Riemann’s zeta function and beyond
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Abstract:
In recent years $L$-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 $L$-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 $L$-functions to establish cases of Langlands functoriality conjectures. We conclude by describing striking recent results which rest upon the analytic properties of $L$-functions.References
-
arthur 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).
- James Arthur and Stephen Gelbart, Lectures on automorphic $L$-functions, $L$-functions and arithmetic (Durham, 1989) London Math. Soc. Lecture Note Ser., vol. 153, Cambridge Univ. Press, Cambridge, 1991, pp. 1–59. MR 1110389, DOI 10.1017/CBO9780511526053.003 BernsteinPCMI Joseph Bernstein, Meromorphic Continuation of Eisenstein Series, IAS/Park City Lecture Notes, Park City, Utah, 2002. Bernstein-Gelbart Joseph Bernstein and Stephen Gelbart (eds.), An Introduction to the Langlands Program, Birkhauser, Boston, 2003.
- B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves. II, J. Reine Angew. Math. 218 (1965), 79–108. MR 179168, DOI 10.1515/crll.1965.218.79 Booker Andrew Booker, Poles of Artin $L$-functions and the strong Artin conjecture, Ann. of Math. (2) (to appear), http://www.math.princeton.edu/~arbooker/papers.
- Armand Borel, Automorphic forms on $\textrm {SL}_2(\textbf {R})$, Cambridge Tracts in Mathematics, vol. 130, Cambridge University Press, Cambridge, 1997. MR 1482800, DOI 10.1017/CBO9780511896064
- 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. MR 1839918, DOI 10.1090/S0894-0347-01-00370-8 Brumley Farrell Brumley, Maass cusp forms with quadratic integer coefficients, Int. Math. Res. Not. (2003), no. 18, 983–997.
- Daniel Bump, Automorphic forms on $\textrm {GL}(3,\textbf {R})$, Lecture Notes in Mathematics, vol. 1083, Springer-Verlag, Berlin, 1984. MR 765698, DOI 10.1007/BFb0100147
- 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
- Daniel Bump, Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, vol. 55, Cambridge University Press, Cambridge, 1997. MR 1431508, DOI 10.1017/CBO9780511609572
- Daniel Bump and David Ginzburg, Symmetric square $L$-functions on $\textrm {GL}(r)$, Ann. of Math. (2) 136 (1992), no. 1, 137–205. MR 1173928, DOI 10.2307/2946548
- Henri Carayol, Preuve de la conjecture de Langlands locale pour $\textrm {GL}_n$: travaux de Harris-Taylor et Henniart, Astérisque 266 (2000), Exp. No. 857, 4, 191–243 (French, with French summary). Séminaire Bourbaki, Vol. 1998/99. MR 1772675 cartier-numeric 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.
- 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 1240754, DOI 10.1097/00006216-199301810-00010
- W. Casselman, $\textrm {GL}_{n}$, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 663–704. MR 0562502
- Olga Taussky, An algebraic property of Laplace’s differential equation, Quart. J. Math. Oxford Ser. 10 (1939), 99–103. MR 83, DOI 10.1093/qmath/os-10.1.99 clozel Laurent Clozel, Spectral Theory of Automorphic Forms, IAS/Park City Lecture Notes, Park City, Utah, 2002. Cogdell J. W. Cogdell, $L$-functions and Converse Theorems for $\textrm {GL}(n)$, IAS/Park City Lecture Notes, Park City, Utah, 2002, http://www.math.okstate.edu/~cogdell/.
- J. W. Cogdell, H. H. Kim, I. I. Piatetski-Shapiro, and F. Shahidi, On lifting from classical groups to $\textrm {GL}_N$, Publ. Math. Inst. Hautes Études Sci. 93 (2001), 5–30. MR 1863734, DOI 10.1007/s10240-001-8187-z C-K-PS-S2 —, Functoriality for the classical groups, preprint.
