Bulletin of the American Mathematical Society

Author(s): Stephen S. Gelbart; Stephen D. Miller
Journal: Bull. Amer. Math. Soc. 41 (2004), 59-112.
MSC (2000): Primary 11-02, 11M06, 11M41, 11F03, 30D15
Posted: October 30, 2003
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Abstract: In recent years

-functions and their analytic properties have assumed a central role in number theory and automorphic forms. In this expository article, we describe the two major methods for proving the analytic continuation and functional equations of

-functions: the method of integral representations, and the method of Fourier expansions of Eisenstein series. Special attention is paid to technical properties, such as boundedness in vertical strips; these are essential in applying the converse theorem, a powerful tool that uses analytic properties of

-functions to establish cases of Langlands functoriality conjectures. We conclude by describing striking recent results which rest upon the analytic properties of

-functions.

1.: James Arthur, The principle of functoriality, Bull. Amer. Math. Soc. (N.S.) 40 (2002), no. 1, 39-53 (electronic), Mathematical challenges of the 21st century (Los Angeles, CA, 2000).
2.: James Arthur and Stephen Gelbart, Lectures on automorphic -functions, -functions and arithmetic (Durham, 1989), London Math. Soc. Lecture Note Ser., vol. 153, Cambridge Univ. Press, Cambridge, 1991, pp. 1-59. MR 92j:11045
3.: Joseph Bernstein, Meromorphic Continuation of Eisenstein Series, IAS/Park City Lecture Notes, Park City, Utah, 2002.
4.: Joseph Bernstein and Stephen Gelbart (eds.), An Introduction to the Langlands Program, Birkhauser, Boston, 2003.
5.: B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves. II, J. Reine Angew. Math. 218 (1965), 79-108. MR 31:3419
6.: Andrew Booker, Poles of Artin -functions and the strong Artin conjecture, Ann. of Math. (2) (to appear), http://www.math.princeton.edu/ arbooker/papers.
7.: Armand Borel, Automorphic forms on ${\rm SL}\sb 2({\bf R})$ , Cambridge Tracts in Mathematics, vol. 130, Cambridge University Press, Cambridge, 1997. MR 98j:11028
8.: 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 2002d:11058
9.: Farrell Brumley, Maass cusp forms with quadratic integer coefficients, Int. Math. Res. Not. (2003), no. 18, 983-997.
10.: Daniel Bump, Automorphic forms on ${\rm GL}(3,{\bf R})$ , Lecture Notes in Mathematics, vol. 1083, Springer-Verlag, Berlin, 1984. MR 86g:11028
11.: -, The Rankin-Selberg method: a survey, Number theory, trace formulas and discrete groups (Oslo, 1987), Academic Press, Boston, MA, 1989, pp. 49-109. MR 90m:11079
12.: -, Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, vol. 55, Cambridge University Press, Cambridge, 1997. MR 97k:11080
13.: Daniel Bump and David Ginzburg, Symmetric square -functions on ${\rm GL}(r)$ , Ann. of Math. (2) 136 (1992), no. 1, 137-205. MR 93i:11058
14.: Henri Carayol, Preuve de la conjecture de Langlands locale pour ${\rm GL}\sb n$ : travaux de Harris-Taylor et Henniart, Astérisque (2000), no. 266, Exp. No. 857, 4, 191-243 (French, with French summary), Séminaire Bourbaki, Vol. 1998/99. MR 2001i:11136
15.: P. Cartier, Some numerical computations relating to automorphic functions (English), Computers in Number Theory (Oxford, 1969), Proc. Atlas Sympos. No. 2, 1971, pp. 37-48.
16.: Pierre Cartier, Des nombres premiers à la géométrie algébrique (une brève histoire de la fonction zéta), Analyse diophantienne et géométrie algébrique, Cahiers Sém. Hist. Math. Sér. 2, vol. 3, Univ. Paris VI, Paris, 1993, pp. 51-77 (French). MR 94i:11060
17.: W. Casselman, ${\rm GL}\sb n$ , Algebraic number fields: -functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), Academic Press, London, 1977, pp. 663-704. MR 58:27777
18.: W. Casselman and J. Shalika, The unramified principal series of -adic groups. II. The Whittaker function, Compositio Math. 41 (1980), no. 2, 207-231. MR 83i:22027
19.: Laurent Clozel, Spectral Theory of Automorphic Forms, IAS/Park City Lecture Notes, Park City, Utah, 2002.
