Abstract
For primes $p \ge 5$, we propose a conjecture that relates the values of cup products in the Galois cohomology of the maximal unramified outside $p$ extension of a cyclotomic field on cyclotomic $p$-units to the values of $p$-adic $L$-functions of cuspidal eigenforms that satisfy mod $p$ congruences with Eisenstein series. Passing up the cyclotomic and Hida towers, we construct an isomorphism of certain spaces that allows us to compare the value of a reciprocity map on a particular norm compatible system of $p$-units to what is essentially the two-variable $p$-adic $L$-function of Mazur and Kitagawa.
-
[as]
A. Ash and G. Stevens, "Modular forms in characteristic $l$ and special values of their $L$-functions," Duke Math. J., vol. 53, iss. 3, pp. 849-868, 1986.@article {as, MRKEY = {860675},
AUTHOR = {Ash, Avner and Stevens, Glenn},
TITLE = {Modular forms in characteristic {$l$} and special values of their {$L$}-functions},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {53},
YEAR = {1986},
NUMBER = {3},
PAGES = {849--868},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {11F75},
MRNUMBER = {860675},
DOI = {10.1215/S0012-7094-86-05346-9},
ZBLNUMBER = {0618.10026},
} -
[busuioc] C. Busuioc, "The Steinberg symbol and special values of $L$-functions," Trans. Amer. Math. Soc., vol. 360, iss. 11, pp. 5999-6015, 2008.
@article {busuioc, MRKEY = {2425699},
AUTHOR = {Busuioc, Cecilia},
TITLE = {The {S}teinberg symbol and special values of {$L$}-functions},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {360},
YEAR = {2008},
NUMBER = {11},
PAGES = {5999--6015},
ISSN = {0002-9947},
CODEN = {TAMTAM},
MRCLASS = {11F67 (19D45 19F15)},
MRNUMBER = {2425699},
MRREVIEWER = {Th{\cfac{o}}ng Nguy{\cftil{e}}n-Quang-{\Dbar}{\cftil{o}}},
DOI = {10.1090/S0002-9947-08-04701-6},
ZBLNUMBER = {05358274},
} -
[ds] F. Diamond and J. Shurman, A First Course in Modular Forms, New York: Springer-Verlag, 2005, vol. 228.
@book {ds, MRKEY = {2112196},
AUTHOR = {Diamond, Fred and Shurman, Jerry},
TITLE = {A {{F}}irst {{C}}ourse in {{M}}odular {{F}}orms},
SERIES = {Grad. Texts in Math.},
VOLUME= {228},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2005},
PAGES = {xvi+436},
ISBN = {0-387-23229-X},
MRCLASS = {11Fxx},
MRNUMBER = {2112196},
MRREVIEWER = {Henri Darmon},
ZBLNUMBER = {1062.11022},
} -
[fukaya] T. Fukaya, "Coleman power series for $K_2$ and $p$-adic zeta functions of modular forms," Doc. Math., pp. 387-442, 2003.
@article {fukaya, MRKEY = {2046604},
AUTHOR = {Fukaya, Takako},
TITLE = {Coleman power series for {$K\sb 2$} and {$p$}-adic zeta functions of modular forms},
NOTE = {extra volume.: Kazuya Kato's fiftieth birthday},
JOURNAL = {Doc. Math.},
FJOURNAL = {Documenta Mathematica},
YEAR = {2003},
PAGES = {387--442},
ISSN = {1431-0635},
MRCLASS = {11G40 (11F33 11F85 11G55 11R23 11R42 19F27)},
MRNUMBER = {2046604},
MRREVIEWER = {Alexey A. Panchishkin},
ZBLNUMBER = {1142.11338},
} -
[greenberg] R. Greenberg, "Iwasawa theory for motives," in $L$-Functions and Arithmetic, Cambridge: Cambridge Univ. Press, 1991, vol. 153, pp. 211-233.
@incollection {greenberg, MRKEY = {1110394},
AUTHOR = {Greenberg, Ralph},
TITLE = {Iwasawa theory for motives},
BOOKTITLE = {{$L$}-{{F}}unctions and {{A}}rithmetic},
VENUE={{D}urham, 1989)},
SERIES = {London Math. Soc. Lecture Note Ser.},
VOLUME= {153},
PAGES = {211--233},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1991},
MRCLASS = {11R23 (11G09 11G40 11R34)},
MRNUMBER = {1110394},
MRREVIEWER = {Bas Edixhoven},
DOI = {10.1017/CBO9780511526053.008},
ZBLNUMBER = {0727.11043},
} -
[gs] R. Greenberg and G. Stevens, "$p$-adic $L$-functions and $p$-adic periods of modular forms," Invent. Math., vol. 111, iss. 2, pp. 407-447, 1993.
