Duke Mathematical Journal

On the positivity of the central critical values of automorphic L-functions for GL(2)

Jiandong Guo

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Duke Math. J., Volume 83, Number 1 (1996), 157-190.

First available in Project Euclid: 19 February 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Secondary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols


Guo, Jiandong. On the positivity of the central critical values of automorphic $L$ -functions for $GL(2)$. Duke Math. J. 83 (1996), no. 1, 157--190. doi:10.1215/S0012-7094-96-08307-6. https://projecteuclid.org/euclid.dmj/1077244251

Export citation


  • [BFH1] D. Bump, S. Friedberg, and J. 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.
  • [BFH2] D. Bump, S. Friedberg, and J. Hoffstein, Nonvanishing theorems for $L$-functions of modular forms and their derivatives, Invent. Math. 102 (1990), no. 3, 543–618.
  • [C] W. Casselman, On some results of Atkin and Lehner, Math. Ann. 201 (1973), 301–314.
  • [D] P. Deligne, Formes modulaires et représentations de $\rm GL(2)$, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1973, 55–105. Lecture Notes in Math., Vol. 349.
  • [FJ] S. Friedberg and H. Jacquet, Linear periods, J. Reine Angew. Math. 443 (1993), 91–139.
  • [FH] S. Friedberg and J. Hoffstein, Nonvanishing theorems for automorphic $L$-functions on $\mathrm GL (2)$, to appear in Ann. of Math.
  • [Ge] S. Gelbart, Automorphic forms on adele groups, Ann. of Math. Stud., vol. 83, Princeton Univ. Press, Princeton, N.J., 1975.
  • [Go] R. Godement, Notes on Jacquet-Langlands' theory, 1970, mimeographed notes in Institute for Advanced Study, Princeton, N.J.
  • [G] J. Guo, On a generalization of a result of Waldspurger, preprint, 1994.
  • [J1] H. Jacquet, Sur un résultat de Waldspurger, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 2, 185–229.
  • [J2] H. Jacquet, On the nonvanishing of some $L$-functions, Proc. Indian Acad. Sci. Math. Sci. 97 (1987), no. 1-3, 117–155 (1988).
  • [JL] H. Jacquet and R. Langlands, Automorphic forms on $\rm GL(2)$, Lecture Notes in Math., vol. 114, Springer-Verlag, Berlin, 1970.
  • [JS] H. Jacquet and J. Shalika, On Euler products and the classification of automorphic representations I, Amer. J. Math. 103 (1981), no. 3, 499–558.
  • [KS] S. Katok and P. Sarnak, Heegner points, cycles, and Maass forms, Israel J. Math. 84 (1993), no. 1-2, 193–227.
  • [K] W. Kohnen, Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271 (1985), no. 2, 237–268.
  • [KZ] W. Kohnen and D. Zagier, Values of $L$-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), no. 2, 175–198.
  • [M] J. Martinet, Character theory and Artin $L$-functions, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), Academic Press, London, 1977, pp. 1–87.
  • [R] D. Rohrlich, Nonvanishing of $L$-functions for $\rm GL(2)$, Invent. Math. 97 (1989), no. 2, 381–403.
  • [S1] G. Shimura, The special values of the zeta functions associated with Hilbert modular forms, Duke Math. J. 45 (1978), no. 3, 637–679.
  • [S2] G. Shimura, On the critical values of certain Dirichlet series and the periods of automorphic forms, Invent. Math. 94 (1988), no. 2, 245–305.
  • [S3] G. Shimura, On the fundamental periods of automorphic forms of arithmetic type, Invent. Math. 102 (1990), no. 2, 399–428.
  • [S4] G. Shimura, On the Fourier coefficients of Hilbert modular forms of half-integral weight, Duke Math. J. 71 (1993), no. 2, 501–557.
  • [T] J. Tate, Fourier analysis in number fields and Hecke's zeta-functions, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) eds. J. Cassels and A. Frohlich, Thompson, Washington, D.C., 1967, pp. 305–347.
  • [V] M.-F. Vigneras, Valeurs au centre de symetrie des fonctions $L$ associees aux formes modulaires, Seminaire de theorie des Nombres ed. M.-J. Bertin, Progr. Math., vol. 12, Birkhäuser, Boston, 1981, pp. 331–356.
  • [W1] J.-L. Waldspurger, Correspondances de Shimura et quaternions, Forum Math. 3 (1991), no. 3, 219–307.
  • [W2] J.-L. Waldspurger, Sur les valeurs de certaines fonctions $L$ automorphes en leur centre de symétrie, Compositio Math. 54 (1985), no. 2, 173–242.
  • [W3] J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. (9) 60 (1981), no. 4, 375–484.