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  • 2007-06-28Useful Links and Resources on Langlands Program

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    We would like to thank the colleagues who helped us improve this site by contributing notes, providing materials and information, giving comments and suggestions, etc. Thank you for your time and efforts!   Some useful notes for the beginners:Prerequisites for the Langlands ProgramFirst Steps with the Langlands ProgramA (Very Brief) History of the Trace Formula by James ArthurSelected Readings on Langlands Program and Related Topics: 1. The work of Robert Langlands:       http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/ 2. The work of Jim Arthur http://gauss.claymath.org:8888/cw/arthur/index.phpFor a comprehensive introduction to the Arthur-Selberg trace formula, see the article by J.Arthur: An introduction to the trace formula (2005) For another introduction to the Arthur-Selberg trace formula, see Gelbart, Stephen ,  Lectures on the Arthur-Selberg trace formula . University Lecture Series, 9. American Mathematical Society, Providence, RI, 1996. x+99 pp.3. Langlands program Gelbart, Stephen , An elementary introduction to the Langlands program. Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177--219. Knapp, A. W. Introduction to the Langlands program. Representation theory and automorphic forms (Edinburgh, 1996), 245--302 Proc. Sympos. Pure Math., 61, Amer. Math. Soc., Providence, RI, 1997. download Gelbart, Stephen S. Automorphic forms on adele groups. Annals of Mathematics Studies, No. 83. Princeton University Press, Princeton, N.J.; University of  Tokyo Press, Tokyo, 1975. x+267 pp. J. Bernstein and S. Gelbart (Editors), An Introduction to the Langlands program Gelbart, Stephen ; Shahidi, Freydoon Analytic properties of automorphic $L$-functions. Perspectives in Mathematics, 6. Academic Press, Inc., Boston, MA, 1988. viii+131 Stephen S. Gelbart and Stephen D. Miller, Riemann's Zeta Function and Beyond download 4. Langlands and geometric Langlands program
    • Edward Frenkel, Ramifications of the Geometric Langlands Program download
    • Edward Frenkel, Langlands Correspondence For Loop Groups-An Introduction
      http://math.berkeley.edu/~frenkel/NEWBOOK/
    • Edward Frenkel, Lectures on the Langlands Program and Conformal Field Theory download
    • Edward Frenkel, Recent Advances in the Langlands Program download
    • Edward Frenkel, Affine Algebras, Langlands Duality and Bethe Ansatz download
    • Edward Frenkel and Dennis Gaitsgory,  Local Geometric Langlands Correspondence and Affine Kac-Moody Algebras download
    • Michele Vergne, All what I wanted to know about Langlands program and was afraid to ask download
    A. Borel and W. Casselman (Editors),  Automorphic Forms, Representations , and L-Functions, Proceedings of Symposium in Pure Mathematics,
    Vol. 33, AMS, 1979      http://www.ams.org/online_bks/pspum331/ , http://www.ams.org/online_bks/pspum332/     Here are the individual papers of the book, volume I and volume II: I. Reductive groups. Representations
    • Reductive groups by T.A. Springer download
    • Reductive groups over local fields by J. Tits download
    • Representations of reductive Lie groups by N.R.Wallach download
    • Representations of $GL_{2}(R) and GL_{2}(C)$ by A.W.Knapp download
    • Normalizing factors, tempered....by A.W. Knapp and G. Zuckerman download
    • Orbital integrals for $GL_{2}(R)$ by D. Shelstad download
    • Representations of {german p}-adic groups: A survey by P. Cartier download
    • Cuspidal unramified series....by P. Gerardin download
    • Some remarks on the supercuspidal....by G. Lusztig download
    II. Automorphic Forms and Representations
    • Decomposition of representations into tensor products by D. Flath download
    • Classical and adelic automorphic forms.... by I. Piatetski-Shapiro download
    • Automorphic forms and automorphic....by A. Borel and H. Jacquet download
    • On the notion of an automorphic representation.... by R.P. Langlands download
    • Multiplicity one theorems by I. Piatetski-Shapiro download
    • Forms of $GL(2)$ from the analytic...by S. Gelbart and H. Jacquet download
    • Eisenstein series and the trace formula by J. Arthur download
    • {Theta}-series and invariant theory by R. Howe download
    • Examples of dual reductive pairs by S. Gelbart download
    • On a relation between...cusp forms...by S. Rallis download
    • A counterexample to the "generalized ...by R. Howe and I.I. Piatetski-Shapiro download
    III. Automorphic Representations and L-functions
    • Number theoretic background by J. Tate download
    • Automorphic $L$-functions by A. Borel download
    • Principal $L$-functions of the linear group by H. Jacquet download
    • Automorphic $L$-functions for the symplectic group GSp_4 by M.E. Novodvorsky download
    • On liftings of holomorphic cusp forms by T. Shintani download
    • Orbital integrals and base change by R. Kottwitz download
    • The solution of a base change problem... by P. Gerardin and J.P. Labesse download
    • Report on the local Langlands conjecture for GL_2 by J. Tunnell download
    IV. Arithmetical Algebraic Geometry and Automorphic L-functions
    • The Hasse-Weil $\zeta$ -function of some moduli... by W. Casselman download
    • Points on Shimura varieties mod $p$ by J.S. Milne download
    • Combinatorics and Shimura varieties mod $p$... by R. Kottwitz download
    • Notes on $L$-indistinguishability... by D. Shelstad download
    • Automorphic representations, Shimura varieties, ...by R.P. Langlands download
    • Varietes de Shimura: Interpretation....by P. Deligne download