@article {gs, MRKEY = {1198816},
AUTHOR = {Greenberg, Ralph and Stevens, Glenn},
TITLE = {{$p$}-adic {$L$}-functions and {$p$}-adic periods of modular forms},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {111},
YEAR = {1993},
NUMBER = {2},
PAGES = {407--447},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {11G40 (11F67)},
MRNUMBER = {1198816},
MRREVIEWER = {Nigel Boston},
DOI = {10.1007/BF01231294},
ZBLNUMBER = {0778.11034},
} -
[hida-iwa] H. Hida, "Iwasawa modules attached to congruences of cusp forms," Ann. Sci. École Norm. Sup., vol. 19, iss. 2, pp. 231-273, 1986.
@article {hida-iwa, MRKEY = {868300},
AUTHOR = {Hida, Haruzo},
TITLE = {Iwasawa modules attached to congruences of cusp forms},
JOURNAL = {Ann. Sci. École Norm. Sup.},
FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
VOLUME = {19},
YEAR = {1986},
NUMBER = {2},
PAGES = {231--273},
ISSN = {0012-9593},
CODEN = {ASENAH},
MRCLASS = {11F33 (11G18 11R23)},
MRNUMBER = {868300},
MRREVIEWER = {Yasutaka Ihara},
URL = {http://www.numdam.org/item?id=ASENS_1986_4_19_2_231_0},
ZBLNUMBER = {0607.10022},
} -
[hida] H. Hida, "Galois representations into ${ GL}_2({\bf Z}_p[[X]])$ attached to ordinary cusp forms," Invent. Math., vol. 85, iss. 3, pp. 545-613, 1986.
@article {hida, MRKEY = {848685},
AUTHOR = {Hida, Haruzo},
TITLE = {Galois representations into {${\rm GL}\sb 2({\bf Z}\sb p[[X]])$} attached to ordinary cusp forms},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {85},
YEAR = {1986},
NUMBER = {3},
PAGES = {545--613},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {11F11 (11F33 11F85 11R23)},
MRNUMBER = {848685},
MRREVIEWER = {Ernst-Wilhelm Zink},
DOI = {10.1007/BF01390329},
ZBLNUMBER = {0612.10021},
} -
[iwasawa] K. Iwasawa, "On some modules in the theory of cyclotomic fields," J. Math. Soc. Japan, vol. 16, pp. 42-82, 1964.
@article {iwasawa, MRKEY = {0215811},
AUTHOR = {Iwasawa, Kenkichi},
TITLE = {On some modules in the theory of cyclotomic fields},
JOURNAL = {J. Math. Soc. Japan},
FJOURNAL = {Journal of the Mathematical Society of Japan},
VOLUME = {16},
YEAR = {1964},
PAGES = {42--82},
ISSN = {0025-5645},
MRCLASS = {10.66},
MRNUMBER = {35 \#6646},
MRREVIEWER = {K. Masuda},
DOI = {10.2969/jmsj/01610042},
ZBLNUMBER = {0125.29207},
} -
[kato] K. Kato, "$p$-adic Hodge theory and values of zeta functions of modular forms," in Cohomologies $p$-Adiques et Applications Arithmétiques. III, , 2004, vol. 295, p. ix, 117-290.
@incollection {kato, MRKEY = {2104361},
AUTHOR = {Kato, Kazuya},
TITLE = {{$p$}-adic {H}odge theory and values of zeta functions of modular forms},
BOOKTITLE= {Cohomologies $p$-{{A}}diques et {{A}}pplications {{A}}rithm{é}tiques. {{{\rm III}}}},
SERIES = {Astérisque},
FJOURNAL = {Astérisque},
VOLUME= {295},
YEAR = {2004},
PAGES = {ix, 117--290},
ISSN = {0303-1179},
MRCLASS = {11F85 (11F67 11G40 11R33 11S80 14G10 14G35)},
MRNUMBER = {2104361},
MRREVIEWER = {Fabrizio Andreatta},
ZBLNUMBER={1142.11336},
} -
[kitagawa] K. Kitagawa, "On standard $p$-adic $L$-functions of families of elliptic cusp forms," in $p$-Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture, Providence, RI: Amer. Math. Soc., 1994, vol. 165, pp. 81-110.
