On some applications of automorphic forms to number theory
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- by Daniel Bump, Solomon Friedberg and Jeffrey Hoffstein PDF
- Bull. Amer. Math. Soc. 33 (1996), 157-175 Request permission
Abstract:
A basic idea of Dirichlet is to study a collection of interesting quantities $\{a_n\}_{n\geq 1}$ by means of its Dirichlet series in a complex variable $w$: $\sum _{n\geq 1}a_nn^{-w}$. In this paper we examine this construction when the quantities $a_n$ are themselves infinite series in a second complex variable $s$, arising from number theory or representation theory. We survey a body of recent work on such series and present a new conjecture concerning them.References
- James Arthur, A trace formula for reductive groups. II. Applications of a truncation operator, Compositio Math. 40 (1980), no. 1, 87–121. MR 558260
- Laure Barthel and Dinakar Ramakrishnan, A nonvanishing result for twists of $L$-functions of $\textrm {GL}(n)$, Duke Math. J. 74 (1994), no. 3, 681–700. MR 1277950, DOI 10.1215/S0012-7094-94-07425-5
- A. Borel, Automorphic $L$-functions, 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. 27–61. MR 546608
- 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
- D. Bump, S. Friedberg, and M. Furusawa, Explicit formulas for the Waldspurger and Bessel models, preprint.
- Daniel Bump, Solomon Friedberg, and Jeffrey Hoffstein, A nonvanishing theorem for derivatives of automorphic $L$-functions with applications to elliptic curves, Bull. Amer. Math. Soc. (N.S.) 21 (1989), no. 1, 89–93. MR 983456, DOI 10.1090/S0273-0979-1989-15771-6
- Daniel Bump, Solomon Friedberg, and Jeffrey Hoffstein, Eisenstein series on the metaplectic group and nonvanishing theorems for automorphic $L$-functions and their derivatives, Ann. of Math. (2) 131 (1990), no. 1, 53–127. MR 1038358, DOI 10.2307/1971508
- Daniel Bump, Solomon Friedberg, and Jeffrey Hoffstein, Nonvanishing theorems for $L$-functions of modular forms and their derivatives, Invent. Math. 102 (1990), no. 3, 543–618. MR 1074487, DOI 10.1007/BF01233440
- Daniel Bump, Solomon Friedberg, and Jeffrey Hoffstein, $p$-adic Whittaker functions on the metaplectic group, Duke Math. J. 63 (1991), no. 2, 379–397. MR 1115113, DOI 10.1215/S0012-7094-91-06316-7
- Daniel Bump and David Ginzburg, Spin $L$-functions on symplectic groups, Internat. Math. Res. Notices 8 (1992), 153–160. MR 1177328, DOI 10.1155/S1073792892000175
- D. Bump, D. Ginzburg, and J. Hoffstein, The symmetric cube, To appear in Inventiones Math.
- Daniel Bump and Jeffrey Hoffstein, Cubic metaplectic forms on $\textrm {GL}(3)$, Invent. Math. 84 (1986), no. 3, 481–505. MR 837524, DOI 10.1007/BF01388743
- J. L. B. Cooper, The absolute Cesàro summability of Fourier integrals, Proc. London Math. Soc. 45 (1939), 425–439 (French). MR 0000309, DOI 10.1112/plms/s2-45.1.425
- D. A. Burgess, On character sums and $L$-series. II, Proc. London Math. Soc. (3) 13 (1963), 524–536. MR 148626, DOI 10.1112/plms/s3-13.1.524
- Nelson Dunford and B. J. Pettis, Linear operations among summable functions, Proc. Nat. Acad. Sci. U.S.A. 25 (1939), 544–550. MR 343, DOI 10.1073/pnas.25.10.544
- M. Furusawa, On the theta lift from $SO_{2n+1}$ to $\widetilde {Sp}_n$, J. Reine Angew. Math. 466 (1995), 87–110.
- 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, Ilya Piatetski-Shapiro, and Stephen Rallis, Explicit constructions of automorphic $L$-functions, Lecture Notes in Mathematics, vol. 1254, Springer-Verlag, Berlin, 1987. MR 892097, DOI 10.1007/BFb0078125
- D. Ginzburg, Fax to Daniel Bump (1994).
- Dorian Goldfeld and Jeffrey Hoffstein, Eisenstein series of ${1\over 2}$-integral weight and the mean value of real Dirichlet $L$-series, Invent. Math. 80 (1985), no. 2, 185–208. MR 788407, DOI 10.1007/BF01388603
- T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
- S. Gupta, Average value of quadratic twists of $L$-functions over function fields, Thesis, Brown University (1995).
