Langlands base change for $\mathrm{GL}(2)$

Luis Dieulefait

doi:https://doi.org/10.4007/annals.2012.176.2.7

Abstract

Let $F$ be a totally real Galois number field. We prove the existence of base change relative to the extension $F/\mathbb{Q}$ for every holomorphic newform of weight at least $2$ and odd level, under simple local assumptions on the field $F$.

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[AC] J. Arthur and L. Clozel, Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Princeton, NJ: Princeton Univ. Press, 1989, vol. 120.

Show bibtex

@book {AC, MRKEY = {1007299}, AUTHOR = {Arthur, James and Clozel, Laurent}, TITLE = {Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula}, SERIES = {Ann. of Math. Stud.}, VOLUME = {120}, PUBLISHER = {Princeton Univ. Press}, ADDRESS = {Princeton, NJ}, YEAR = {1989}, PAGES = {xiv+230}, ISBN = {0-691-08517-X; 0-691-08518-8}, MRCLASS = {22E55 (11F70 11F72 11R39)}, MRNUMBER = {1007299}, MRREVIEWER = {Stephen Gelbart}, ZBLNUMBER = {0682.10022}, }
[BLGG] T. Barnet-Lamb, T. Gee, and D. Geraghty, Congruences between Hilbert modular forms: constructing ordinary lifts.

Show bibtex

@misc{BLGG, author={Barnet-Lamb, T. and Gee, T. and Geraghty, D.}, TITLE={Congruences between {H}ilbert modular forms: constructing ordinary lifts}, ARXIV = {1006.0466}, }
[BLGGT] T. Barnet-Lamb, T. Gee, D. Geraghty, and R. Taylor, Potential automorphy and change of weight.

Show bibtex

@misc{BLGGT, author={Barnet-Lamb, T. and Gee, T. and Geraghty, D. and Taylor, R.}, TITLE={Potential automorphy and change of weight}, ARXIV = {1010.2561}, }
[BLZ] $Go to document$ L. Berger, H. Li, and H. J. Zhu, "Construction of some families of 2-dimensional crystalline representations," Math. Ann., vol. 329, iss. 2, pp. 365-377, 2004.

Show bibtex

@article {BLZ, MRKEY = {2060368}, AUTHOR = {Berger, Laurent and Li, Hanfeng and Zhu, Hui June}, TITLE = {Construction of some families of 2-dimensional crystalline representations}, JOURNAL = {Math. Ann.}, FJOURNAL = {Mathematische Annalen}, VOLUME = {329}, YEAR = {2004}, NUMBER = {2}, PAGES = {365--377}, ISSN = {0025-5831}, CODEN = {MAANA}, MRCLASS = {11F80 (11F33 11F85 14F30)}, MRNUMBER = {2060368}, MRREVIEWER = {Abdellah Mokrane}, DOI = {10.1007/s00208-004-0529-y}, ZBLNUMBER = {1085.11028}, }
[C] $Go to document$ X. Caruso, "Représentations semi-stables de torsion dans le case $er<p-1$," J. Reine Angew. Math., vol. 594, pp. 35-92, 2006.

Show bibtex

@article {C, MRKEY = {2248152}, AUTHOR = {Caruso, Xavier}, TITLE = {Représentations semi-stables de torsion dans le case {$er JOURNAL = {J. Reine Angew. Math.}, FJOURNAL = {Journal für die Reine und Angewandte Mathematik}, VOLUME = {594}, YEAR = {2006}, PAGES = {35--92}, ISSN = {0075-4102}, CODEN = {JRMAA8}, MRCLASS = {11F85 (11G25 11S20 14F20 14F30 14F40)}, MRNUMBER = {2248152}, MRREVIEWER = {Yuichiro Taguchi}, DOI = {10.1515/CRELLE.2006.035}, ZBLNUMBER = {1134.14013}, }
[Cl] $Go to document$ A. H. Clifford, "Representations induced in an invariant subgroup," Ann. of Math., vol. 38, iss. 3, pp. 533-550, 1937.