    • Congruence relations and Shimura curves by Y. Ihara download
    • Valeurs de fonctions $L$ et periodes....by P. Deligne with Appendix download
    • An introduction to Drinfeld's "Shtuka" by D.A. Kazhdan download
    • Automorphic forms on GL_2 over function fields by G. Harder and D.A. Kazhdan download
    Algebraic Groups and Discontinuous Subgroups , Editors: Armand Borel and George D. Mostow
    Available at     http://www.ams.org/online_bks/pspum9/
    • Algebraic Groups, Arithmetic Groups download
    • Arithmetic Properties of Algebraic Groups. Adele Groups download
    • Automorphic Functions and Decomposition of ... download
    • Bounded Symmetric Domains, Holomorphic Automorphic Forms, Moduli download
    • Quotients of Symmetric Spaces. Deformations download
    5. Langlands program and physics
    • Edward Witten, Gauge theory and the geometric Langlands program,
      talk at the Third Simons Workshop in Mathematics and Physics
      SUNY at Stony Brook.       download
    • Edward Witten, Gauge theory and the geometric Langlands program
      2006 Bowen Lectures at Berkeley.       slides 1 , slides 2 , slides 3 .
    • Edward Witten, Gauge theory and the geometric Langlands program
      minicourse at the 9th anual Workshop in Algebraic Geometry and Physics
      at University of Pennsylvania       Slides
    • Anton Kapustin and Edward Witten, Electric-Magnetic Duality
      And The Geometric Langlands Program       download
    • Edward Witten, Gauge Theory and Geometric Langlands
      Strings 2006, Beijing       download
    • Sergei Gukov and Edward Witten, Gauge theory, ramification, and the
      geometric Langlands program.
    • N. Hitchin, Langlands duality and G_2 spectral curves , math.AG/0611524 download
    • T. Hausel & M. Thaddeus, Mirror symmetry, Langlands duality, and the
      Hitchin system       , Invent. Math. 153 (2003) 197-229       download
    • M. Thaddeus, Mirror symmetry, Langlands duality, and commuting elements
      of Lie groups       , Int. Math. Res. Notices (2001) No. 22       download
    6. Infinite-dimensional Lie algebras Minoru Wakimoto, Lectures on infinite-dimensional Lie algebras , World Scientific, 2001.               Victor Kac, Infinite dimensional Lie algebras , Cambridge University Press, 1990.    Igor Frenkel, James Lepowsky, Arne Meurman, Vertex operator algebras and the Monster,
    Pure and Applied Mathematics, Vol. 134, Academic Press, 1988.     7. The theory of vertex operator algebras or chiral algebras    1) Yi-Zhi Huang, James Lepowsky and Lin Zhang, Logarithmic tensor product theory for
    generalized modules for a conformal vertex algebra     .Part I:     download    2) Yi-Zhi Huang, James Lepowsky and Lin Zhang, A logarithmic generalization of tensor
    product theory for modules for a vertex operator algebra     , Internat. J. Math. 17 (2006), 975-1012.     download    3) Yi-Zhi Huang, Vertex operator algebras, the Verlinde conjecture, and modular tensor categories,
    Proc. Natl. Acad. Sci. USA 102 (2005), 5352--5356.    http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=556239    4) James Lepowsky, From the representation theory of vertex operator algebras to modular tensor
    categories in conformal field theory     , Proc. Natl. Acad. Sci. USA 102(2005), 5304--5305.     http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=556255    5) Edward Frenkel and David Ben-Zvi, Vertex algebras and algebraic curves, Mathematical
    Surveys and Monographs    , Vol. 88, American Mathematical Society, 2001.     6) Alexander Beilinson and Vladimir Drinfeld, Chiral algebras, Colloquium Publications, Vol. 51, AMS, 2000.
    Preliminary version:     http://www.math.uchicago.edu/~arinkin/langlands/chiral/     7) Yi-Zhi Huang, Two-dimensional conformal geometry and vertex operator algebras,
    Progress in Mathematics, Vol. 148, Birkhauser, 1997.     8) Yi-Zhi Huang and James, Lepowsky, On the D-module and formal variable approaches to vertex algebras,
    in: Topics in Geometry: In Memory of Joseph D'Atri, ed.    9) S. Gindikin, Progress in Nonlinear Differential Equations , Vol. 20, Birkhaser, 1996, 175--202. q-alg/9603020.10) Igor B. Frenkel, Yi-Zhi Huang and James Lepowsky, On axiomatic approaches to vertex operator
    algebras and modules     , Mem. AMS, vol. 104, no. 494, 1993.    11) Igor Frenkel, James Lepowsky and Arne Meurman, Vertex operator algebras and the Monster,
    Pure and Applied Mathematics, Vol. 134, Academic Press, 1988.    12) Yi-Zhi Huang, Review of Vertex Algebras and Algebraic Curves by E. Frenkel and D. Ben-Zvi ,
    Bull. Amer. Math. Soc. * 39 *(2002), 585-591.     http://www.ams.org/journal-getitem?pii=S0273-0979-02-00955-2
    (This book review contains a brief nontechnical survey of the geometry of vertex operator algebras and conformal field theories.)    13) Yi-Zhi Huang, Riemann surfaces with boundaries and the theory of vertex operator algebras, in: /Vertex Operator Algebras
    in Mathematics and Physics/, ed. S. Berman, Y. Billig, Y.-Z. Huang and J. Lepowsky, Fields Institute Communications,
    Vol. 39, Amer. Math. Soc., Providence, 2003, 109--125. math.QA/0212308.     download    Selection recommended by Dipendra Prasad:Langlands program needs background in the theory of Algebraic groups, their representations, as well as Algebraic Number Theory and modular forms. Below are certain standard text and articles on these topics. For a beginner, this smaller list might already be quite daunting. Perhaps, then one could read the expository articles mentioned below of Knapp, Gelbart and Arthur, and then learn the relevant material in greater detail from the below mentioned books according to one's fancy.