20.: J. W. Cogdell, -functions and Converse Theorems for ${\rm GL}(n)$ , IAS/Park City Lecture Notes, Park City, Utah, 2002, http://www.math.okstate.edu/ cogdell/.
21.: J. W. Cogdell, H. H. Kim, I. I. Piatetski-Shapiro, and F. Shahidi, On lifting from classical groups to ${\rm GL}\sb N$ , Publ. Math. Inst. Hautes Études Sci. (2001), no. 93, 5-30. MR 2002i:11048
22.: -, Functoriality for the classical groups, preprint.
23.: J. W. Cogdell and I. I. Piatetski-Shapiro, Converse theorems for ${\rm GL}\sb n$ , Inst. Hautes Études Sci. Publ. Math. (1994), no. 79, 157-214. MR 95m:22009
24.: -, A converse theorem for ${\rm GL}\sb 4$ , Math. Res. Lett. 3 (1996), no. 1, 67-76. MR 97d:22019
25.: -, Converse theorems for ${\rm GL}\sb n$ . II, J. Reine Angew. Math. 507 (1999), 165-188. MR 2000a:22029
26.: -, Converse theorems, functoriality, and applications to number theory, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 119-128.
27.: Paul Cohen and Peter Sarnak, Notes on Eisenstein Series, Stanford University, 1980.
28.: J. Brian Conrey, The Riemann hypothesis, Notices Amer. Math. Soc. 50 (2003), no. 3, 341-353.
29.: J. B. Conrey and D. W. Farmer, An extension of Hecke's converse theorem, Internat. Math. Res. Notices (1995), no. 9, 445-463. MR 96i:11101
30.: Harold Davenport, Multiplicative number theory, 3rd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000, Revised and with a preface by Hugh L. Montgomery. MR 2001f:11001
31.: Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. (1974), no. 43, 273-307 (French). MR 49:5013
32.: W. Duke, Hyperbolic distribution problems and half-integral weight Maass forms, Invent. Math. 92 (1988), no. 1, 73-90. MR 89d:11033
33.: -, Rational points on the sphere, Ramanujan Journal, Rankin Volume (to appear), http://www.math.ucla.edu/ duke.
34.: W. Duke, J. Friedlander, and H. Iwaniec, Bounds for automorphic -functions, Invent. Math. 112 (1993), no. 1, 1-8. MR 94c:11043
35.: W. Duke and H. Iwaniec, Estimates for coefficients of -functions. I, Automorphic forms and analytic number theory (Montreal, PQ, 1989), Univ. Montréal, Montreal, QC, 1990, pp. 43-47. MR 92f:11068
36.: H. M. Edwards, Riemann's zeta function, Dover Publications Inc., Mineola, NY, 2001, Reprint of the 1974 original [Academic Press, New York; MR 57:5922]. MR 2002g:11129
37.: Edward Frenkel, Recent advances in the Langlands program, http://arxiv.org/PS_cache/math/pdf/0303/0303074.pdf.
38.: Wee Teck Gan, Benedict Gross, and Gordan Savin, Fourier coefficients of modular forms on $G\sb 2$ , Duke Math. J. 115 (2002), no. 1, 105-169.
39.: Paul B. Garrett, Decomposition of Eisenstein series: Rankin triple products, Ann. of Math. (2) 125 (1987), no. 2, 209-235. MR 88m:11033
40.: Stephen S. Gelbart, Automorphic forms on adèle groups, Princeton University Press, Princeton, N.J., 1975, Annals of Mathematics Studies, No. 83. MR 52:280
41.: Stephen Gelbart, An elementary introduction to the Langlands program, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177-219. MR 85e:11094
42.: -, Lectures on the Arthur-Selberg trace formula, University Lecture Series, vol. 9, American Mathematical Society, Providence, RI, 1996. MR 98d:22017
43.: 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 81e:10025
44.: Stephen Gelbart and Freydoon Shahidi, Analytic properties of automorphic -functions, Perspectives in Mathematics, vol. 6, Academic Press Inc., Boston, MA, 1988. MR 89f:11077
45.: -, Boundedness of automorphic -functions in vertical strips, J. Amer. Math. Soc. 14 (2001), no. 1, 79-107 (electronic). MR 2003a:11056
46.: David Ginzburg and Stephen Rallis, The exterior cube -function for ${\rm GL}(6)$ , Compositio Math. 123 (2000), no. 3, 243-272. MR 2001j:11025
47.: David Ginzburg, Stephen Rallis, and David Soudry, Generic automorphic forms on ${\rm SO}(2n+1)$ : functorial lift to ${\rm GL}(2n)$ , endoscopy, and base change, Internat. Math. Res. Notices (2001), no. 14, 729-764. MR 2002g:11065
48.: Roger Godement, Notes on Jacquet-Langlands' theory (mimeographed notes), The Institute for Advanced Study, 1970.