- J. W. Cogdell and I. I. Piatetski-Shapiro, Converse theorems for $\textrm {GL}_n$, Inst. Hautes Études Sci. Publ. Math. 79 (1994), 157–214. MR 1307299, DOI 10.1007/BF02698889
- J. W. Cogdell and I. I. Piatetski-Shapiro, A converse theorem for $\textrm {GL}_4$, Math. Res. Lett. 3 (1996), no. 1, 67–76. MR 1393384, DOI 10.4310/MRL.1996.v3.n1.a7
- J. W. Cogdell and I. I. Piatetski-Shapiro, Converse theorems for $\textrm {GL}_n$. II, J. Reine Angew. Math. 507 (1999), 165–188. MR 1670207, DOI 10.1515/crll.1999.507.165 cogdell-icm —, 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. Cohen-Sarnak Paul Cohen and Peter Sarnak, Notes on Eisenstein Series, Stanford University, 1980. MR1954010 J. Brian Conrey, The Riemann hypothesis, Notices Amer. Math. Soc. 50 (2003), no. 3, 341–353.
- J. B. Conrey and D. W. Farmer, An extension of Hecke’s converse theorem, Internat. Math. Res. Notices 9 (1995), 445–463. MR 1360623, DOI 10.1155/S1073792895000328
- 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 1790423
- Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 340258, DOI 10.1007/BF02684373
- W. Duke, Hyperbolic distribution problems and half-integral weight Maass forms, Invent. Math. 92 (1988), no. 1, 73–90. MR 931205, DOI 10.1007/BF01393993 Duke-Rankin —, Rational points on the sphere, Ramanujan Journal, Rankin Volume (to appear), http://www.math.ucla.edu/~duke.
- W. Duke, J. Friedlander, and H. Iwaniec, Bounds for automorphic $L$-functions, Invent. Math. 112 (1993), no. 1, 1–8. MR 1207474, DOI 10.1007/BF01232422
- W. Duke and H. Iwaniec, Estimates for coefficients of $L$-functions. I, Automorphic forms and analytic number theory (Montreal, PQ, 1989) Univ. Montréal, Montreal, QC, 1990, pp. 43–47. MR 1111010
- S. Bergmann and J. Marcinkiewicz, Sur les fonctions analytiques de deux variables complexes, Fund. Math. 33 (1939), 75–94 (French). MR 57, DOI 10.4064/fm-33-1-75-94 edfrenkel Edward Frenkel, Recent advances in the Langlands program, http://arxiv.org/PS_cache/math/pdf/0303/0303074.pdf. MR1932327 Wee Teck Gan, Benedict Gross, and Gordan Savin, Fourier coefficients of modular forms on $G_ 2$, Duke Math. J. 115 (2002), no. 1, 105–169.
- Paul B. Garrett, Decomposition of Eisenstein series: Rankin triple products, Ann. of Math. (2) 125 (1987), no. 2, 209–235. MR 881269, DOI 10.2307/1971310
- Stephen S. Gelbart, Automorphic forms on adèle groups, Annals of Mathematics Studies, No. 83, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1975. MR 0379375, DOI 10.1515/9781400881611
- Stephen Gelbart, An elementary introduction to the Langlands program, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219. MR 733692, DOI 10.1090/S0273-0979-1984-15237-6
- Stephen Gelbart, Lectures on the Arthur-Selberg trace formula, University Lecture Series, vol. 9, American Mathematical Society, Providence, RI, 1996. MR 1410260, DOI 10.1090/ulect/009
- Stephen Gelbart and Hervé Jacquet, A relation between automorphic representations of $\textrm {GL}(2)$ and $\textrm {GL}(3)$, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 471–542. MR 533066, DOI 10.24033/asens.1355
- Stephen Gelbart and Freydoon Shahidi, Analytic properties of automorphic $L$-functions, Perspectives in Mathematics, vol. 6, Academic Press, Inc., Boston, MA, 1988. MR 951897
- Stephen Gelbart and Freydoon Shahidi, Boundedness of automorphic $L$-functions in vertical strips, J. Amer. Math. Soc. 14 (2001), no. 1, 79–107. MR 1800349, DOI 10.1090/S0894-0347-00-00351-9
- David Ginzburg and Stephen Rallis, The exterior cube $L$-function for $\textrm {GL}(6)$, Compositio Math. 123 (2000), no. 3, 243–272. MR 1795291, DOI 10.1023/A:1002461508749
- David Ginzburg, Stephen Rallis, and David Soudry, Generic automorphic forms on $\textrm {SO}(2n+1)$: functorial lift to $\textrm {GL}(2n)$, endoscopy, and base change, Internat. Math. Res. Notices 14 (2001), 729–764. MR 1846354, DOI 10.1155/S1073792801000381 Godemondsnotes Roger Godement, Notes on Jacquet-Langlands’ theory (mimeographed notes), The Institute for Advanced Study, 1970.