Show bibtex

@article {Cl, MRKEY = {1503352}, AUTHOR = {Clifford, A. H.}, TITLE = {Representations induced in an invariant subgroup}, JOURNAL = {Ann. of Math.}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {38}, YEAR = {1937}, NUMBER = {3}, PAGES = {533--550}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {Contributed Item}, MRNUMBER = {1503352}, DOI = {10.2307/1968599}, ZBLNUMBER = {0017.29705}, }
[Clo] $Go to document$ L. Clozel, "Sur la théorie de Wiles et le changement de base nonabélien," Internat. Math. Res. Notices, vol. 1995, p. no. 9, 437-444.

Show bibtex

@article {Clo, MRKEY = {1360622}, AUTHOR = {Clozel, Laurent}, TITLE = {Sur la théorie de {W}iles et le changement de base nonabélien}, JOURNAL = {Internat. Math. Res. Notices}, FJOURNAL = {International Mathematics Research Notices}, VOLUME = {1995}, PAGES = {no.~9, 437--444}, ISSN = {1073-7928}, MRCLASS = {11R39 (11F80)}, MRNUMBER = {1360622}, MRREVIEWER = {Richard Taylor}, DOI = {10.1155/S1073792895000316}, ZBLNUMBER = {0846.11031}, }
[Co] P. Colmez, "Représentations de $\mathrm{GL}_2(\Bbb Q_p)$ et $(\phi,\Gamma)$-modules," Astérisque, vol. 330, pp. 281-509, 2010.

Show bibtex

@article {Co, MRKEY = {2642409}, AUTHOR = {Colmez, Pierre}, TITLE = {Représentations de {$\mathrm{GL}\sb 2(\bold Q\sb p)$} et {$(\phi,\Gamma)$}-modules}, JOURNAL = {Astérisque}, FJOURNAL = {Astérisque}, VOLUME = {330}, YEAR = {2010}, PAGES = {281--509}, ISSN = {0303-1179}, ISBN = {978-2-85629-281-5}, MRCLASS = {11S37 (11F80)}, MRNUMBER = {2642409}, MRREVIEWER = {Fabrizio Andreatta}, ZBLNUMBER = {1235.11107}, ZBLNUMBER = {1218.11107}, ZBLNUMBER = {1192.11001}, }
[D] $Go to document$ L. Dieulefait, Remarks on Serre’s modularity conjecture.

Show bibtex

@misc {D, MRKEY = {2603902}, AUTHOR = {Dieulefait, Luis}, TITLE= {Remarks on {S}erre's modularity conjecture}, NOTE={{\it Manuscripta Math.}, to appear}, DOI = {10.1007/s00229-011-0503-4}, }
[E] $Go to document$ B. Edixhoven, "The weight in Serre’s conjectures on modular forms," Invent. Math., vol. 109, iss. 3, pp. 563-594, 1992.

Show bibtex

@article {E, MRKEY = {1176206}, AUTHOR = {Edixhoven, Bas}, TITLE = {The weight in {S}erre's conjectures on modular forms}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {109}, YEAR = {1992}, NUMBER = {3}, PAGES = {563--594}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {11R39 (11F80 11G18)}, MRNUMBER = {1176206}, MRREVIEWER = {Douglas L. Ulmer}, DOI = {10.1007/BF01232041}, ZBLNUMBER = {0777.11013}, }
[El] $Go to document$ J. S. Ellenberg, "Serre’s conjecture over $\Bbb F_9$," Ann. of Math., vol. 161, iss. 3, pp. 1111-1142, 2005.