49.: Roger Godement and Hervé Jacquet, Zeta functions of simple algebras, Lecture Notes in Mathematics, Vol. 260, Springer-Verlag, Berlin, 1972. MR 49:7241
50.: D. Goldfeld and P. Sarnak, Sums of Kloosterman sums, Invent. Math. 71 (1983), no. 2, 243-250. MR 84e:10037
51.: I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 5th ed., Academic Press Inc., Boston, MA, 1994, Translated from the fourth Russian edition. Translation edited and with a preface by Alan Jeffrey. MR 94g:00008
52.: Benedict H. Gross, Algebraic modular forms, Israel J. Math. 113 (1999), 61-93. MR 2001b:11037
53.: Benedict H. Gross and Stephen S. Kudla, Heights and the central critical values of triple product -functions, Compositio Math. 81 (1992), no. 2, 143-209. MR 93g:11047
54.: Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of -series, Invent. Math. 84 (1986), no. 2, 225-320. MR 87j:11057
55.: H. Hamburger, Über die Funktionalgleichung der $\zeta$ -Funktion, Math. Z. 10 (1921), 240-258, 11 (1921), 224-245, 13 (1922), 283-311.
56.: Harish-Chandra, Automorphic forms on semisimple Lie groups, Notes by J. G. M. Mars. Lecture Notes in Mathematics, No. 62, Springer-Verlag, Berlin, 1968. MR 38:1216
57.: Michael Harris, On the local Langlands correspondence, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 583-597.
58.: Michael Harris and Stephen S. Kudla, The central critical value of a triple product -function, Ann. of Math. (2) 133 (1991), no. 3, 605-672. MR 93a:11043
59.: 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 2002m:11050
60.: Erich Hecke, Lectures on Dirichlet series, modular functions and quadratic forms, Vandenhoeck & Ruprecht, Göttingen, 1983, Edited by Bruno Schoeneberg; With the collaboration of Wilhelm Maak. MR 85c:11042
61.: -, Mathematische Werke, 3rd ed., Vandenhoeck & Ruprecht, Göttingen, 1983 (German), With introductory material by B. Schoeneberg, C. L. Siegel and J. Nielsen. MR 86a:01049
62.: Guy Henniart, Une preuve simple des conjectures de Langlands pour ${\rm GL}(n)$ sur un corps -adique, Invent. Math. 139 (2000), no. 2, 439-455 (French, with English summary). MR 2001e:11052
63.: -, Sur la conjecture de Langlands locale pour ${\rm GL}\sb n$ , J. Théor. Nombres Bordeaux 13 (2001), no. 1, 167-187 (French), 21st Journées Arithmétiques (Rome, 2001). MR 2002f:11178
64.: G. Henniart, Progrés rècents en fonctorialitè de Langlands, Seminaire Bourbaki Exposé 890 (Juin 2001), 890-1 to 890-21.
65.: Guy Henniart, Une caractérisation de la correspondance de Langlands locale pour ${\rm GL}(n)$ , Bull. Soc. Math. France 130 (2002), no. 4, 587-602 (French).
66.: Tamotsu Ikeda, On the gamma factor of the triple -function. I, Duke Math. J. 97 (1999), no. 2, 301-318. MR 2000a:11080
67.: Henryk Iwaniec, Fourier coefficients of modular forms of half-integral weight, Invent. Math. 87 (1987), no. 2, 385-401. MR 88b:11024
68.: -, Topics in classical automorphic forms, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, Providence, RI, 1997. MR 98e:11051
69.: -, Spectral methods of automorphic forms, 2nd ed., Graduate Studies in Mathematics, vol. 53, American Mathematical Society, Providence, RI, 2002.