- Roger Godement and Hervé Jacquet, Zeta functions of simple algebras, Lecture Notes in Mathematics, Vol. 260, Springer-Verlag, Berlin-New York, 1972. MR 0342495, DOI 10.1007/BFb0070263
- D. Goldfeld and P. Sarnak, Sums of Kloosterman sums, Invent. Math. 71 (1983), no. 2, 243–250. MR 689644, DOI 10.1007/BF01389098
- I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 5th ed., Academic Press, Inc., Boston, MA, 1994. Translation edited and with a preface by Alan Jeffrey. MR 1243179
- Benedict H. Gross, Algebraic modular forms, Israel J. Math. 113 (1999), 61–93. MR 1729443, DOI 10.1007/BF02780173
- Benedict H. Gross and Stephen S. Kudla, Heights and the central critical values of triple product $L$-functions, Compositio Math. 81 (1992), no. 2, 143–209. MR 1145805
- Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), no. 2, 225–320. MR 833192, DOI 10.1007/BF01388809 Ham H. Hamburger, Über die Funktionalgleichung der $\zeta$-Funktion, Math. Z. 10 (1921), 240–258, 11 (1921), 224–245, 13 (1922), 283–311.
- Harish-Chandra, Automorphic forms on semisimple Lie groups, Lecture Notes in Mathematics, No. 62, Springer-Verlag, Berlin-New York, 1968. Notes by J. G. M. Mars. MR 0232893, DOI 10.1007/BFb0098434 harris-icm 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.
- Michael Harris and Stephen S. Kudla, The central critical value of a triple product $L$-function, Ann. of Math. (2) 133 (1991), no. 3, 605–672. MR 1109355, DOI 10.2307/2944321
- 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
- 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 693092
- Erich Hecke, Mathematische Werke, 3rd ed., Vandenhoeck & Ruprecht, Göttingen, 1983 (German). With introductory material by B. Schoeneberg, C. L. Siegel and J. Nielsen. MR 749754
- Guy Henniart, Une preuve simple des conjectures de Langlands pour $\textrm {GL}(n)$ sur un corps $p$-adique, Invent. Math. 139 (2000), no. 2, 439–455 (French, with English summary). MR 1738446, DOI 10.1007/s002220050012
- Guy Henniart, Sur la conjecture de Langlands locale pour $\textrm {GL}_n$, J. Théor. Nombres Bordeaux 13 (2001), no. 1, 167–187 (French, with English and French summaries). 21st Journées Arithmétiques (Rome, 2001). MR 1838079, DOI 10.5802/jtnb.313 henniartexpose G. Henniart, Progrés rècents en fonctorialitè de Langlands, Seminaire Bourbaki Exposé 890 (Juin 2001), 890–1 to 890–21. MR1947454 Guy Henniart, Une caractérisation de la correspondance de Langlands locale pour $\textrm {GL}(n)$, Bull. Soc. Math. France 130 (2002), no. 4, 587–602 (French).
- Tamotsu Ikeda, On the gamma factor of the triple $L$-function. I, Duke Math. J. 97 (1999), no. 2, 301–318. MR 1682237, DOI 10.1215/S0012-7094-99-09713-2
- Henryk Iwaniec, Fourier coefficients of modular forms of half-integral weight, Invent. Math. 87 (1987), no. 2, 385–401. MR 870736, DOI 10.1007/BF01389423
- Henryk Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, Providence, RI, 1997. MR 1474964, DOI 10.1090/gsm/017 greeniwaniec —, Spectral methods of automorphic forms, 2nd ed., Graduate Studies in Mathematics, vol. 53, American Mathematical Society, Providence, RI, 2002.
