Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

Book Review

The AMS does not provide abstracts of book reviews. You may download the entire review from the links below.


Full text of review: PDF   This review is available free of charge.
Book Information:

Authors: Dorian Goldfeld and with exercises and a preface by Xander Faber Joseph Hundley
Title: Automorphic representations and L-functions for the general linear group. Volume I
Additional book information: Cambridge Studies in Advanced Mathematics, Vol. 129, Cambridge University Press, Cambridge, 2011, xx+550 pp., ISBN 978-0-521-47423-8, US $105.00

Authors: Dorian Goldfeld and with exercises and a preface by Xander Faber Joseph Hundley
Title: Automorphic representations and L-functions for the general linear group. Volume II
Additional book information: Cambridge Studies in Advanced Mathematics, Vol. 130, Cambridge University Press, Cambridge, 2011, xx+188 pp., ISBN 978-1-107-00794-4, US $82.00

References [Enhancements On Off] (What's this?)

  • [1] James Arthur, The work of Ngô Bao Châu, Proceedings of the International Congress of Mathematicians. Volume I, Hindustan Book Agency, New Delhi, 2010, pp. 57-70. MR 2827882
  • [2] James Arthur and Laurent Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Annals of Mathematics Studies, vol. 120, Princeton University Press, Princeton, NJ, 1989. MR 1007299 (90m:22041)
  • [3] Emil Artin and John Tate, Class field theory, AMS Chelsea Publishing, Providence, RI, 2009. Reprinted with corrections from the 1967 original. MR 2467155 (2009k:11001)
  • [4] Mahdi Asgari and Freydoon Shahidi, Generic transfer for general spin groups, Duke Math. J. 132 (2006), no. 1, 137-190. MR 2219256 (2007d:11055a), https://doi.org/10.1215/S0012-7094-06-13214-3
  • [5] Mahdi Asgari and Freydoon Shahidi, Generic transfer from $ \rm GSp(4)$ to $ \rm GL(4)$, Compos. Math. 142 (2006), no. 3, 541-550. MR 2231191 (2007d:11055b), https://doi.org/10.1112/S0010437X06001904
  • [6] 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 546608 (81m:10056)
  • [7] Armand Borel and W. Casselman (eds.), Automorphic forms, representations and $ L$-functions. Part 1, Proceedings of Symposia in Pure Mathematics, XXXIII, Providence, R.I., American Mathematical Society, 1979. MR 546586 (80g:10002a)
  • [8] Armand Borel and W. Casselman (eds.), Automorphic forms, representations, and $ L$-functions. Part 2, Proceedings of Symposia in Pure Mathematics, XXXIII, Providence, R.I., American Mathematical Society, 1979. MR 546606 (80g:10002b)
  • [9] Armand Borel and George D. Mostow (eds.), Proceedings of Symposia in Pure Mathematics. Vol. IX: Algebraic groups and discontinuous subgroups, Proceedings of the Symposium in Pure Mathematics of the American Mathematical Society held at the University of Colorado, Boulder, Colorado (July 5-August 6, vol. 1965, American Mathematical Society, Providence, R.I., 1966. MR 0202512 (34:2381)
  • [10] Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, On the modularity of elliptic curves over $ \mathbf {Q}$: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843-939 (electronic). MR 1839918 (2002d:11058), https://doi.org/10.1090/S0894-0347-01-00370-8
  • [11] D. Bump, J. W. Cogdell, E. de Shalit, D. Gaitsgory, E. Kowalski, and S. S. Kudla, An introduction to the Langlands program, Birkhäuser Boston Inc., Boston, MA, 2003. Lectures presented at the Hebrew University of Jerusalem, Jerusalem, March 12-16, 2001; Edited by Joseph Bernstein and Stephen Gelbart. MR 1990371 (2004g:11037)