70.: H. Iwaniec and P. Sarnak, Perspectives on the analytic theory of -functions, Geom. Funct. Anal. (2000), no. Special Volume, 705-741, GAFA 2000 (Tel Aviv, 1999). MR 2002b:11117
71.: Hervé Jacquet, Dirichlet series for the group ${\rm GL}(n)$ , Automorphic forms, representation theory and arithmetic (Bombay, 1979), Tata Inst. Fund. Res. Studies in Math., vol. 10, Tata Inst. Fundamental Res., Bombay, 1981, pp. 155-163. MR 83j:10032
72.: H. Jacquet and R. P. Langlands, Automorphic forms on ${\rm GL}(2)$ , Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin, 1970, http://sunsite.ubc.ca/DigitalMathArchive/Langlands/JL.html. MR 53:5481
73.: Hervé Jacquet, Ilja Iosifovitch Piatetski-Shapiro, and Joseph Shalika, Automorphic forms on ${\rm GL}(3)$ , Ann. of Math. (2) 109 (1979), no. 1 and 2, 169-258. MR 80i:10034b
74.: H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika, Rankin-Selberg convolutions, Amer. J. Math. 105 (1983), no. 2, 367-464. MR 85g:11044
75.: Hervé Jacquet and Joseph A. Shalika, A non-vanishing theorem for zeta functions of ${\rm GL}\sb{n}$ , Invent. Math. 38 (1976/77), no. 1, 1-16. MR 55:5583
76.: H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic representations. I, Amer. J. Math. 103 (1981), no. 3, 499-558, http://www.jstor.org/view/00029327/di994444/99p0113q/0. MR 82m:10050a
77.: Hervé Jacquet and Joseph Shalika, Exterior square -functions, Automorphic forms, Shimura varieties, and -functions, Vol. II (Ann Arbor, MI, 1988), Perspect. Math., vol. 11, Academic Press, Boston, MA, 1990, pp. 143-226. MR 91g:11050
78.: D. Jiang and D. Soudry, Generic representations and local Langlands reciprocity law for -adic $SO_{2n+1}$ , preprint, 2001.
79.: N. M. Katz, The work of Pierre Deligne, Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, pp. 47-52. MR 82g:01060
80.: D. Kazhdan, B. Mazur, and C.-G. Schmidt, Relative modular symbols and Rankin-Selberg convolutions, J. Reine Angew. Math. 519 (2000), 97-141. MR 2001j:11026
81.: Henry H. Kim, Langlands-Shahidi method and poles of automorphic -functions: application to exterior square -functions, Canad. J. Math. 51 (1999), no. 4, 835-849. MR 2000f:11058
82.: -, Applications of Langlands' functorial lift of odd orthogonal groups, Trans. Amer. Math. Soc. 354 (2002), no. 7, 2775-2796 (electronic). MR 2003c:22025
83.: -, Functoriality for the exterior square of ${\rm GL}\sb 4$ and the symmetric fourth of ${\rm GL}\sb 2$ , 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.
84.: Henry H. Kim and Freydoon Shahidi, Symmetric cube -functions for $\rm GL\sb 2$ are entire, Ann. of Math. (2) 150 (1999), no. 2, 645-662. MR 2000k:11065
85.: -, Functorial products for $\rm GL\sb 2\times GL\sb 3$ and functorial symmetric cube for $\rm GL\sb 2$ , C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), no. 8, 599-604 (English, with English and French summaries). MR 2002i:11049
86.: -, Cuspidality of symmetric powers with applications, Duke Math. J. 112 (2002), no. 1, 177-197. MR 2003a:11057
87.: -, Functorial products for ${\rm GL}\sb 2\times{\rm GL}\sb 3$ and the symmetric cube for ${\rm GL}\sb 2$ , Ann. of Math. (2) 155 (2002), no. 3, 837-893, With an appendix by Colin J. Bushnell and Guy Henniart.
