89},
      YEAR = {1983},
      PAGES = {77--87},
      ISSN = {0027-7630},
      CODEN = {NGMJA2},
      MRCLASS = {11R42 (11R23)},
      MRNUMBER = {0692344},
      MRREVIEWER = {Leslie Jane Federer},
      URL = {http://projecteuclid.org/getRecord?id=euclid.nmj/1118787106},
      ZBLNUMBER = {0513.12012},
      }
  • [greenberg-bu] R. Greenberg, "Trivial zeros of $p$-adic $L$-functions," in $p$-adic Monodromy and the Birch and Swinnerton-Dyer Conjecture, Providence, RI, 1994, pp. 149-174.
    @inproceedings {greenberg-bu, MRKEY = {1279608},
      AUTHOR = {Greenberg, Ralph},
      TITLE = {Trivial zeros of {$p$}-adic {$L$}-functions},
      BOOKTITLE = {{$p$}-adic Monodromy and the {B}irch and {S}winnerton-{D}yer Conjecture},
      VENUE={{B}oston, {MA},
      1991},
      SERIES = {Contemp. Math.},
      VOLUME = {165},
      PAGES = {149--174},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1994},
      MRCLASS = {11G40 (11F67 11S40)},
      MRNUMBER = {1279608},
      MRREVIEWER = {K. Shiratani},
      ZBLNUMBER = {0838.11070},
      }
  • [greenberg-park] R. Greenberg, "Introduction to Iwasawa theory for elliptic curves," in Arithmetic Algebraic Geometry, Providence, RI, 2001, pp. 407-464.
    @inproceedings {greenberg-park,
      author = {Greenberg, Ralph},
      TITLE = {Introduction to {I}wasawa theory for elliptic curves},
      BOOKTITLE = {Arithmetic Algebraic Geometry},
      VENUE={{P}ark {C}ity, {UT},
      1999},
      SERIES = {IAS/Park City Math. Ser.},
      VOLUME = {9},
      PAGES = {407--464},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2001},
      MRCLASS = {11G05 (11G40 11R23 11R34)},
      MRNUMBER = {1860044},
      ZBLNUMBER={1002.11048},
      MRREVIEWER = {Massimo Bertolini},
      }
  • [gross] B. H. Gross, "$p$-adic $L$-series at $s=0$," J. Fac. Sci. Univ. Tokyo Sect. IA Math., vol. 28, iss. 3, pp. 979-994 (1982), 1981.
    @article {gross, MRKEY = {0656068},
      AUTHOR = {Gross, Benedict H.},
      TITLE = {{$p$}-adic {$L$}-series at {$s=0$}},
      JOURNAL = {J. Fac. Sci. Univ. Tokyo Sect. IA Math.},
      FJOURNAL = {Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics},
      VOLUME = {28},
      YEAR = {1981},
      NUMBER = {3},
      PAGES = {979--994 (1982)},
      ISSN = {0040-8980},
      CODEN = {JFTMAT},
      MRCLASS = {12B30},
      MRNUMBER = {0656068},
      MRREVIEWER = {Lawrence Washington},
      ZBLNUMBER = {0507.12010},
      }
  • [katz-cm] Go to document N. M. Katz, "$p$-adic $L$-functions for CM fields," Invent. Math., vol. 49, iss. 3, pp. 199-297, 1978.
    @article {katz-cm, MRKEY = {0513095},
      AUTHOR = {Katz, Nicholas M.},
      TITLE = {{$p$}-adic {$L$}-functions for {CM} fields},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {49},
      YEAR = {1978},
      NUMBER = {3},
      PAGES = {199--297},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {10D25 (12A65 12A67 14K22)},
      MRNUMBER = {0513095},
      MRREVIEWER = {V. V. Shokurov},
      DOI = {10.1007/BF01390187},
      ZBLNUMBER = {0417.12003},
      }
  • [miyake] T. Miyake, Modular Forms, New York: Springer-Verlag, 1989.
    @book {miyake, MRKEY = {1021004},
      AUTHOR = {Miyake, Toshitsune},
      TITLE = {Modular Forms},
      NOTE = {translated from the Japanese by Yoshitaka Maeda},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1989},
      PAGES = {x+335},
      ISBN = {3-540-50268-8},
      MRCLASS = {11F11 (11F25 11F72)},
      MRNUMBER = {1021004},
      MRREVIEWER = {Harvey Cohn},
      ZBLNUMBER = {0701.11014},
      }
  • [ribet] Go to document K. A. Ribet, "A modular construction of unramified $p$-extensions of $\mathbf{Q}(\mu_p)$," Invent. Math., vol. 34, iss. 3, pp. 151-162, 1976.
    @article {ribet, MRKEY = {0419403},
      AUTHOR = {Ribet, Kenneth A.},
      TITLE = {A modular construction of unramified {$p$}-extensions of {$\mathbf{Q}(\mu_p)$}},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {34},
      YEAR = {1976},
      NUMBER = {3},
      PAGES = {151--162},
      ISSN = {0020-9910},
      MRCLASS = {12A35 (10D05)},
      MRNUMBER = {0419403},
      MRREVIEWER = {V. V. Sokurov},
      DOI = {10.1007/BF01403065},
      ZBLNUMBER = {0338.12003},
      }
  • [shim] Go to document G. Shimura, "The special values of the zeta functions associated with Hilbert modular forms," Duke Math. J., vol. 45, iss. 3, pp. 637-679, 1978.