Show bibtex

@article {El, MRKEY = {2180399}, AUTHOR = {Ellenberg, Jordan S.}, TITLE = {Serre's conjecture over {$\Bbb F\sb 9$}}, JOURNAL = {Ann. of Math.}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {161}, YEAR = {2005}, NUMBER = {3}, PAGES = {1111--1142}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {11F80 (11F33 11G18)}, MRNUMBER = {2180399}, MRREVIEWER = {Gabor Wiese}, DOI = {10.4007/annals.2005.161.1111}, ZBLNUMBER = {1153.11312}, }
[G] D. Geraghty, Modularity lifting theorems for ordinary Galois representations.

Show bibtex

@misc{G, author={Geraghty, D.}, TITLE={Modularity lifting theorems for ordinary Galois representations}, NOTE={preprint}, }
[H] $Go to document$ H. Hida, "Serre’s conjecture and base change for $GL(2)$," Pure Appl. Math. Q., vol. 5, iss. 1, pp. 81-125, 2009.

Show bibtex

@article {H, MRKEY = {2520456}, AUTHOR = {Hida, Haruzo}, TITLE = {Serre's conjecture and base change for {$GL(2)$}}, JOURNAL = {Pure Appl. Math. Q.}, FJOURNAL = {Pure and Applied Mathematics Quarterly}, VOLUME = {5}, YEAR = {2009}, NUMBER = {1}, PAGES = {81--125}, ISSN = {1558-8599}, MRCLASS = {11R39 (11F41 11F80 11R52)}, MRNUMBER = {2520456}, MRREVIEWER = {Jerzy Browkin}, ZBLNUMBER = {05543145}, URL = {http://www.intlpress.com/JPAMQ/p/2009/81-125.pdf}, }
[HM] $Go to document$ H. Hida and Y. Maeda, "Non-abelian base change for totally real fields," Pacific J. Math., vol. 181, pp. 189-217, 1997.

Show bibtex

@article {HM, MRKEY = {1610859}, AUTHOR = {Hida, Haruzo and Maeda, Yoshitaka}, TITLE = {Non-abelian base change for totally real fields}, NOTE = {Olga Taussky-Todd memorial issue}, JOURNAL = {Pacific J. Math.}, FJOURNAL = {Pacific Journal of Mathematics}, YEAR = {1997}, VOLUME = {181}, PAGES = {189--217}, ISSN = {0030-8730}, CODEN = {PJMAAI}, MRCLASS = {11F80 (11F85)}, MRNUMBER = {1610859}, MRREVIEWER = {Alexey A. Panchishkin}, DOI = {10.2140/pjm.1997.181.189}, ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER}, ZBLNUMBER = {0942.11026}, }
[Kh] $Go to document$ C. Khare, "A local analysis of congruences in the $(p,p)$ case. II," Invent. Math., vol. 143, iss. 1, pp. 129-155, 2001.

Show bibtex

@article {Kh, MRKEY = {1802794}, AUTHOR = {Khare, Chandrashekhar}, TITLE = {A local analysis of congruences in the {$(p,p)$} case. {II}}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {143}, YEAR = {2001}, NUMBER = {1}, PAGES = {129--155}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {11F33 (11F32 11F80 11F85)}, MRNUMBER = {1802794}, MRREVIEWER = {Alexey A. Panchishkin}, DOI = {10.1007/s002220000103}, ZBLNUMBER = {0971.11028}, }
[KW1] $Go to document$ C. Khare and J. Wintenberger, "Serre’s modularity conjecture. I," Invent. Math., vol. 178, iss. 3, pp. 485-504, 2009.

Show bibtex

@article {KW1, MRKEY = {2551763}, AUTHOR = {Khare, Chandrashekhar and Wintenberger, Jean-Pierre}, TITLE = {Serre's modularity conjecture. {I}}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {178}, YEAR = {2009}, NUMBER = {3}, PAGES = {485--504}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {11F80 (11F11 11F33 11R39)}, MRNUMBER = {2551763}, MRREVIEWER = {Gabor Wiese}, DOI = {10.1007/s00222-009-0205-7}, ZBLNUMBER = {05636295}, }
[KW2] $Go to document$ C. Khare and J. Wintenberger, "Serre’s modularity conjecture. II," Invent. Math., vol. 178, iss. 3, pp. 505-586, 2009.