- H. Iwaniec and P. Sarnak, Perspectives on the analytic theory of $L$-functions, Geom. Funct. Anal. Special Volume (2000), 705–741. GAFA 2000 (Tel Aviv, 1999). MR 1826269, DOI 10.1007/978-3-0346-0425-3_{6}
- Hervé Jacquet, Dirichlet series for the group $\textrm {GL}(n)$, Automorphic forms, representation theory and arithmetic (Bombay, 1979), Tata Inst. Fund. Res. Studies in Math., vol. 10, Tata Institute of Fundamental Research, Bombay, 1981, pp. 155–163. MR 633661
- H. Jacquet and R. P. Langlands, Automorphic forms on $\textrm {GL}(2)$, Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin-New York, 1970. MR 0401654, DOI 10.1007/BFb0058988
- Hervé Jacquet, Ilja Iosifovitch Piatetski-Shapiro, and Joseph Shalika, Automorphic forms on $\textrm {GL}(3)$. I, Ann. of Math. (2) 109 (1979), no. 1, 169–212. MR 519356, DOI 10.2307/1971270
- H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika, Rankin-Selberg convolutions, Amer. J. Math. 105 (1983), no. 2, 367–464. MR 701565, DOI 10.2307/2374264
- Hervé Jacquet and Joseph A. Shalika, A non-vanishing theorem for zeta functions of $\textrm {GL}_{n}$, Invent. Math. 38 (1976/77), no. 1, 1–16. MR 432596, DOI 10.1007/BF01390166
- 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. MR 618323, DOI 10.2307/2374103
- Hervé Jacquet and Joseph Shalika, Exterior square $L$-functions, Automorphic forms, Shimura varieties, and $L$-functions, Vol. II (Ann Arbor, MI, 1988) Perspect. Math., vol. 11, Academic Press, Boston, MA, 1990, pp. 143–226. MR 1044830 jiang-soudry D. Jiang and D. Soudry, Generic representations and local Langlands reciprocity law for $p$-adic $SO_{2n+1}$, preprint, 2001.
- 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 562594
- D. Kazhdan, B. Mazur, and C.-G. Schmidt, Relative modular symbols and Rankin-Selberg convolutions, J. Reine Angew. Math. 519 (2000), 97–141. MR 1739728, DOI 10.1515/crll.2000.019
- Henry H. Kim, Langlands-Shahidi method and poles of automorphic $L$-functions: application to exterior square $L$-functions, Canad. J. Math. 51 (1999), no. 4, 835–849. MR 1701344, DOI 10.4153/CJM-1999-036-0
- Henry H. Kim, Applications of Langlands’ functorial lift of odd orthogonal groups, Trans. Amer. Math. Soc. 354 (2002), no. 7, 2775–2796. MR 1895203, DOI 10.1090/S0002-9947-02-02969-0 Kim-Sha4 —, Functoriality for the exterior square of $\textrm {GL}_ 4$ and the symmetric fourth of $\textrm {GL}_ 2$, 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.
- Henry H. Kim and Freydoon Shahidi, Symmetric cube $L$-functions for $\rm GL_2$ are entire, Ann. of Math. (2) 150 (1999), no. 2, 645–662. MR 1726704, DOI 10.2307/121091
- Henry H. Kim and Freydoon Shahidi, Functorial products for $\rm GL_2\times GL_3$ and functorial symmetric cube for $\rm GL_2$, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), no. 8, 599–604 (English, with English and French summaries). MR 1799096, DOI 10.1016/S0764-4442(00)01677-3
- Henry H. Kim and Freydoon Shahidi, Cuspidality of symmetric powers with applications, Duke Math. J. 112 (2002), no. 1, 177–197. MR 1890650, DOI 10.1215/S0012-9074-02-11215-0 Kim-Sha3 —, Functorial products for $\textrm {GL}_ 2\times \textrm {GL}_ 3$ and the symmetric cube for $\textrm {GL}_ 2$, Ann. of Math. (2) 155 (2002), no. 3, 837–893, With an appendix by Colin J. Bushnell and Guy Henniart.
- 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 1476501, DOI 10.1090/pspum/061/1476501
- 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 1476507, DOI 10.1090/pspum/061/1476507
- 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); English transl., Leningrad Math. J. 1 (1990), no. 5, 1229–1253. MR 1036843
- Tomio Kubota, Elementary theory of Eisenstein series, Kodansha, Ltd., Tokyo; Halsted Press [John Wiley & Sons, Inc.], New York-London-Sydney, 1973. MR 0429749
- 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, DOI 10.1007/s002220100174
- Serge Lang, Algebraic number theory, 2nd ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR 1282723, DOI 10.1007/978-1-4612-0853-2
- Serge Lang, Complex analysis, 4th ed., Graduate Texts in Mathematics, vol. 103, Springer-Verlag, New York, 1999. MR 1659317, DOI 10.1007/978-1-4757-3083-8
- R. P. Langlands, Eisenstein series, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 235–252. MR 0249539
- Robert P. Langlands, Euler products, Yale Mathematical Monographs, vol. 1, Yale University Press, New Haven, Conn.-London, 1971. A James K. Whittemore Lecture in Mathematics given at Yale University, 1967. MR 0419366
- Robert P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, Vol. 544, Springer-Verlag, Berlin-New York, 1976. MR 0579181, DOI 10.1007/BFb0079929
- R. P. Langlands, Automorphic representations, Shimura varieties, and motives. Ein Märchen, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 205–246. MR 546619
- Robert P. Langlands, Base change for $\textrm {GL}(2)$, Annals of Mathematics Studies, No. 96, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980. MR 574808
- 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. MR 993313
- R. P. Langlands, 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 1011897, DOI 10.1090/surv/031/03
- 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 1476510, DOI 10.1090/pspum/061/1476510
- Robert P. Langlands, The trace formula and its applications: an introduction to the work of James Arthur, Canad. Math. Bull. 44 (2001), no. 2, 160–209. MR 1827854, DOI 10.4153/CMB-2001-020-8 beyondendoscopy 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.