    @article {shim, MRKEY = {0507462},
      AUTHOR = {Shimura, Goro},
      TITLE = {The special values of the zeta functions associated with {H}ilbert modular forms},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {45},
      YEAR = {1978},
      NUMBER = {3},
      PAGES = {637--679},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {10D20 (10H10)},
      MRNUMBER = {0507462},
      MRREVIEWER = {Hiroshi Saito},
      URL = {http://projecteuclid.org/getRecord?id=euclid.dmj/1077312955},
      ZBLNUMBER = {0394.10015},
      }
  • [siegel] C. L. Siegel, "Über die Fourierschen Koeffizienten von Modulformen," Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, vol. 1970, pp. 15-56, 1970.
    @article {siegel, MRKEY = {0285488},
      AUTHOR = {Siegel, Carl Ludwig},
      TITLE = {Über die {F}ourierschen {K}oeffizienten von {M}odulformen},
      JOURNAL = {Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II},
      FJOURNAL = {Nachrichten der Akademie der Wissenschaften in Göttingen. II. Mathematisch-Physikalische Klasse},
      VOLUME = {1970},
      YEAR = {1970},
      PAGES = {15--56},
      ISSN = {0065-5295},
      MRCLASS = {10.20},
      MRNUMBER = {0285488},
      MRREVIEWER = {J. Spilker},
      ZBLNUMBER = {0225.10031},
      }
  • [tate-book] J. Tate, Les Conjectures de Stark sur les Fonctions $L$ d’Artin en $s=0$, Lecture notes edited by Dominique Bernardi and Norbert Schappacher, Boston, MA: Birkhäuser Boston Inc., 1984, vol. 47.
    @book {tate-book,
      author = {Tate, John},
      TITLE = {Les Conjectures de {S}tark sur les Fonctions {$L$} d'{A}rtin en {$s=0$},
      {\rm Lecture notes edited by Dominique Bernardi and Norbert Schappacher}},
      SERIES = {Progr. Math.},
      VOLUME = {47},
      PUBLISHER = {Birkhäuser Boston Inc.},
      ADDRESS = {Boston, MA},
      YEAR = {1984},
      PAGES = {143},
      ISBN = {0-8176-3188-7},
      MRCLASS = {11R42},
      MRNUMBER = {0782485},
      ZBLNUMBER={0545.12009},
      MRREVIEWER = {Leslie Jane Federer},
      }
  • [wiles-reps] Go to document A. Wiles, "On $p$-adic representations for totally real fields," Ann. of Math., vol. 123, iss. 3, pp. 407-456, 1986.
    @article {wiles-reps, MRKEY = {0840720},
      AUTHOR = {Wiles, A.},
      TITLE = {On {$p$}-adic representations for totally real fields},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {123},
      YEAR = {1986},
      NUMBER = {3},
      PAGES = {407--456},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11R23 (11F80 11R18)},
      MRNUMBER = {0840720},
      MRREVIEWER = {Jean-Fran{ç}ois Jaulent},
      DOI = {10.2307/1971332},
      ZBLNUMBER = {0613.12013},
      }
  • [wileslambda] Go to document A. Wiles, "On ordinary $\lambda$-adic representations associated to modular forms," Invent. Math., vol. 94, iss. 3, pp. 529-573, 1988.
    @article {wileslambda, MRKEY = {0969243},
      AUTHOR = {Wiles, A.},
      TITLE = {On ordinary {$\lambda$}-adic representations associated to modular forms},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {94},
      YEAR = {1988},
      NUMBER = {3},
      PAGES = {529--573},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11F41 (11F80 11R23 11R80)},
      MRNUMBER = {0969243},
      MRREVIEWER = {Sheldon Kamienny},
      DOI = {10.1007/BF01394275},
      ZBLNUMBER = {0664.10013},
      }
  • [wiles] Go to document A. Wiles, "The Iwasawa conjecture for totally real fields," Ann. of Math., vol. 131, iss. 3, pp. 493-540, 1990.
    @article {wiles, MRKEY = {1053488},
      AUTHOR = {Wiles, A.},
      TITLE = {The {I}wasawa conjecture for totally real fields},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {131},
      YEAR = {1990},
      NUMBER = {3},
      PAGES = {493--540},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11R42 (11F67 11R23)},
      MRNUMBER = {1053488},
      MRREVIEWER = {Alexey A. Panchishkin},
      DOI = {10.2307/1971468},
      ZBLNUMBER = {0719.11071},
      }

Authors

Samit Dasgupta

Mathematics Department
University of California Santa Cruz
Santa Cruz, CA 95064

Henri Darmon

The Department of Mathematics and Statistics
McGill University
805 Sherbrook Street West
Montreal, Quebec
Canada H3A 2K6

Robert Pollack

Department of Mathematics and Statistics
Boston University
111 Cummington Street
Boston, MA 02215