@incollection {kitagawa, MRKEY = {1279604},
AUTHOR = {Kitagawa, Koji},
TITLE = {On standard {$p$}-adic {$L$}-functions of families of elliptic cusp forms},
BOOKTITLE = {{$p$}-{{A}}dic {{M}}onodromy and the {B}irch and {S}winnerton-{D}yer {{C}}onjecture},
VENUE={{B}oston, {MA},
1991)},
SERIES = {Contemp. Math.},
VOLUME = {165},
PAGES = {81--110},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1994},
MRCLASS = {11F33 (11F67 11F85 11G20 19F15)},
MRNUMBER = {1279604},
MRREVIEWER = {Alexey A. Panchishkin},
ZBLNUMBER = {0841.11028},
} -
[manin] J. I. Manin, "Parabolic points and zeta functions of modular curves," Izv. Akad. Nauk SSSR Ser. Mat., vol. 36, pp. 19-66, 1972.
@article {manin, MRKEY = {0314846},
AUTHOR = {Manin, Ju. I.},
TITLE = {Parabolic points and zeta functions of modular curves},
JOURNAL = {Izv. Akad. Nauk SSSR Ser. Mat.},
FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
VOLUME = {36},
YEAR = {1972},
PAGES = {19--66},
ISSN = {0373-2436},
MRCLASS = {14G10 (10D15 14H25)},
MRNUMBER = {47 \#3396},
MRREVIEWER = {K.-B. Gundlach},
ZBLNUMBER = {0243.14008},
DOI={10.1070/IM1972v006n01ABEH001867},
} -
[mazur-arith] B. Mazur, "On the arithmetic of special values of $L$ functions," Invent. Math., vol. 55, iss. 3, pp. 207-240, 1979.
@article {mazur-arith, MRKEY = {553997},
AUTHOR = {Mazur, B.},
TITLE = {On the arithmetic of special values of {$L$} functions},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {55},
YEAR = {1979},
NUMBER = {3},
PAGES = {207--240},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {14G10 (10D24 12A70 14G25 14K07)},
MRNUMBER = {553997},
MRREVIEWER = {V. V. Shokurov},
DOI = {10.1007/BF01406841},
ZBLNUMBER = {0426.14009},
} -
[mazur-anom] B. Mazur, Anomalous eigenforms and the two-variable $p$-adic $L$-function.
@misc{mazur-anom,
author={Mazur, B.},
TITLE={Anomalous eigenforms and the two-variable $p$-adic {$L$}-function},
NOTE={unpublished note},
} -
[MW84] B. Mazur and A. Wiles, "Class fields of abelian extensions of $\mathbf{Q}$," Invent. Math., vol. 76, pp. 179-330, 1984.
@article{MW84,
author= {Mazur, B. and Wiles, A.},
TITLE = {Class fields of abelian extensions of $\mathbf{Q}$},
JOURNAL={Invent. Math.},
VOLUME={76},
YEAR={1984},
PAGES={179--330},
MRNUMBER={742853},
ZBLNUMBER={0545.12005},
DOI={10.1007/BF01388599},
} -
[mw] B. Mazur and A. Wiles, "On $p$-adic analytic families of Galois representations," Compositio Math., vol. 59, iss. 2, pp. 231-264, 1986.
@article {mw, MRKEY = {860140},
AUTHOR = {Mazur, B. and Wiles, A.},
TITLE = {On {$p$}-adic analytic families of {G}alois representations},
JOURNAL = {Compositio Math.},
FJOURNAL = {Compositio Mathematica},
VOLUME = {59},
YEAR = {1986},
NUMBER = {2},
PAGES = {231--264},
ISSN = {0010-437X},
CODEN = {CMPMAF},
MRCLASS = {11G18 (11F11 11F33 11R23 11R39)},
MRNUMBER = {860140},
MRREVIEWER = {Ernst-Wilhelm Zink},
ZBLNUMBER = {0654.12008},
} -
[mcs] W. G. McCallum and R. T. Sharifi, "A cup product in the Galois cohomology of number fields," Duke Math. J., vol. 120, iss. 2, pp. 269-310, 2003.