- Gérard Laumon, La correspondance de Langlands sur les corps de fonctions (d’après Laurent Lafforgue), Astérisque 276 (2002), 207–265 (French). Séminaire Bourbaki, Vol. 1999/2000. MR 1886762
- Jian-Shu Li and Joachim Schwermer, Automorphic representations and cohomology of arithmetic groups, Challenges for the 21st century (Singapore, 2000) World Sci. Publ., River Edge, NJ, 2001, pp. 102–137. MR 1875016
- 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 1308046, DOI 10.1007/978-3-0346-0332-4
- A. Lubotzky, R. Phillips, and P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988), no. 3, 261–277. MR 963118, DOI 10.1007/BF02126799
- W. Luo, Z. Rudnick, and P. Sarnak, On Selberg’s eigenvalue conjecture, Geom. Funct. Anal. 5 (1995), no. 2, 387–401. MR 1334872, DOI 10.1007/BF01895672
- Wenzhi Luo, Zeév Rudnick, and Peter Sarnak, On the generalized Ramanujan conjecture for $\textrm {GL}(n)$, 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 1703764
- Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
- G. A. Margulis, Explicit constructions of expanders, Problemy Peredači Informacii 9 (1973), no. 4, 71–80 (Russian). MR 0484767
- G. A. Margulis, 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 1090825, DOI 10.1007/978-3-642-51445-6 mil 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. ms-expos 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. ms-voronoi —, Automorphic Distributions, $L$-functions, and Voronoi Summation for $GL(3)$, http://www.math.rutgers.edu/~sdmiller/voronoi. ms-inforder —, Distributions and Analytic Continuation of Dirichlet Series, http://www.math.rutgers.edu/~sdmiller/voronoi.
- 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, DOI 10.1017/CBO9780511470905 mordell L. J. Mordell, On Mr. Ramanujan’s Empirical Expansions of Modular Functions, Proc. Cambridge Phil. Soc. 19 (1917), 117–124.
- Carlos J. Moreno, Analytic proof of the strong multiplicity one theorem, Amer. J. Math. 107 (1985), no. 1, 163–206. MR 778093, DOI 10.2307/2374461
- Goran Muić, Some results on square integrable representations; irreducibility of standard representations, Internat. Math. Res. Notices 14 (1998), 705–726. MR 1637097, DOI 10.1155/S1073792898000427
- Goran Muić, A proof of Casselman-Shahidi’s conjecture for quasi-split classical groups, Canad. Math. Bull. 44 (2001), no. 3, 298–312. MR 1847492, DOI 10.4153/CMB-2001-030-4
- Werner Müller, The trace class conjecture in the theory of automorphic forms, Ann. of Math. (2) 130 (1989), no. 3, 473–529. MR 1025165, DOI 10.2307/1971453 Murty M. Ram Murty, Ramanujan Graphs, Journal of the Ramanujan Mathematical Society (to appear).