Show bibtex

@article {KW2, MRKEY = {2551764}, AUTHOR = {Khare, Chandrashekhar and Wintenberger, Jean-Pierre}, TITLE = {Serre's modularity conjecture. {II}}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {178}, YEAR = {2009}, NUMBER = {3}, PAGES = {505--586}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {11F80 (11F11 11F33 11R39)}, MRNUMBER = {2551764}, MRREVIEWER = {Gabor Wiese}, DOI = {10.1007/s00222-009-0206-6}, ZBLNUMBER = {05636296}, }
[KiBT] $Go to document$ M. Kisin, "Moduli of finite flat group schemes, and modularity," Ann. of Math., vol. 170, iss. 3, pp. 1085-1180, 2009.

Show bibtex

@article {KiBT, MRKEY = {2600871}, AUTHOR = {Kisin, Mark}, TITLE = {Moduli of finite flat group schemes, and modularity}, JOURNAL = {Ann. of Math.}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {170}, YEAR = {2009}, NUMBER = {3}, PAGES = {1085--1180}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {11F80 (11R39 14L15)}, MRNUMBER = {2600871}, MRREVIEWER = {David L. Savitt}, DOI = {10.4007/annals.2009.170.1085}, ZBLNUMBER = {1201.14034}, }
[KiFM] $Go to document$ M. Kisin, "The Fontaine-Mazur conjecture for ${ GL}_2$," J. Amer. Math. Soc., vol. 22, iss. 3, pp. 641-690, 2009.

Show bibtex

@article {KiFM, MRKEY = {2505297}, AUTHOR = {Kisin, Mark}, TITLE = {The {F}ontaine-{M}azur conjecture for {${\rm GL}\sb 2$}}, JOURNAL = {J. Amer. Math. Soc.}, FJOURNAL = {Journal of the American Mathematical Society}, VOLUME = {22}, YEAR = {2009}, NUMBER = {3}, PAGES = {641--690}, ISSN = {0894-0347}, MRCLASS = {11F80 (11F33)}, MRNUMBER = {2505297}, MRREVIEWER = {David L. Savitt}, DOI = {10.1090/S0894-0347-09-00628-6}, ZBLNUMBER = {05859417}, }
[Ki2] $Go to document$ M. Kisin, "Modularity of 2-adic Barsotti-Tate representations," Invent. Math., vol. 178, iss. 3, pp. 587-634, 2009.

Show bibtex

@article {Ki2, MRKEY = {2551765}, AUTHOR = {Kisin, Mark}, TITLE = {Modularity of 2-adic {B}arsotti-{T}ate representations}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {178}, YEAR = {2009}, NUMBER = {3}, PAGES = {587--634}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {11F80 (11F41 14L15 14L30)}, MRNUMBER = {2551765}, MRREVIEWER = {Gabor Wiese}, DOI = {10.1007/s00222-009-0207-5}, ZBLNUMBER = {05636297}, }
[L] R. P. Langlands, Base Change for ${ GL}(2)$, Princeton, N.J.: Princeton Univ. Press, 1980, vol. 96.

Show bibtex

@book {L, MRKEY = {0574808}, AUTHOR = {Langlands, Robert P.}, TITLE = {Base Change for {${\rm GL}(2)$}}, SERIES = {Ann. of Math. Stud.}, VOLUME = {96}, PUBLISHER = {Princeton Univ. Press}, ADDRESS = {Princeton, N.J.}, YEAR = {1980}, PAGES = {vii+237}, ISBN = {0-691-08263-4; 0-691-08272-3}, MRCLASS = {10D40 (10-02 12A67 22E55)}, MRNUMBER = {0574808}, MRREVIEWER = {Stephen Gelbart}, ZBLNUMBER = {0444.22007}, }
[M] J. Manoharmayum, "On the modularity of certain ${ GL}_2(\Bbb F_7)$ Galois representations," Math. Res. Lett., vol. 8, iss. 5-6, pp. 703-712, 2001.