- Michael Harris, David Soudry, and Richard Taylor, $l$-adic representations associated to modular forms over imaginary quadratic fields. I. Lifting to $\textrm {GSp}_4(\textbf {Q})$, Invent. Math. 112 (1993), no. 2, 377–411. MR 1213108, DOI 10.1007/BF01232440
- Jeff Hoffstein, Eisenstein series and theta functions on the metaplectic group, Theta functions: from the classical to the modern, CRM Proc. Lecture Notes, vol. 1, Amer. Math. Soc., Providence, RI, 1993, pp. 65–104. MR 1224051, DOI 10.1090/crmp/001/02
- Lars Hörmander, An introduction to complex analysis in several variables, 3rd ed., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990. MR 1045639
- Henryk Iwaniec, On the order of vanishing of modular $L$-functions at the critical point, Sém. Théor. Nombres Bordeaux (2) 2 (1990), no. 2, 365–376. MR 1081731, DOI 10.5802/jtnb.33
- Hervé Jacquet, Principal $L$-functions of the linear group, 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. 63–86. MR 546609
- M. Jutila, On the mean value of $L({1\over 2},\,\chi )$ for real characters, Analysis 1 (1981), no. 2, 149–161. MR 632705, DOI 10.1524/anly.1981.1.2.149
- D. A. Kazhdan and S. J. Patterson, Metaplectic forms, Inst. Hautes Études Sci. Publ. Math. 59 (1984), 35–142. MR 743816, DOI 10.1007/BF02698770
- V. A. Kolyvagin, The Mordell-Weil and Shafarevich-Tate groups for Weil elliptic curves, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 6, 1154–1180, 1327 (Russian); English transl., Math. USSR-Izv. 33 (1989), no. 3, 473–499. MR 984214, DOI 10.1070/IM1989v033n03ABEH000853
- R. P. Langlands, Problems in the theory of automorphic forms, Lectures in modern analysis and applications, III, Lecture Notes in Math., Vol. 170, Springer, Berlin, 1970, pp. 18–61. MR 0302614
- Daniel B. Lieman, The $\textrm {GL}(2)$ Rankin-Selberg convolution for higher level non-cuspidal forms, The Rademacher legacy to mathematics (University Park, PA, 1992) Contemp. Math., vol. 166, Amer. Math. Soc., Providence, RI, 1994, pp. 83–92. MR 1284052, DOI 10.1090/conm/166/01640
- Daniel B. Lieman, Nonvanishing of $L$-series associated to cubic twists of elliptic curves, Ann. of Math. (2) 140 (1994), no. 1, 81–108. MR 1289492, DOI 10.2307/2118541
- T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
- H. Maass, Konstruktion ganzer Modulformen halbzahliger Dimension, Abh. Math. Semin. Univ. Hamburg 12 (1937), 133–162.
- Hideya Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. École Norm. Sup. (4) 2 (1969), 1–62 (French). MR 240214, DOI 10.24033/asens.1174
- M. Ram Murty and V. Kumar Murty, Mean values of derivatives of modular $L$-series, Ann. of Math. (2) 133 (1991), no. 3, 447–475. MR 1109350, DOI 10.2307/2944316
- Mark E. Novodvorsky, Automorphic $L$-functions for symplectic group $\textrm {GSp}(4)$, 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. 87–95. MR 546610
- X. She, On the nonvanishing of cubic twists of automorphic $L$-series, Thesis, Brown University (1995).
- Goro Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440–481. MR 332663, DOI 10.2307/1970831
- C. L. Siegel, The average measure of quadratic forms with given determinant and signature, in Gesammelte Abhandlungen, Band II, Springer-Verlag, 1966, pp. 473–491.
- Richard Taylor, $l$-adic representations associated to modular forms over imaginary quadratic fields. II, Invent. Math. 116 (1994), no. 1-3, 619–643. MR 1253207, DOI 10.1007/BF01231575
- Don Zagier, The Rankin-Selberg method for automorphic functions which are not of rapid decay, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 415–437 (1982). MR 656029
Additional Information
- Daniel Bump
- Affiliation: Department of Mathematics, Stanford University, Stanford, CA 94305-2125
- Email: bump@gauss.stanford.edu
- Solomon Friedberg
- Affiliation: Department of Mathematics, University of California Santa Cruz, Santa Cruz, CA 95064
- MR Author ID: 192407
- ORCID: 0000-0002-1246-7738
- Email: friedbe@cats.ucsc.edu
- Jeffrey Hoffstein
- Affiliation: Department of Mathematics, Brown University, Providence, RI 02912
- MR Author ID: 87085
- Email: jhoff@gauss.math.brown.edu
- Additional Notes: Research supported by NSA grant MDA904-95-H-1053 (Friedberg) and by NSF grants DMS-9346517 (Bump) and DMS-9322150 (Hoffstein).
- © Copyright 1996 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 33 (1996), 157-175
- MSC (1991): Primary 11F66; Secondary 11F70, 11M41, 11N75
- DOI: https://doi.org/10.1090/S0273-0979-96-00654-4
- MathSciNet review: 1359575