@article {mcs, MRKEY = {2019977},
AUTHOR = {McCallum, William G. and Sharifi, Romyar T.},
TITLE = {A cup product in the {G}alois cohomology of number fields},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {120},
YEAR = {2003},
NUMBER = {2},
PAGES = {269--310},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {11R34 (11R23 11R70)},
MRNUMBER = {2019977},
MRREVIEWER = {Th{\cfac{o}}ng Nguy{\cftil{e}}n-Quang-{\Dbar}{\cftil{o}}},
DOI = {10.1215/S0012-7094-03-12023-2},
ZBLNUMBER = {1047.11106},
} -
[nsw] J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of Number Fields, Second ed., New York: Springer-Verlag, 2008, vol. 323.
@book {nsw, MRKEY = {2392026},
AUTHOR = {Neukirch, J{ü}rgen and Schmidt, Alexander and Wingberg, Kay},
TITLE = {Cohomology of {{N}}umber {{F}}ields},
SERIES = {Grundl. Math. Wissen.},
VOLUME = {323},
EDITION = {Second},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2008},
PAGES = {xvi+825},
ISBN = {978-3-540-37888-4},
MRCLASS = {11R34 (11-02 11G45 11R23 11S20 11S25 11S31 12G05)},
MRNUMBER = {2392026},
ZBLNUMBER = {1136.11001},
} -
[ohta-eich] M. Ohta, "On the $p$-adic Eichler-Shimura isomorphism for $\Lambda$-adic cusp forms," J. Reine Angew. Math., vol. 463, pp. 49-98, 1995.
@article {ohta-eich, MRKEY = {1332907},
AUTHOR = {Ohta, Masami},
TITLE = {On the {$p$}-adic {E}ichler-{S}himura isomorphism for {$\Lambda$}-adic cusp forms},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
VOLUME = {463},
YEAR = {1995},
PAGES = {49--98},
ISSN = {0075-4102},
CODEN = {JRMAA8},
MRCLASS = {11F33 (11F67)},
MRNUMBER = {1332907},
MRREVIEWER = {Andrea Mori},
DOI = {10.1515/crll.1995.463.49},
ZBLNUMBER = {0827.11025},
} -
[ohta-ord] M. Ohta, "Ordinary $p$-adic étale cohomology groups attached to towers of elliptic modular curves," Compositio Math., vol. 115, iss. 3, pp. 241-301, 1999.
@article {ohta-ord, MRKEY = {1674001},
AUTHOR = {Ohta, Masami},
TITLE = {Ordinary {$p$}-adic étale cohomology groups attached to towers of elliptic modular curves},
JOURNAL = {Compositio Math.},
FJOURNAL = {Compositio Mathematica},
VOLUME = {115},
YEAR = {1999},
NUMBER = {3},
PAGES = {241--301},
ISSN = {0010-437X},
CODEN = {CMPMAF},
MRCLASS = {11F33 (11F67 11G18)},
MRNUMBER = {1674001},
MRREVIEWER = {Andrea Mori},
DOI = {10.1023/A:1000556212097},
ZBLNUMBER = {0967.11015},
} -
[ohta-ord2] M. Ohta, "Ordinary $p$-adic étale cohomology groups attached to towers of elliptic modular curves. II," Math. Ann., vol. 318, iss. 3, pp. 557-583, 2000.
@article {ohta-ord2, MRKEY = {1800769},
AUTHOR = {Ohta, Masami},
TITLE = {Ordinary {$p$}-adic étale cohomology groups attached to towers of elliptic modular curves. {II}},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {318},
YEAR = {2000},
NUMBER = {3},
PAGES = {557--583},
ISSN = {0025-5831},
CODEN = {MAANA},
MRCLASS = {11F33 (11F67 11G18 11R23)},
MRNUMBER = {2369998},
MRREVIEWER = {Andrea Mori},
DOI = {10.1007/s002080000119},
ZBLNUMBER = {0967.11016},
} -
[ohta-cong] M. Ohta, "Congruence modules related to Eisenstein series," Ann. Sci. École Norm. Sup., vol. 36, iss. 2, pp. 225-269, 2003.
@article {ohta-cong, MRKEY = {1980312},
AUTHOR = {Ohta, Masami},
TITLE = {Congruence modules related to {E}isenstein series},
JOURNAL = {Ann. Sci. École Norm. Sup.},
FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
VOLUME = {36},
YEAR = {2003},
NUMBER = {2},
PAGES = {225--269},
ISSN = {0012-9593},
CODEN = {ASENAH},
MRCLASS = {11F85 (11F33 11F80)},
MRNUMBER = {1980312},
MRREVIEWER = {Jacques Tilouine},
DOI = {10.1016/S0012-9593(03)00009-0},
ZBLNUMBER = {1047.11046},
} -
[ohta-comp] M. Ohta, "Companion forms and the structure of $p$-adic Hecke algebras," J. Reine Angew. Math., vol. 585, pp. 141-172, 2005.