    General exposition to Langlands program:
    A.W. Knapp: Introduction to Langlands program, Proceedings of Symposia in Pure Mathematics, vol. 61 (1997), pages 245-302.
    S. Gelbart: An Elementary Introduction to Langlands Program, Bulletin of the AMS (1984) 177-219.
    J. Arthur: Automorphic Representations and Number Theory, CMS Conference Proceeding, vol. 1, AMS (1981), pages 3-51.

    For Algebraic Groups, here are some books:
    T.A. Springer: Linear Algebraic Groups, Progress in Mathematics, vol. 9, Birkhauser.
    A. Borel: Linear Algebraic Groups, Graduate Text in Mathematics, Springer Verlag.

    For Automorphic Forms which is the central concern of the Langlands program, here is the recommended book:
    D. Bump: Automorphic Forms and Representations, Cambridge University Press, Cambridge.

    For classical theory of modular forms, and the connection to elliptic curves:
    N. Koblitz: Introduction to Elliptic curves and modular forms, Graduate Text in Mathematics, vol. 97, Springer Verlag.

    For Number Theory:
    J. Neukirch: Algebraic Number Theory, Springer Verlag 1999.
    J. Frohlich and M. Taylor: Algebraic Number Theory, Cambridge studies in Advanced Math, vol. 27, Cambridge University Press (1993).

