Skip to main content Accessibility help
×
Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-25T15:26:15.201Z Has data issue: false hasContentIssue false

5 - The conjectural connections between automorphic representations and Galois representations

Published online by Cambridge University Press:  05 October 2014

Kevin Buzzard
Affiliation:
Imperial College London
Toby Gee
Affiliation:
Imperial College London
Fred Diamond
Affiliation:
King's College London
Payman L. Kassaei
Affiliation:
King's College London
Minhyong Kim
Affiliation:
University of Oxford
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[ABD+66] M., Artin, J. E., Bertin, M., Demazure, P., Gabriel, A., Grothendieck, M., Raynaud, and J.-P., Serre, Schémas en groupes. Fasc. 7: Exposés 23 à 26, Séminaire de Géométrie Algébrique de l'Institut des Hautes Études Scientifiques, vol. 1963/64, Institut des Hautes Études Scientifiques, Paris, 1965/1966. MR 0207710 (34 #7525)
[ABV92] Jeffrey, Adams, Dan, Barbasch, and David A., Vogan Jr., The Lang-lands classification and irreducible characters for real reductive groups, Progress in Mathematics, vol. 104, Birkhäuser Boston Inc., Boston, MA, 1992. MR MR1162533 (93j:22001)
[Art02] James, Arthur, A note on the automorphic Langlands group, Canad. Math. Bull. 45 (2002), no. 4, 466–482, Dedicated to Robert V. Moody. MR MR1941222 (2004a:11120)Google Scholar
[Bor79] A., Borel, Automorphic L-functions, Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 27–61. MR MR546608 (81m:10056)
[BR94] Don, Blasius and Jonathan D., Rogawski, Zeta functions of Shimura varieties, Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 525–571. MR MR1265563 (95e:11051)
[Bru03] Farrell, Brumley, Maass cusp forms with quadratic integer coefficients, Int. Math.Res.Not. (2003), no. 18, 983–997. MR 1962012 (2004d:11025)Google Scholar
[BW00] A., Borel and N., Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, second edn., Mathematical Surveys and Monographs, vol. 67, American Mathematical Society, Providence, RI, 2000. MR MR1721403 (2000j:22015)
[BZ77] I. N., Bernstein and A. V., Zelevinsky, Induced representations of reductive p-adic groups. I, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 4, 441–472. MR MR0579172 (58 #28310)Google Scholar
[Car79] P., Cartier, Representations of p-adic groups: a survey, Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 111–155. MR MR546593 (81e:22029)
[CHT08] Laurent, Clozel, Michael, Harris, and Richard, Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois representations, Pub. Math. IHES 108 (2008), 1–181.Google Scholar
[Clo90] Laurent, Clozel, Motifs et formes automorphes: applications du principe de fonctorialité, Automorphic forms, Shimura varieties, and L-functions, Vol. I (Ann Arbor, MI, 1988), Perspect. Math., vol. 10, Academic Press, Boston, MA, 1990, pp. 77–159. MR MR1044819 (91k:11042)
[Gro99] Benedict H., Gross, Algebraic modular forms, Israel J. Math. 113 (1999), 61–93. MR MR1729443 (2001b:11037)Google Scholar
[Gro07] Benedict H., Gross, Odd Galois representations, preprint, 2007.
[Jan03] Jens Carsten, Jantzen, Representations of algebraic groups, second edn., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR MR2015057 (2004h:20061)
[JS81] H., Jacquet and J. A., Shalika, On Euler products and the classification of automorphic forms. II, Amer. J. Math. 103 (1981), no. 4, 777–815. MR MR623137 (82m:10050b)Google Scholar
[Kna01] Anthony W., Knapp, Representation theory of semisimple groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2001, An overview based on examples, Reprint of the 1986 original. MR MR1880691 (2002k:22011)
[Kna02] Anthony W., Knapp, Lie groups beyond an introduction, second edn., Progress in Mathematics, vol. 140, Birkhäuser Boston Inc., Boston, MA, 2002. MR MR1920389 (2003c:22001)
[Kot84] Robert E., Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. J. 51 (1984), no. 3, 611–650. MR MR757954 (85m:11080)Google Scholar
[Lan79] R. P., Langlands, On the notion of an automorphic representation, Auto-morphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 203–207.
[Lan97] R. P., Langlands, Representations of abelian algebraic groups, Pacific J. Math. (1997), Special Issue, 231–250, Olga Taussky-Todd: in memoriam. MR MR1610871 (99b:11125)Google Scholar
[Sar02] Peter, Sarnak, Maass cusp forms with integer coefficients, A panorama of number theory or the view from Baker' garden (Zürich, 1999), Cambridge University Press, Cambridge, 2002, pp. 121–127. MR 1975448 (2004c:11053)
[Ser79] Jean-Pierre, Serre, Groupes algébriques associés aux modules de Hodge-Tate, Journées de Géométrie Algébrique de Rennes. (Rennes, 1978), Vol. III, Astérisque, vol. 65, Soc. Math. France, Paris, 1979, pp. 155–188. MR MR563476 (81j:14027)
[Spr79] T. A., Springer, Reductive groups, Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–27. MR MR546587 (80h:20062)
[Tan57] Yutaka, Taniyama, L-functions of number fields and zeta functions of abelian varieties, J. Math. Soc. Japan 9 (1957), 330–366. MR 0095161 (20 #1667)Google Scholar
[Tay08] Richard, Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois representations. II, Pub. Math. IHES 108 (2008), 183–239.Google Scholar
[Tit79] J., Tits, Reductive groups over local fields, Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 29–69. MR MR546588 (80h:20064)
[Vog93] David A., Vogan Jr., The local Langlands conjecture, Representation theory of groups and algebras, Contemp. Math., vol. 145, Amer. Math. Soc., Providence, RI, 1993, pp. 305–379. MR MR1216197 (94e:22031)
[Wal81] Michel, Waldschmidt, Transcendance et exponentielles en plusieurs variables, Invent. Math. 63 (1981), no. 1, 97–127. MR MR608530 (82k:10042)Google Scholar
[Wal82] Michel, Waldschmidt, Sur certains caractères du groupe des classes d'idèles d'un corps de nombres, Seminar on Number Theory, Paris 1980-81 (Paris, 1980/1981), Progr. Math., vol. 22, Birkhäuser Boston, Boston, MA, 1982, pp. 323–335. MR MR693328 (85c:11097)
[Wal85] J.-L., Waldspurger, Quelques propriétés arithmétiques de certaines formes automorphes sur GL(2), Compositio Math. 54 (1985), no. 2, 121–171. MR MR783510 (87g:11061a)Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×