A reciprocity map and the two-variable $p$-adic $L$-function

Romyar Sharifi

doi:http://doi.org/10.4007/annals.2011.173.1.7

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.

Show/hide bibliography for this article

[as] $Go to document$ 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.

Show bibtex

@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] $Go to document$ C. Busuioc, "The Steinberg symbol and special values of $L$-functions," Trans. Amer. Math. Soc., vol. 360, iss. 11, pp. 5999-6015, 2008.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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] $Go to document$ R. Greenberg, "Iwasawa theory for motives," in $L$-Functions and Arithmetic, Cambridge: Cambridge Univ. Press, 1991, vol. 153, pp. 211-233.

Show bibtex

@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] $Go to document$ 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.

Show bibtex

@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] $Go to document$ H. Hida, "Iwasawa modules attached to congruences of cusp forms," Ann. Sci. École Norm. Sup., vol. 19, iss. 2, pp. 231-273, 1986.

Show bibtex

@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] $Go to document$ 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.

Show bibtex

@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] $Go to document$ K. Iwasawa, "On some modules in the theory of cyclotomic fields," J. Math. Soc. Japan, vol. 16, pp. 42-82, 1964.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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] $Go to document$ J. I. Manin, "Parabolic points and zeta functions of modular curves," Izv. Akad. Nauk SSSR Ser. Mat., vol. 36, pp. 19-66, 1972.

Show bibtex

@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] $Go to document$ B. Mazur, "On the arithmetic of special values of $L$ functions," Invent. Math., vol. 55, iss. 3, pp. 207-240, 1979.

Show bibtex

@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.

Show bibtex

@misc{mazur-anom, author={Mazur, B.}, TITLE={Anomalous eigenforms and the two-variable $p$-adic {$L$}-function}, NOTE={unpublished note}, }
[MW84] $Go to document$ B. Mazur and A. Wiles, "Class fields of abelian extensions of $\mathbf{Q}$," Invent. Math., vol. 76, pp. 179-330, 1984.

Show bibtex

@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.

Show bibtex

@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] $Go to document$ 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.

Show bibtex

@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.

Show bibtex

@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] $Go to document$ M. Ohta, "On the $p$-adic Eichler-Shimura isomorphism for $\Lambda$-adic cusp forms," J. Reine Angew. Math., vol. 463, pp. 49-98, 1995.

Show bibtex

@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] $Go to document$ 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.

Show bibtex

@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] $Go to document$ 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.

Show bibtex

@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] $Go to document$ M. Ohta, "Congruence modules related to Eisenstein series," Ann. Sci. École Norm. Sup., vol. 36, iss. 2, pp. 225-269, 2003.

Show bibtex

@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] $Go to document$ M. Ohta, "Companion forms and the structure of $p$-adic Hecke algebras," J. Reine Angew. Math., vol. 585, pp. 141-172, 2005.

Show bibtex

@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] $Go to document$ 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.

Show bibtex

@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] $Go to document$ R. T. Sharifi, "Iwasawa theory and the Eisenstein ideal," Duke Math. J., vol. 137, iss. 1, pp. 63-101, 2007.

Show bibtex

@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.

Show bibtex

@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] $Go to document$ 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.

Show bibtex

@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}, }

A reciprocity map and the two-variable $p$-adic $L$-function

Abstract

Authors