    For articles at higher level of sophistication, there is a large collection of them in,
    Proceedings Symposia in Pure Mathematics of the AMS, vol. 33, edited by Borel and Casselman.
    some are quite readable too, and many are essential reading even now, such as the article of A. Borel on "Automorphic L-functions."    Seletion recommended by Solomon Friedberg:Connections between L-functions, multiple Dirichlet series, and analytic number theory:G. Chinta, S. Friedberg, J. Hoffstein, Multiple Dirichlet series and automorphic forms, Proc. Symp. Pure Math. 75 (2006), 3--41. download B. Brubaker, D. Bump, G. Chinta, S. Friedberg, J. Hoffstein, Weyl group multiple Dirichlet series I, Proc. Symp. Pure Math. 75 (2006), 91--114. download B. Brubaker, S. Friedberg, D. Bump, Weyl group multiple Dirichlet series II: the stable case, Inventiones Math 165 (2006), 325--355. For a .pdf file, click here. For an older .pdf version of the paper that contains a more coordinatized treatment, click here.B. Brubaker, D. Bump, S. Friedberg, J. Hoffstein, Weyl group multiple Dirichlet series III: Eisenstein series and twisted unstable A_r, to appear in Annals of Mathematics.downloadG. Chinta, S. Friedberg, J. Hoffstein, Asypmtotics for sums of twisted L-functions and applications, in: Automorphic Representations, L-functions and Applications: Progress and Prospects, Ohio State University Mathematical Research Institute Publications 11, de Gruyter, 2005, pp. 75--94.B. Brubaker, S. Friedberg, J. Hoffstein, Cubic twists of GL(2) automorphic L-functions, Inventiones Math. 160 (2005), 31--58.downloadMultiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory, edited by S. Friedberg (Managing Editor), D. Bump, D. Goldfeld, J. Hoffstein, Proceedings of Symposia in Pure Mathematics, Volume 75, American Mathematical Society (Providence, RI), 2006.Bump, Daniel The Rankin-Selberg method: an introduction and survey. Automorphic representations, $L$-functions and applications: progress and prospects, 41--73, Ohio State Univ. Math. Res. Inst. Publ., 11, de Gruyter, Berlin, 2005.Shahidi, Freydoon, Automorphic $L$-functions and functoriality. Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 655--666, Higher Ed. Press, Beijing, 2002. 11F70 (11R39 11S37)Shahidi, Freydoon, Functoriality and small eigenvalues of Laplacian on Riemann surfaces. Surveys in differential geometry. Vol. IX, 385--400, Surv. Differ. Geom., IX, Int. Press, Somerville, MA, 2004. 11F72 (11F70 11R39 11R42)Arthur, James(3-TRNT), The principle of functoriality. Mathematical challenges of the 21st century (Los Angeles, CA, 2000).Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 1, 39--53 (electronic)F.Shahid, Automorphic $L$-functions: a survey. Automorphic forms, Shimura varieties, and $L$-functions, Vol. I (Ann Arbor, MI, 1988), 415--437, Perspect. Math., 10, Academic Press, Boston, MA, 1990. [This article is necessarily out-of-date, but may still be useful as an introduction.]Cogdell, James W.; Kim, Henry H.; Murty, M.Ram, Lectures on automorphic $L$-functions. Fields Institute Monographs 20. (2004) Goldfeld, Dorian, Automorphic Forms and L-Functions for the Group GL(n,R),  Series: Cambridge Studies in Advanced Mathematics (No. 99)
    http://www.cambridge.org/us/catalogue/catalogue.asp?isbn=0521837715     Iwaniec, Henryk; Kowalski, Emmanuel, Analytic number theory. American Mathematical Society Colloquium Publications, 53. American Mathematical Society, Providence, RI, 2004. xii+615 pp. ISBN: 0-8218-3633-1Iwaniec, Henryk, Spectral methods of automorphic forms. Second edition. Graduate Studies in Mathematics, 53. American Mathematical Society, Providence, RI; Revista Matemathica Iberoamericana, Madrid, 2002. xii+220 pp. ISBN: 0-8218-3160-7Moreno, Carlos Julio, Advanced analytic number theory: $L$-functions. Mathematical Surveys and Monographs, 115. American Mathematical Society, Providence, RI, 2005. xx+291 pp. ISBN: 0-8218-3641-2.An introduction to the Langlands program. Lectures presented at the Hebrew University of Jerusalem, Jerusalem, March 12--16, 2001. Edited by Joseph Bernstein and Stephen Gelbart. Birkhauser Boston, Inc., Boston, MA, 2003. x+281 pp. ISBN: 0-8176-3211-5.Automorphic representations, $L$-functions and applications: progress and prospects. Proceedings of a conference honoring Steve Rallis on the occasion of his 60th birthday held at Ohio State University, Columbus, OH, March 27--30, 2003. Edited by James W. Cogdell, Dihua Jiang, Stephen S. Kudla, David Soudry and Robert Stanton. Ohio State University Mathematical Research Institute Publications, 11. Walter de Gruyter & Co., Berlin, 2005.Erez Lapid's lecture notes: More publications on Langlands program:
    • Roman Bezrukavnikov, Noncommutative Counterparts of the Springer Resolution download
    • A. W. Knapp , Introduction to the Langlands Program download
    • P. Mezo, Notes on the local Langlands Program download
    • Matt Szczesny, The Langlands Program and Physics download
    • I. Mirkovic and K. Vilonen, Geometric Langlands Duality and Representations
    •  of Algebraic Groups Over Commutative Rings download
    • The February 2005 Talbot workshop "Geometric Langlands retreat" download
    • J. Bernstein, Algebraic Theory of D-modules download
    • A. Beilinson and V. Drinfeld , Quantization of Hitchin's integrable system
      and Hecke eigensheaves       1-100 , 101-200 , 201-300 , 301-384
    James Arthur, The principle of functoriality. Mathematical  challenges of the 21st century (Los Angeles, CA, 2000). Bull. Amer.  Math. Soc. (N.S.) 40 (2003), no. 1, 39--53 available electronically at http://www.ams.org/journal-getitem?pii=S0273-0979-02-00963-1 On the Corvallis-Proceedings: A. Borel and W. Casselman (Editors), Automorphic Forms, Representations, and L-Functions, Proceedings of Symposium in Pure Mathematics, Vol.  33, AMS, 1979. The Corvallis-Proceedings is surely excellent and I wouldn't miss it, but in many parts it is also obsolete by now. There are more recent accounts, like the Ann Arbor Proceedings (which officially was the successor of the Corvallis-meeting) or the Edinburgh conference proceedings: Representation theory and automorphic forms. Papers from the Instructional Conference held in Edinburgh, March 17--29, 1996. Edited by T. N. Bailey and A. W. Knapp. Proceedings of Symposia in Pure Mathematics, 61. American Mathematical Society, Providence, RI; International Centre for Mathematical Sciences (ICMS), Edinburgh, 1997. viii+479 pp. ISBN: 0-8218-0609-2 Knapp, Anthony W, Representation theory of semisimple  groups. An overview based on examples. Reprint of the 1986 original. Princeton Landmarks in Mathematics. Princeton University Press,  Princeton, NJ, 2001. xx+773 pp. ISBN: 0-691-09089-0   The articles of Cartier, Arthur, Borel in the Corvallis proceedings Continue with the Edinburgh proceedings.              The information posted on this page are collected by PAMQ.
    The copyrights belong to the original sites or authors.    Last update: May 16, 2007   