  • [12] Daniel Bump, The Rankin-Selberg method: a survey, Number theory, trace formulas and discrete groups (Oslo, 1987) Academic Press, Boston, MA, 1989, pp. 49-109. MR 993311 (90m:11079)
  • [13] Daniel Bump, Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, vol. 55, Cambridge University Press, Cambridge, 1997. MR 1431508 (97k:11080)
  • [14] Daniel Bump, The Rankin-Selberg method: an introduction and survey, Automorphic representations, $ L$-functions and applications: progress and prospects, Ohio State Univ. Math. Res. Inst. Publ., vol. 11, de Gruyter, Berlin, 2005, pp. 41-73. MR 2192819 (2006k:11097)
  • [15] William Casselman, Introduction to the representation theory of $ p$-adic groups, http://www. math.ubc.ca/~cass/research/pdf/p-adic-book.pdf.
  • [16] Laurent Clozel, Michael Harris, and Richard Taylor, Automorphy for some $ l$-adic lifts of automorphic mod $ l$ Galois representations, Publ. Math. Inst. Hautes Études Sci. (2008), no. 108, 1-181, With Appendix A, summarizing unpublished work of Russ Mann, and Appendix B by Marie-France Vignéras. MR 2470687 (2010j:11082)
  • [17] J. W. Cogdell, Dual groups and Langlands functoriality, An introduction to the Langlands program (Jerusalem, 2001) Birkhäuser Boston, Boston, MA, 2003, pp. 251-268. MR 1990382
  • [18] J. W. Cogdell, H. H. Kim, I. I. Piatetski-Shapiro, and F. Shahidi, On lifting from classical groups to $ {\rm GL}_N$, Publ. Math. Inst. Hautes Études Sci. 93 (2001), 5-30. MR 1863734 (2002i:11048), https://doi.org/10.1007/s10240-001-8187-z
  • [19] J. W. Cogdell, H. H. Kim, I. I. Piatetski-Shapiro, and F. Shahidi, Functoriality for the classical groups, Publ. Math. Inst. Hautes Études Sci. 99 (2004), 163-233. MR 2075885 (2006a:22010), https://doi.org/10.1007/s10240-004-0020-z
  • [20] J. W. Cogdell and I. I. Piatetski-Shapiro, Converse theorems for $ {\rm GL}_n$, Inst. Hautes Études Sci. Publ. Math. 79 (1994), 157-214. MR 1307299 (95m:22009)
  • [21] J. W. Cogdell and I. I. Piatetski-Shapiro, Converse theorems for $ {\rm GL}_n$. II, J. Reine Angew. Math. 507 (1999), 165-188. MR 1670207 (2000a:22029), https://doi.org/10.1515/crll.1999.507.165
  • [22] James W. Cogdell, $ L$-functions and converse theorems for $ {\rm GL}_n$, Automorphic forms and applications, IAS/Park City Math. Ser., vol. 12, Amer. Math. Soc., Providence, RI, 2007, pp. 97-177. MR 2331345 (2008e:11060)
  • [23] James W. Cogdell, Henry H. Kim, and M. Ram Murty, Lectures on automorphic $ L$-functions, Fields Institute Monographs, vol. 20, American Mathematical Society, Providence, RI, 2004. MR 2071722 (2005h:11104)
  • [24] Wee Teck Gan and Shuichiro Takeda, The local Langlands conjecture for $ {\rm GSp}(4)$, Ann. of Math. (2) 173 (2011), no. 3, 1841-1882. MR 2800725 (2012c:22019), https://doi.org/10.4007/annals.2011.173.3.12
  • [25] Stephen Gelbart and Hervé Jacquet, A relation between automorphic representations of $ {\rm GL}(2)$ and $ {\rm GL}(3)$, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 471-542. MR 533066 (81e:10025)
  • [26] Stephen Gelbart, Ilya Piatetski-Shapiro, and Stephen Rallis, Explicit constructions of automorphic $ L$-functions, Lecture Notes in Mathematics, vol. 1254, Springer-Verlag, Berlin, 1987. MR 892097 (89k:11038)