88.: A. W. Knapp, Introduction to the Langlands program, Representation theory and automorphic forms (Edinburgh, 1996), Proc. Sympos. Pure Math., vol. 61, Amer. Math. Soc., Providence, RI, 1997, pp. 245-302. MR 99d:11123
89.: A. W. Knapp and J. D. Rogawski, Applications of the trace formula, Representation theory and automorphic forms (Edinburgh, 1996), Proc. Sympos. Pure Math., vol. 61, Amer. Math. Soc., Providence, RI, 1997, pp. 413-431. MR 98j:11105
90.: V. A. Kolyvagin and D. Yu. Logachëv, Finiteness of the Shafarevich-Tate group and the group of rational points for some modular abelian varieties, Algebra i Analiz 1 (1989), no. 5, 171-196 (Russian). MR 91c:11032
91.: Tomio Kubota, Elementary theory of Eisenstein series, Kodansha Ltd., Tokyo, 1973. MR 55:2759
92.: Laurent Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math. 147 (2002), no. 1, 1-241 (French). MR 2002m:11039
93.: Serge Lang, Algebraic number theory, 2nd ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR 95f:11085
94.: -, Complex analysis, 4th ed., Graduate Texts in Mathematics, vol. 103, Springer-Verlag, New York, 1999. MR 99i:30001
95.: R. P. Langlands, Eisenstein series, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, RI, 1966, pp. 235-252, http://sunsite.ubc.ca/DigitalMathArchive/Langlands/automorphic.html#es boulder. MR 40:2784
96.: Robert P. Langlands, Euler products, a James K. Whittemore Lecture in Mathematics given at Yale University, 1967; Yale Mathematical Monographs, 1, Yale University Press, New Haven, Conn., 1971, http://sunsite.ubc.ca/DigitalMathArchive/Langlands/automorphic.html# eulerproducts. MR 54:7387
97.: -, On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, Vol. 544, Springer-Verlag, Berlin, 1976, http://sunsite.ubc.ca/ DigitalMathArchive/Langlands/automorphic.html#esspringer. MR 58:28319
98.: R. P. Langlands, Automorphic representations, Shimura varieties, and motives. Ein Märchen, Automorphic forms, representations and -functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, RI, 1979, pp. 205-246. MR 83f:12010
99.: Robert P. Langlands, Base change for ${\rm GL}(2)$ , Annals of Mathematics Studies, vol. 96, Princeton University Press, Princeton, NJ, 1980, http://sunsite.ubc.ca/DigitalMathArchive/Langlands/basechange.html#book. MR 82a:10032
100.: R. P. Langlands, Eisenstein series, the trace formula, and the modern theory of automorphic forms, Number theory, trace formulas and discrete groups (Oslo, 1987), Academic Press, Boston, MA, 1989, pp. 125-155, http://sunsite.ubc.ca/DigitalMathArchive/Langlands/endoscopy.html#oslo. MR 90e:11077
101.: -, On the classification of irreducible representations of real algebraic groups, Representation theory and harmonic analysis on semisimple Lie groups, Math. Surveys Monogr., vol. 31, Amer. Math. Soc., Providence, RI, 1989, pp. 101-170. MR 91e:22017
102.: Robert P. Langlands, Where stands functoriality today?, Representation theory and automorphic forms (Edinburgh, 1996), Proc. Sympos. Pure Math., vol. 61, Amer. Math. Soc., Providence, RI, 1997, pp. 457-471. MR 99c:11140
103.: -, The trace formula and its applications: an introduction to the work of James Arthur, Canad. Math. Bull. 44 (2001), no. 2, 160-209. MR 2002e:22024
104.: Robert Langlands, Beyond endoscopy, Contributions to Automorphic Forms, Geometry and Number Theory: Shalika Fest 2002 (Haruzo Hida, Dinakar Ramakrishnan, and Freydoon Shahidi, eds.), to appear.
105.: Gérard Laumon, La correspondance de Langlands sur les corps de fonctions (d'après Laurent Lafforgue), Astérisque (2002), no. 276, 207-265 (French), Séminaire Bourbaki, Vol. 1999/2000. MR 2003b:11052
106.: Jian-Shu Li and Joachim Schwermer, Automorphic representations and cohomology of arithmetic groups, Challenges for the 21st century (Singapore, 2000), World Sci. Publishing, River Edge, NJ, 2001, pp. 102-137. MR 2003c:11051
107.: Alexander Lubotzky, Discrete groups, expanding graphs and invariant measures, Progress in Mathematics, vol. 125, Birkhäuser Verlag, Basel, 1994, With an appendix by Jonathan D. Rogawski. MR 96g:22018
108.: A. Lubotzky, R. Phillips, and P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988), no. 3, 261-277. MR 89m:05099
109.: W. Luo, Z. Rudnick, and P. Sarnak, On Selberg's eigenvalue conjecture, Geom. Funct. Anal. 5 (1995), no. 2, 387-401. MR 96h:11045
110.: Wenzhi Luo, Zeév Rudnick, and Peter Sarnak, On the generalized Ramanujan conjecture for ${\rm GL}(n)$ , Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996), Proc. Sympos. Pure Math., vol. 66, Amer. Math. Soc., Providence, RI, 1999, pp. 301-310. MR 2000e:11072
111.: Hans Maass, Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 121 (1949), 141-183 (German). MR 11:163c
112.: G. A. Margulis, Explicit constructions of expanders, Problemy Peredaci Informacii 9 (1973), no. 4, 71-80 (Russian). MR 58:4643
113.: -, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991. MR 92h:22021
114.: Stephen D. Miller, Cusp Forms on $SL_3(\mathbb{Z} )\backslash SL_3(\mathbb{R} )/SO_3(\mathbb{R} )$ , Ph.D. thesis, Princeton University, 1997, http://www.math.rutgers.edu/ sdmiller/thesis.html.