     


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    Tag: 自守形式 郎兰兹纲领
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    发表于19:16:56 | 编辑 | 继续话题 | 转发 | 分享　0

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  • 下学期的代数数论课，我们都可以讨论这些东西的。可以接着张贤科教授的数论课，我们接着讲Tate's thesis，class field theory，然后我们再接着叶老师讲Notes on Jacquet-Langlands' Theory这本书。其实像Tate's thesis大家早已经学过了。关于类域论经典的内容，可以用ideal，idele，cohomology，simple algebra等语言叙述。对我们来说cohomology不是很熟悉，可以考虑看用ideal，idele的语言来叙述的书，想Lang的，和Larry Joel Goldstein的Analytic Number Theory这本书其实讲的就是代数数论。将类域论突出了L-functions的作用。去年我们讲过，不过我没有讲完。
    jgh | 发表于2007-06-30 11:07:03 [回复]
  • 大家注意看纪师兄给大家发的这个资源，尤其是附带的两篇文章。下个学期要学习代数数论，建议大家把第一篇中代数数论中的介绍仔细看一下。文章里列举的知识点是非常全面的，从基本概念，到代数数论中的基本问题；然后是为了深入了解问题定义的ARTIN L FUNCTION ；而同时HECKE定义的 HECKE L FUNCTION 恰恰是一种自守形式。然后TATE在他的博士论文中重新诠释了 HECKE L FUNCTION，使之与表示论（自守表示）联系起来了。



    由实际问题出发的ARTIN L FUNCTION，

    我们下学期重点学习自守形式的Hecke L function，

    算上已经稍有接触的自守表示，



    还有所谓的“实际问题“：class field theory。

    这个地方包含了很多精彩的知识点的。





    这些东西我只熟悉一部分，基本概念以及以后的TATE论文部分。对了CLASS FIELD THEORY 以及 ARTIN，HECHE L FUNCTION不怎么熟悉。大家下学期一起学习吧
    qhpi | 发表于2007-06-30 07:38:59 [回复]