  • [27] Stephen Gelbart and Freydoon Shahidi, Analytic properties of automorphic $ L$-functions, Perspectives in Mathematics, vol. 6, Academic Press Inc., Boston, MA, 1988. MR 951897 (89f:11077)
  • [28] Stephen S. Gelbart, Automorphic forms on adèle groups, Princeton University Press, Princeton, N.J., 1975. Annals of Mathematics Studies, No. 83. MR 0379375 (52 #280)
  • [29] I. M. Gelfand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Representation theory and automorphic functions, Generalized Functions, vol. 6, Academic Press Inc., Boston, MA, 1990. Translated from the Russian by K. A. Hirsch; Reprint of the 1969 edition. MR 1071179 (91g:11052)
  • [30] Roger Godement and Hervé Jacquet, Zeta functions of simple algebras, Lecture Notes in Mathematics, Vol. 260, Springer-Verlag, Berlin, 1972. MR 0342495 (49 #7241)
  • [31] Michael Harris, Nick Shepherd-Barron, and Richard Taylor, A family of Calabi-Yau varieties and potential automorphy, Ann. of Math. (2) 171 (2010), no. 2, 779-813. MR 2630056 (2011g:11106), https://doi.org/10.4007/annals.2010.171.779
  • [32] Michael Harris and Richard Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001. With an appendix by Vladimir G. Berkovich. MR 1876802 (2002m:11050)
  • [33] Guy Henniart, Une preuve simple des conjectures de Langlands pour $ {\rm GL}(n)$ sur un corps $ p$-adique, Invent. Math. 139 (2000), no. 2, 439-455 (French, with English summary). MR 1738446 (2001e:11052), https://doi.org/10.1007/s002220050012
  • [34] H. Jacquet and R. P. Langlands, Automorphic forms on $ {\rm GL}(2)$, Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin, 1970. MR 0401654 (53 #5481)
  • [35] Lizhen Ji, Kefeng Liu, Shing-Tung Yau, and Zhu-Jun Zheng (eds.), Automorphic forms and the Langlands program, Advanced Lectures in Mathematics (ALM), vol. 9, International Press, Somerville, MA, 2010, Including papers from the International Conference ``Langlands and Geometric Langlands Program'' held in Guangzhou, June 18-21, 2007. MR 2640472 (2011i:22003)
  • [36] Henry H. Kim, Functoriality for the exterior square of $ {\rm GL}_4$ and the symmetric fourth of $ {\rm GL}_2$, J. Amer. Math. Soc. 16 (2003), no. 1, 139-183 (electronic). With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. MR 1937203 (2003k:11083), https://doi.org/10.1090/S0894-0347-02-00410-1
  • [37] Henry H. Kim and Freydoon Shahidi, Functorial products for $ {\rm GL}_2\times {\rm GL}_3$ and the symmetric cube for $ {\rm GL}_2$, Ann. of Math. (2) 155 (2002), no. 3, 837-893. With an appendix by Colin J. Bushnell and Guy Henniart. MR 1923967 (2003m:11075), https://doi.org/10.2307/3062134
  • [38] Laurent Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math. 147 (2002), no. 1, 1-241 (French, with English and French summaries). MR 1875184 (2002m:11039), https://doi.org/10.1007/s002220100174
  • [39] R. P. Langlands, Problems in the theory of automorphic forms, Lectures in modern analysis and applications, III, Springer, Berlin, 1970, pp. 18-61. Lecture Notes in Math., Vol. 170. MR 0302614 (46 #1758)
  • [40] Robert P. Langlands, Euler products, Yale University Press, New Haven, Conn., 1971. A James K. Whittemore Lecture in Mathematics given at Yale University, 1967; Yale Mathematical Monographs, 1. MR 0419366 (54 #7387)
  • [41] Robert P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, Vol. 544, Springer-Verlag, Berlin, 1976. MR 0579181 (58:28319)