- Andrew Ogg, Modular forms and Dirichlet series, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0256993
- A. P. Ogg, A remark on the Sato-Tate conjecture, Invent. Math. 9 (1969/70), 198–200. MR 258835, DOI 10.1007/BF01404324
- S. J. Patterson and I. I. Piatetski-Shapiro, The symmetric-square $L$-function attached to a cuspidal automorphic representation of $\textrm {GL}_3$, Math. Ann. 283 (1989), no. 4, 551–572. MR 990589, DOI 10.1007/BF01442854
- R. S. Phillips and P. Sarnak, On cusp forms for co-finite subgroups of $\textrm {PSL}(2,\textbf {R})$, Invent. Math. 80 (1985), no. 2, 339–364. MR 788414, DOI 10.1007/BF01388610
- R. S. Phillips and P. Sarnak, The Weyl theorem and the deformation of discrete groups, Comm. Pure Appl. Math. 38 (1985), no. 6, 853–866. MR 812352, DOI 10.1002/cpa.3160380614
- R. Phillips and P. Sarnak, Perturbation theory for the Laplacian on automorphic functions, J. Amer. Math. Soc. 5 (1992), no. 1, 1–32. MR 1127079, DOI 10.1090/S0894-0347-1992-1127079-X
- I. I. Piatetski-Shapiro, Multiplicity one theorems, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 209–212. MR 546599
- 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 1364456, DOI 10.1090/trans2/169/08
- I. Piatetski-Shapiro and Stephen Rallis, Rankin triple $L$ functions, Compositio Math. 64 (1987), no. 1, 31–115. MR 911357
- I. Piatetski-Shapiro and S. Rallis, A new way to get Euler products, J. Reine Angew. Math. 392 (1988), 110–124. MR 965059, DOI 10.1515/crll.1988.392.110
- I. Piatetski-Shapiro, S. Rallis, and G. Schiffmann, $L$ functions for the group $G_2$, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 2, 389–399. MR 1031583, DOI 10.1090/S0273-0979-1990-15942-7
- I. Piatetski-Shapiro, S. Rallis, and G. Schiffmann, Rankin-Selberg integrals for the group $G_2$, Amer. J. Math. 114 (1992), no. 6, 1269–1315. MR 1198304, DOI 10.2307/2374763
- I. I. Pjateckij-Šapiro, 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 0406934
- 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 1650237, DOI 10.1016/S0764-4442(98)80138-9
- Dinakar Ramakrishnan, On the coefficients of cusp forms, Math. Res. Lett. 4 (1997), no. 2-3, 295–307. MR 1453061, DOI 10.4310/MRL.1997.v4.n2.a10
- Dinakar Ramakrishnan, Modularity of the Rankin-Selberg $L$-series, and multiplicity one for $\textrm {SL}(2)$, Ann. of Math. (2) 152 (2000), no. 1, 45–111. MR 1792292, DOI 10.2307/2661379 Ramakrishnan —, Existence of Ramanujan primes for $GL(3)$, Contributions to Automorphic Forms, Geometry and Number Theory: Shalika Fest 2002 (Haruzo Hida, Dinakar Ramakrishnan, and Freydoon Shahidi, eds.), to appear.
- Dinakar Ramakrishnan and Robert J. Valenza, Fourier analysis on number fields, Graduate Texts in Mathematics, vol. 186, Springer-Verlag, New York, 1999. MR 1680912, DOI 10.1007/978-1-4757-3085-2
- T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
- Michael J. Razar, Modular forms for $G_{0}(N)$ and Dirichlet series, Trans. Amer. Math. Soc. 231 (1977), no. 2, 489–495. MR 444576, DOI 10.1090/S0002-9947-1977-0444576-9 Riem 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/.
- 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, DOI 10.1515/9781400882441
- Jonathan D. Rogawski, 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 1476504, DOI 10.1090/pspum/061/1476504
- 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 1159118
- Peter Sarnak, Some applications of modular forms, Cambridge Tracts in Mathematics, vol. 99, Cambridge University Press, Cambridge, 1990. MR 1102679, DOI 10.1017/CBO9780511895593
- Peter Sarnak, Selberg’s eigenvalue conjecture, Notices Amer. Math. Soc. 42 (1995), no. 11, 1272–1277. MR 1355461 Sarlet —, Letter to Stephen Gelbart and Freydoon Shahidi, 2001. sarint —, 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. sar-shalfest —, Nonvanishing of $L$-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. Sar-balt —, Spectra of hyperbolic surfaces, Bull. Amer. Math. Soc. (N.S.) 40 (2003), 441–478, http://www.math.princeton.edu/~sarnak.