115.: Stephen D. Miller and Wilfried Schmid, Summation Formulas, from Poisson and Voronoi to the Present, Noncommutative Analysis, in Honor of Jacques Carmona, Progress in Mathematics, vol. 220, Birkhäuser, 2003, http://www.math.rutgers.edu/ sdmiller/voronoi.
116.: -, Automorphic Distributions, -functions, and Voronoi Summation for , http://www.math.rutgers.edu/ sdmiller/voronoi.
117.: -, Distributions and Analytic Continuation of Dirichlet Series, http://www.math.rutgers.edu/ sdmiller/voronoi.
118.: C. M $\oe$ glin 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 97d:11083
119.: L. J. Mordell, On Mr. Ramanujan's Empirical Expansions of Modular Functions, Proc. Cambridge Phil. Soc. 19 (1917), 117-124.
120.: Carlos J. Moreno, Analytic proof of the strong multiplicity one theorem, Amer. J. Math. 107 (1985), no. 1, 163-206. MR 86m:22027
121.: Goran Muic, Some results on square integrable representations; irreducibility of standard representations, Internat. Math. Res. Notices (1998), no. 14, 705-726. MR 99f:22031
122.: -, A proof of Casselman-Shahidi's conjecture for quasi-split classical groups, Canad. Math. Bull. 44 (2001), no. 3, 298-312. MR 2002f:22015
123.: Werner Müller, The trace class conjecture in the theory of automorphic forms, Ann. of Math. (2) 130 (1989), no. 3, 473-529. MR 90m:11083
124.: M. Ram Murty, Ramanujan Graphs, Journal of the Ramanujan Mathematical Society (to appear).
125.: Andrew Ogg, Modular forms and Dirichlet series, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 41:1648
126.: A. P. Ogg, A remark on the Sato-Tate conjecture, Invent. Math. 9 (1969/1970), 198-200. MR 41:3481
127.: S. J. Patterson and I. I. Piatetski-Shapiro, The symmetric-square -function attached to a cuspidal automorphic representation of ${\rm GL}\sb 3$ , Math. Ann. 283 (1989), no. 4, 551-572. MR 90d:11070
128.: R. S. Phillips and P. Sarnak, On cusp forms for co-finite subgroups of ${\rm PSL}(2,{\bf R})$ , Invent. Math. 80 (1985), no. 2, 339-364. MR 86m:11037
129.: -, The Weyl theorem and the deformation of discrete groups, Comm. Pure Appl. Math. 38 (1985), no. 6, 853-866. MR 87f:11035
130.: R. Phillips and P. Sarnak, Perturbation theory for the Laplacian on automorphic functions, J. Amer. Math. Soc. 5 (1992), no. 1, 1-32. MR 92g:11056
131.: I. I. Piatetski-Shapiro, Multiplicity one theorems, Automorphic forms, representations and -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. 209-212. MR 81m:22027
132.: I. Piatetski-Shapiro and Ravi Raghunathan, On Hamburger's theorem, Lie groups and Lie algebras: E. B. Dynkin's Seminar, Amer. Math. Soc. Transl. Ser. 2, vol. 169, Amer. Math. Soc., Providence, RI, 1995, pp. 109-120. MR 97a:11134
133.: I. Piatetski-Shapiro and Stephen Rallis, Rankin triple functions, Compositio Math. 64 (1987), no. 1, 31-115. MR 89k:11037
134.: I. Piatetski-Shapiro and S. Rallis, A new way to get Euler products, J. Reine Angew. Math. 392 (1988), 110-124. MR 90c:11032
135.: I. Piatetski-Shapiro, S. Rallis, and G. Schiffmann, functions for the group $G\sb 2$ , Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 2, 389-399. MR 91a:11023
136.: -, Rankin-Selberg integrals for the group $G\sb 2$ , Amer. J. Math. 114 (1992), no. 6, 1269-1315. MR 93m:22022
137.: I. I. Pjateckij-Sapiro, On the Weil-Jacquet-Langlands theorem, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), Halsted, New York, 1975, pp. 583-595. MR 53:10719
138.: Ravi Raghunathan, A converse theorem for Dirichlet series with poles, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), no. 3, 231-235 (English, with English and French summaries). MR 99i:11034
139.: Dinakar Ramakrishnan, On the coefficients of cusp forms, Math. Res. Lett. 4 (1997), no. 2-3, 295-307. MR 98e:11064
140.: -, Modularity of the Rankin-Selberg -series, and multiplicity one for ${\rm SL}(2)$ , Ann. of Math. (2) 152 (2000), no. 1, 45-111. MR 2001g:11077
141.: -, Existence of Ramanujan primes for , Contributions to Automorphic Forms, Geometry and Number Theory: Shalika Fest 2002 (Haruzo Hida, Dinakar Ramakrishnan, and Freydoon Shahidi, eds.), to appear.