  • [42] Robert P. Langlands, Base change for $ {\rm GL}(2)$, Annals of Mathematics Studies, vol. 96, Princeton University Press, Princeton, N.J., 1980. MR 574808 (82a:10032)
  • [43] C. Mœglin and J.-L. Waldspurger, Spectral decomposition and Eisenstein series, Cambridge Tracts in Mathematics, vol. 113, Cambridge University Press, Cambridge, 1995, Une paraphrase de l'Écriture [A paraphrase of Scripture]. MR 1361168 (97d:11083)
  • [44] Bao Châu Ngô, Le lemme fondamental pour les algèbres de Lie, Publ. Math. Inst. Hautes Études Sci. 111 (2010), 1-169 (French). MR 2653248 (2011h:22011), https://doi.org/10.1007/s10240-010-0026-7
  • [45] Stephen Rallis, Langlands' functoriality and the Weil representation, Amer. J. Math. 104 (1982), no. 3, 469-515. MR 658543 (84c:10025), https://doi.org/10.2307/2374151
  • [46] Dinakar Ramakrishnan, Modularity of the Rankin-Selberg $ L$-series, and multiplicity one for $ {\rm SL}(2)$, Ann. of Math. (2) 152 (2000), no. 1, 45-111. MR 1792292 (2001g:11077), https://doi.org/10.2307/2661379
  • [47] Jonathan D. Rogawski, Automorphic representations of unitary groups in three variables, Annals of Mathematics Studies, vol. 123, Princeton University Press, Princeton, NJ, 1990. MR 1081540 (91k:22037)
  • [48] Yiannis Sakellaridis, Spherical varieties and integral representations of L-functions., Algebra and Number Theory 6 (2012), no. 4, 611-667. MR 1333035 (96d:11071)
  • [49] Peter Sarnak and Freydoon Shahidi (eds.), Automorphic forms and applications, IAS/Park City Mathematics Series, vol. 12, American Mathematical Society, Providence, RI, 2007, Lecture notes from the IAS/Park City Mathematics Institute held in Park City, UT, July 1-20, 2002. MR 2331351 (2008b:00004)
  • [50] Freydoon Shahidi, Eisenstein series and automorphic $ L$-functions, American Mathematical Society Colloquium Publications, vol. 58, American Mathematical Society, Providence, RI, 2010. MR 2683009 (2012d:11119)
  • [51] J. T. Tate, Fourier analysis in number fields, and Hecke's zeta-functions, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 305-347. MR 0217026 (36 #121)
  • [52] Richard Taylor, Automorphy for some $ l$-adic lifts of automorphic mod $ l$ Galois representations. II, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 183-239. MR 2470688 (2010j:11085), https://doi.org/10.1007/s10240-008-0015-2
  • [53] Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553-572. MR 1333036 (96d:11072), https://doi.org/10.2307/2118560
  • [54] Andrew Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443-551. MR 1333035 (96d:11071), https://doi.org/10.2307/2118559

Review Information:

Reviewer: Ramin Takloo-Bighash
Affiliation: University of Illinois at Chicago
Email: rtakloo@math.uic.edu
Journal: Bull. Amer. Math. Soc. 50 (2013), 645-654
MSC (2010): Primary 11F66, 11F41, 11F70
DOI: https://doi.org/10.1090/S0273-0979-2013-01399-5
Published electronically: January 17, 2013
Additional Notes: While writing this article the author was partially supported by a grant from the National Security Agency (Award number 111011) and a grant from the Simons Foundation (Award number 245977). I wish to thank Dorian Goldfeld, Joseph Hundley, Peter Kuchment, Kimball Martin, Dipendra Prasad, and Yiannis Sakellaridis for useful communications.
Review copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
American Mathematical Society