Show bibtex

@article {M, MRKEY = {1879814}, AUTHOR = {Manoharmayum, J.}, TITLE = {On the modularity of certain {${\rm GL}\sb 2(\Bbb F\sb 7)$} {G}alois representations}, JOURNAL = {Math. Res. Lett.}, FJOURNAL = {Mathematical Research Letters}, VOLUME = {8}, YEAR = {2001}, NUMBER = {5-6}, PAGES = {703--712}, ISSN = {1073-2780}, MRCLASS = {11G18 (11F33 11F80 11G05)}, MRNUMBER = {1879814}, MRREVIEWER = {Gebhard B{ö}ckle}, ZBLNUMBER = {1001.11023}, }
[R1] $Go to document$ K. A. Ribet, "On $l$-adic representations attached to modular forms. II," Glasgow Math. J., vol. 27, pp. 185-194, 1985.

Show bibtex

@article {R1, MRKEY = {0819838}, AUTHOR = {Ribet, Kenneth A.}, TITLE = {On {$l$}-adic representations attached to modular forms. {II}}, JOURNAL = {Glasgow Math. J.}, FJOURNAL = {Glasgow Mathematical Journal}, VOLUME = {27}, YEAR = {1985}, PAGES = {185--194}, ISSN = {0017-0895}, CODEN = {GLMJAS}, MRCLASS = {11F11 (11F33 11G10)}, MRNUMBER = {0819838}, MRREVIEWER = {G. Frey}, DOI = {10.1017/S0017089500006170}, ZBLNUMBER = {0596.10027}, }
[R2] K. A. Ribet, "Raising the levels of modular representations," in Séminaire de Théorie des Nombres, Paris 1987–88, Boston, MA: Birkhäuser, 1990, vol. 81, pp. 259-271.

Show bibtex

@incollection {R2, MRKEY = {1042773}, AUTHOR = {Ribet, Kenneth A.}, TITLE = {Raising the levels of modular representations}, BOOKTITLE = {Séminaire de {T}héorie des {N}ombres, {P}aris 1987--88}, SERIES = {Progr. Math.}, VOLUME = {81}, PAGES = {259--271}, PUBLISHER = {Birkhäuser}, ADDRESS = {Boston, MA}, YEAR = {1990}, MRCLASS = {11F80 (11F11 11G18 11S37)}, MRNUMBER = {1042773}, MRREVIEWER = {Glenn Stevens}, ZBLNUMBER = {0705.11030}, }
[RS] K. A. Ribet and W. A. Stein, "Lectures on Serre’s conjectures," in Arithmetic Algebraic Geometry, Providence, RI: Amer. Math. Soc., 2001, vol. 9, pp. 143-232.

Show bibtex

@incollection {RS, MRKEY = {1860042}, AUTHOR = {Ribet, Kenneth A. and Stein, William A.}, TITLE = {Lectures on {S}erre's conjectures}, BOOKTITLE = {Arithmetic Algebraic Geometry}, VENUE={{P}ark {C}ity, {UT}, 1999}, SERIES = {IAS/Park City Math. Ser.}, VOLUME = {9}, PAGES = {143--232}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, RI}, YEAR = {2001}, MRCLASS = {11F80 (11F66 11G05)}, MRNUMBER = {1860042}, MRREVIEWER = {Chandrashekhar Khare}, ZBLNUMBER = {1160.11327}, }
[S] $Go to document$ J. Serre, "Sur les représentations modulaires de degré $2$ de ${ Gal}(\overline{\bf Q}/{\bf Q})$," Duke Math. J., vol. 54, iss. 1, pp. 179-230, 1987.