142.: Dinakar Ramakrishnan and Robert J. Valenza, Fourier analysis on number fields, Graduate Texts in Mathematics, vol. 186, Springer-Verlag, New York, 1999. MR 2000d:11002
143.: R. A. Rankin, Contributions to the theory of Ramanujan's function $\tau(n)$ and similar arithmetical functions. I. The zeros of the function $\sum\sp \infty\sb {n=1}\tau(n)/n\sp s$ on the line ${\mathfrak R}s=13/2$ . II. The order of the Fourier coefficients of integral modular forms, Proc. Cambridge Philos. Soc. 35 (1939), 351-372. MR 1:69d
144.: Michael J. Razar, Modular forms for $G\sb{0}(N)$ and Dirichlet series, Trans. Amer. Math. Soc. 231 (1977), no. 2, 489-495. MR 56:2926
145.: B. Riemann, Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Mon. Not. Berlin Akad (Nov. 1859), 671-680. See also http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/.
146.: Jonathan D. Rogawski, Automorphic representations of unitary groups in three variables, Annals of Mathematics Studies, vol. 123, Princeton University Press, Princeton, NJ, 1990. MR 91k:22037
147.: -, Functoriality and the Artin conjecture, Representation theory and automorphic forms (Edinburgh, 1996), Proc. Sympos. Pure Math., vol. 61, Amer. Math. Soc., Providence, RI, 1997, pp. 331-353. MR 98j:11106
148.: P. Sarnak, On cusp forms. II, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part II (Ramat Aviv, 1989), Israel Math. Conf. Proc., vol. 3, Weizmann, Jerusalem, 1990, pp. 237-250. MR 93e:11068
149.: Peter Sarnak, Some applications of modular forms, Cambridge Tracts in Mathematics, vol. 99, Cambridge University Press, Cambridge, 1990. MR 92k:11045
150.: -, Selberg's eigenvalue conjecture, Notices Amer. Math. Soc. 42 (1995), no. 11, 1272-1277. MR 97c:11059
151.: -, Letter to Stephen Gelbart and Freydoon Shahidi, 2001.
152.: -, Maass cusp forms with integer coefficients, A Panorama of Number Theory or The View from Baker's Garden (G. Wüstholz, ed.), Cambridge University Press, 2002, pp. 121-128.
153.: -, Nonvanishing of -functions on Re(s)=1, Contributions to Automorphic Forms, Geometry and Number Theory: Shalika Fest 2002 (Haruzo Hida, Dinakar Ramakrishnan, and Freydoon Shahidi, eds.), to appear.
154.: -, Spectra of hyperbolic surfaces, Bull. Amer. Math. Soc. (N.S.) 40 (2003), 441-478, http://www.math.princeton.edu/ sarnak.