@article {ohta-comp, MRKEY = {2164625},
AUTHOR = {Ohta, Masami},
TITLE = {Companion forms and the structure of {$p$}-adic {H}ecke algebras},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
VOLUME = {585},
YEAR = {2005},
PAGES = {141--172},
ISSN = {0075-4102},
CODEN = {JRMAA8},
MRCLASS = {11F85 (11F33)},
MRNUMBER = {2164625},
MRREVIEWER = {Jacques Tilouine},
DOI = {10.1515/crll.2005.2005.585.141},
ZBLNUMBER = {1081.11035},
} -
[ohta-comp2] M. Ohta, "Companion forms and the structure of $p$-adic Hecke algebras. II," J. Math. Soc. Japan, vol. 59, iss. 4, pp. 913-951, 2007.
@article {ohta-comp2, MRKEY = {2369998},
AUTHOR = {Ohta, Masami},
TITLE = {Companion forms and the structure of {$p$}-adic {H}ecke algebras. {II}},
JOURNAL = {J. Math. Soc. Japan},
FJOURNAL = {Journal of the Mathematical Society of Japan},
VOLUME = {59},
YEAR = {2007},
NUMBER = {4},
PAGES = {913--951},
ISSN = {0025-5645},
CODEN = {NISUBC},
MRCLASS = {11F33 (11F80 11F85)},
MRNUMBER = {2369998},
MRREVIEWER = {M. Ram Murty},
DOI = {10.2969/jmsj/05940913},
ZBLNUMBER = {1187.11014},
} -
[me-paireis] R. T. Sharifi, "Iwasawa theory and the Eisenstein ideal," Duke Math. J., vol. 137, iss. 1, pp. 63-101, 2007.
@article {me-paireis, MRKEY = {2309144},
AUTHOR = {Sharifi, Romyar T.},
TITLE = {Iwasawa theory and the {E}isenstein ideal},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {137},
YEAR = {2007},
NUMBER = {1},
PAGES = {63--101},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {11R23 (11F33 11R34)},
MRNUMBER = {2309144},
MRREVIEWER = {Ramdorai Sujatha},
DOI = {10.1215/S0012-7094-07-13713-X},
ZBLNUMBER = {1131.11068},
} -
[stevens] G. Stevens, Arithmetic on Modular Curves, Boston, MA: Birkhäuser, 1982, vol. 20.
@book {stevens, MRKEY = {670070},
AUTHOR = {Stevens, Glenn},
TITLE = {Arithmetic on {{M}}odular {{C}}urves},
SERIES = {Progr. Math.},
VOLUME = {20},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston, MA},
YEAR = {1982},
PAGES = {xvii+214},
ISBN = {3-7643-3088-0},
MRCLASS = {11G40 (11G30 14G10 14G13)},
MRNUMBER = {670070},
MRREVIEWER = {Ernst Kani},
ZBLNUMBER = {0529.10028},
} -
[tilouine] J. Tilouine, "Un sous-groupe $p$-divisible de la jacobienne de $X_1(Np^r)$ comme module sur l’algèbre de Hecke," Bull. Soc. Math. France, vol. 115, iss. 3, pp. 329-360, 1987.
@article {tilouine, MRKEY = {926532},
AUTHOR = {Tilouine, Jacques},
TITLE = {Un sous-groupe {$p$}-divisible de la jacobienne de {$X\sb 1(Np\sp r)$} comme module sur l'algèbre de {H}ecke},
JOURNAL = {Bull. Soc. Math. France},
FJOURNAL = {Bulletin de la Société Mathématique de France},
VOLUME = {115},
YEAR = {1987},
NUMBER = {3},
PAGES = {329--360},
ISSN = {0037-9484},
CODEN = {BSMFAA},
MRCLASS = {11G18 (11F99 11R23)},
MRNUMBER = {926532},
MRREVIEWER = {Noburo Ishii},
URL = {http://www.numdam.org/item?id=BSMF_1987__115__329_0},
ZBLNUMBER = {0677.14006},
}
Print version