Show bibtex

@article {S, MRKEY = {0885783}, AUTHOR = {Serre, Jean-Pierre}, TITLE = {Sur les représentations modulaires de degré {$2$} de {${\rm Gal}(\overline{\bf Q}/{\bf Q})$}}, JOURNAL = {Duke Math. J.}, FJOURNAL = {Duke Mathematical Journal}, VOLUME = {54}, YEAR = {1987}, NUMBER = {1}, PAGES = {179--230}, ISSN = {0012-7094}, CODEN = {DUMJAO}, MRCLASS = {11F11 (11G05 14G15 14G25 14K15)}, MRNUMBER = {0885783}, MRREVIEWER = {M. A. Kenku}, DOI = {10.1215/S0012-7094-87-05413-5}, ZBLNUMBER = {0641.10026}, }
[SW] $Go to document$ C. M. Skinner and A. J. Wiles, "Nearly ordinary deformations of irreducible residual representations," Ann. Fac. Sci. Toulouse Math., vol. 10, iss. 1, pp. 185-215, 2001.

Show bibtex

@article {SW, MRKEY = {1928993}, AUTHOR = {Skinner, C. M. and Wiles, Andrew J.}, TITLE = {Nearly ordinary deformations of irreducible residual representations}, JOURNAL = {Ann. Fac. Sci. Toulouse Math.}, FJOURNAL = {Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série 6}, VOLUME = {10}, YEAR = {2001}, NUMBER = {1}, PAGES = {185--215}, ISSN = {0240-2963}, MRCLASS = {11F80 (11F11 11F33)}, MRNUMBER = {1928993}, MRREVIEWER = {Gebhard B{ö}ckle}, ZBLNUMBER = {1024.11036}, DOI = {10.5802/afst.988}, }
[T] $Go to document$ R. Taylor, "On icosahedral Artin representations. II," Amer. J. Math., vol. 125, iss. 3, pp. 549-566, 2003.

Show bibtex

@article {T, MRKEY = {1981033}, AUTHOR = {Taylor, Richard}, TITLE = {On icosahedral {A}rtin representations. {II}}, JOURNAL = {Amer. J. Math.}, FJOURNAL = {American Journal of Mathematics}, VOLUME = {125}, YEAR = {2003}, NUMBER = {3}, PAGES = {549--566}, ISSN = {0002-9327}, CODEN = {AJMAAN}, MRCLASS = {11F80 (11F33 11G18 11R42)}, MRNUMBER = {1981033}, MRREVIEWER = {Chandrashekhar Khare}, DOI = {10.1353/ajm.2003.0021}, ZBLNUMBER = {1031.11031}, }
[W] $Go to document$ A. Wiles, "Modular elliptic curves and Fermat’s last theorem," Ann. of Math., vol. 141, iss. 3, pp. 443-551, 1995.

Show bibtex

@article {W, MRKEY = {1333035}, AUTHOR = {Wiles, Andrew}, TITLE = {Modular elliptic curves and {F}ermat's last theorem}, JOURNAL = {Ann. of Math.}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {141}, YEAR = {1995}, NUMBER = {3}, PAGES = {443--551}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {11G05 (11D41 11F11 11F80 11G18)}, MRNUMBER = {1333035}, MRREVIEWER = {Karl Rubin}, DOI = {10.2307/2118559}, ZBLNUMBER = {0823.11029}, }
[Wi] J. -P. Wintenberger, Sur les représentations $p$-adiques géométriques de conducteur $1$ et de dimension $2$ de $G_{\mathbb Q}$.

Show bibtex

@misc{Wi, author={Wintenberger, J.-P.}, TITLE={Sur les repr{é}sentations $p$-adiques g{é}om{é}triques de conducteur $1$ et de dimension $2$ de {$G_{\mathbb Q}$}}, ARXIV= {math.NT/0406576}, }

Langlands base change for $\mathrm{GL}(2)$

Abstract

Authors