155.: Atle Selberg, Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist, Arch. Math. Naturvid. 43 (1940), 47-50 (German). MR 2:88a
156.: A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47-87. MR 19:531g
157.: Atle Selberg, Discontinuous groups and harmonic analysis, Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, pp. 177-189. MR 31:372
158.: -, On the estimation of Fourier coefficients of modular forms, Proc. Sympos. Pure Math., Vol. VIII, Amer. Math. Soc., Providence, RI, 1965, pp. 1-15. MR 32:93
159.: Jean-Pierre Serre, Abelian -adic representations and elliptic curves, McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 41:8422
160.: J-P. Serre, A course in arithmetic, Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York, 1973, translated from the French. MR 49:8956
161.: Freydoon Shahidi, On certain -functions, Amer. J. Math. 103 (1981), no. 2, 297-355. MR 82i:10030
162.: -, Local coefficients as Artin factors for real groups, Duke Math. J. 52 (1985), no. 4, 973-1007. MR 87m:11049
163.: -, On the Ramanujan conjecture and finiteness of poles for certain -functions, Ann. of Math. (2) 127 (1988), no. 3, 547-584. MR 89h:11021
164.: -, A proof of Langlands' conjecture on Plancherel measures; complementary series for -adic groups, Ann. of Math. (2) 132 (1990), no. 2, 273-330. MR 91m:11095
165.: -, Symmetric power -functions for ${\rm GL}(2)$ , Elliptic curves and related topics, CRM Proc. Lecture Notes, vol. 4, Amer. Math. Soc., Providence, RI, 1994, pp. 159-182. MR 95c:11066
166.: -, Intertwining Operators, -functions, and Representation Theory, Lecture Notes of the Elevent KAIST Mathematics Workshop (Ja Kyung Koo, ed.), 1996, pp. 1-63, http://www.math.rutgers.edu/ sdmiller/L-functions/.
167.: -, Automorphic -functions and functoriality, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 655-666.
168.: -, Langlands-Shahidi Method and Converse Theorems, IAS/Park City Lecture Notes, Park City, Utah, 2002.
169.: J. A. Shalika, The multiplicity one theorem for ${\rm GL}\sb{n}$ , Ann. of Math. (2) 100 (1974), 171-193. MR 50:545
170.: Goro Shimura, On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. (3) 31 (1975), no. 1, 79-98. MR 52:3064
171.: Takuro Shintani, On an explicit formula for class- ``Whittaker functions'' on $GL\sb{n}$ over -adic fields, Proc. Japan Acad. 52 (1976), no. 4, 180-182. MR 53:10991
172.: Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 199?, Corrected reprint of the 1986 original. MR 95m:11054
173.: John T. Tate, Algebraic cycles and poles of zeta functions, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), Harper & Row, New York, 1965, pp. 93-110. MR 37:1371
174.: 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 36:121
175.: Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553-572. MR 96d:11072
176.: E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press Oxford University Press, New York, 1986, Edited and with a preface by D. R. Heath-Brown. MR 88c:11049
177.: Jerrold Tunnell, Artin's conjecture for representations of octahedral type, Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 2, 173-175. MR 82j:12015
178.: André Weil, Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168 (1967), 149-156 (German). MR 34:7473
179.: -, On Eisenstein's copy of the Disquisitiones, Algebraic number theory, Adv. Stud. Pure Math., vol. 17, Academic Press, Boston, MA, 1989, pp. 463-469. MR 91m:01016
180.: -, Prehistory of the zeta-function, Number theory, trace formulas and discrete groups (Oslo, 1987), Academic Press, Boston, MA, 1989, pp. 1-9. MR 90e:01029
181.: Andrew Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443-551. MR 96d:11071
182.: Scott A. Wolpert, Spectral limits for hyperbolic surfaces. I, II, Invent. Math. 108 (1992), no. 1, 67-89, 91-129. MR 93b:58160
183.: -, Disappearance of cusp forms in special families, Ann. of Math. (2) 139 (1994), no. 2, 239-291. MR 95e:11062

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 11-02, 11M06, 11M41, 11F03, 30D15

Stephen S. Gelbart
Affiliation: Faculty of Mathematics and Computer Science, Nicki and J. Ira Harris Professorial Chair, The Weizmann Institute of Science, Rehovot 76100, Israel
Email: gelbar@wisdom.weizmann.ac.il

Stephen D. Miller
Affiliation: Department of Mathematics, Hill Center-Busch Campus, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8109
Email: miller@math.rutgers.edu

DOI: 10.1090/S0273-0979-03-00995-9
PII: S 0273-0979(03)00995-9
Received by editor(s): July 15, 2002,
Received by editor(s) in revised form: September 8, 2003
Posted: October 30, 2003
Dedicated: Dedicated to Ilya Piatetski-Shapiro, with admiration
Additional Notes: The first author was partially supported by the Minerva Foundation, and the second author was supported by NSF grant DMS-0122799
Copyright of article: Copyright 2003, American Mathematical Society

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