- Leonard Eugene Dickson, New First Course in the Theory of Equations, John Wiley & Sons, Inc., New York, 1939. MR 0000002
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
- Atle Selberg, Discontinuous groups and harmonic analysis, Proc. Internat. Congr. Mathematicians (Stockholm, 1962) Inst. Mittag-Leffler, Djursholm, 1963, pp. 177–189. MR 0176097
- Atle Selberg, On the estimation of Fourier coefficients of modular forms, Proc. Sympos. Pure Math., Vol. VIII, Amer. Math. Soc., Providence, R.I., 1965, pp. 1–15. MR 0182610
- Jean-Pierre Serre, Abelian $l$-adic representations and elliptic curves, W. A. Benjamin, Inc., New York-Amsterdam, 1968. McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute. MR 0263823
- J.-P. Serre, A course in arithmetic, Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French. MR 0344216, DOI 10.1007/978-1-4684-9884-4
- Freydoon Shahidi, On certain $L$-functions, Amer. J. Math. 103 (1981), no. 2, 297–355. MR 610479, DOI 10.2307/2374219
- Freydoon Shahidi, Local coefficients as Artin factors for real groups, Duke Math. J. 52 (1985), no. 4, 973–1007. MR 816396, DOI 10.1215/S0012-7094-85-05252-4
- Freydoon Shahidi, On the Ramanujan conjecture and finiteness of poles for certain $L$-functions, Ann. of Math. (2) 127 (1988), no. 3, 547–584. MR 942520, DOI 10.2307/2007005
- Freydoon Shahidi, A proof of Langlands’ conjecture on Plancherel measures; complementary series for $p$-adic groups, Ann. of Math. (2) 132 (1990), no. 2, 273–330. MR 1070599, DOI 10.2307/1971524
- Freydoon Shahidi, Symmetric power $L$-functions for $\textrm {GL}(2)$, Elliptic curves and related topics, CRM Proc. Lecture Notes, vol. 4, Amer. Math. Soc., Providence, RI, 1994, pp. 159–182. MR 1260961, DOI 10.1090/crmp/004/11 ShahKore —, Intertwining Operators, $L$-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/. shahidi-icm —, Automorphic $L$-functions and functoriality, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 655–666. ShahidiPCMI —, Langlands-Shahidi Method and Converse Theorems, IAS/Park City Lecture Notes, Park City, Utah, 2002.
- J. A. Shalika, The multiplicity one theorem for $\textrm {GL}_{n}$, Ann. of Math. (2) 100 (1974), 171–193. MR 348047, DOI 10.2307/1971071
- Goro Shimura, On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. (3) 31 (1975), no. 1, 79–98. MR 382176, DOI 10.1112/plms/s3-31.1.79
- Takuro Shintani, On an explicit formula for class-$1$ “Whittaker functions” on $GL_{n}$ over $P$-adic fields, Proc. Japan Acad. 52 (1976), no. 4, 180–182. MR 407208
- Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1992. Corrected reprint of the 1986 original. MR 1329092, DOI 10.1007/978-1-4757-4252-7
- 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 0225778
- 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
- Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553–572. MR 1333036, DOI 10.2307/2118560
- 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 882550
- Jerrold Tunnell, Artin’s conjecture for representations of octahedral type, Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 2, 173–175. MR 621884, DOI 10.1090/S0273-0979-1981-14936-3
- André Weil, Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168 (1967), 149–156 (German). MR 207658, DOI 10.1007/BF01361551
- André Weil, 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 1097628, DOI 10.2969/aspm/01710463
- André Weil, Prehistory of the zeta-function, Number theory, trace formulas and discrete groups (Oslo, 1987) Academic Press, Boston, MA, 1989, pp. 1–9. MR 993308
- Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. MR 1333035, DOI 10.2307/2118559
- Scott A. Wolpert, Spectral limits for hyperbolic surfaces. I, II, Invent. Math. 108 (1992), no. 1, 67–89, 91–129. MR 1156387, DOI 10.1007/BF02100600
- Scott A. Wolpert, Disappearance of cusp forms in special families, Ann. of Math. (2) 139 (1994), no. 2, 239–291. MR 1274093, DOI 10.2307/2946582
Additional Information
- 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
- Received by editor(s): July 15, 2002
- Received by editor(s) in revised form: September 8, 2003
- Published electronically: October 30, 2003
- Additional Notes: The first author was partially supported by the Minerva Foundation, and the second author was supported by NSF grant DMS-0122799
- © Copyright 2003 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 41 (2004), 59-112
- MSC (2000): Primary 11-02, 11M06, 11M41, 11F03, 30D15
- DOI: https://doi.org/10.1090/S0273-0979-03-00995-9
- MathSciNet review: 2015450
Dedicated: Dedicated to Ilya Piatetski-Shapiro, with admiration