Duke Mathematical Journal

On the critical values of $L$-functions of $GL(2)$ and $GL(2) \times GL(2)$

Haruzo Hida

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

Source
Duke Math. J. Volume 74, Number 2 (1994), 431-529.

Dates
First available: 20 February 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1077288205

Mathematical Reviews number (MathSciNet)
MR1272981

Zentralblatt MATH identifier
0838.11036

Digital Object Identifier
doi:10.1215/S0012-7094-94-07417-6

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

Citation

Hida, Haruzo. On the critical values of L -functions of G L ( 2 ) and G L ( 2 ) × G L ( 2 ) . Duke Mathematical Journal 74 (1994), no. 2, 431--529. doi:10.1215/S0012-7094-94-07417-6. http://projecteuclid.org/euclid.dmj/1077288205.


Export citation

References

  • [B] D. Blasius, On the critical values of Hecke $L$-series, Ann. of Math. (2) 124 (1986), no. 1, 23–63.
  • [B1] D. Blasius, Period relations and critical values of $L$-functions, unpublished manuscript, 1987.
  • [BR] D. Blasius and J. D. Rogawski, Motives for Hilbert modular forms, Invent. Math. 114 (1993), no. 1, 55–87.
  • [BW] A. Borel and N. R. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Mathematics Studies, vol. 94, Princeton University Press, Princeton, N.J., 1980.
  • [C] W. Casselman, On some results of Atkin and Lehner, Math. Ann. 201 (1973), 301–314.
  • [C1] L. Clozel, Motifs et formes automorphes: applications du principe de fonctorialité, Automorphic forms, Shimura varieties, and $L$-functions, Vol. I (Ann Arbor, MI, 1988), Perspect. Math., vol. 10, Academic Press, Boston, MA, 1990, pp. 77–159.
  • [D] P. Deligne, Valeurs de fonctions $L$ et périodes d'intégrales, 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. 313–346.
  • [D1] P. Deligne, Hodge cycles on abelian varieties, Lecture Notes in Math., vol. 900, Springer-Verlag, New York, 1982, 9–100.
  • [DM] P. Deligne and J. S. Milne, Tannakian categories, Lecture Notes in Math., vol. 900, Springer-Verlag, New York, 1982.
  • [GJ] S. Gelbart and H. 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.
  • [Ha1] G. Harder, Eisenstein cohomology of arithmetic groups. The case $\rm GL\sb 2$, Invent. Math. 89 (1987), no. 1, 37–118.
  • [Ha2] G. Harder, General aspects in the theory of modular symbols, Seminar on number theory, Paris 1981–82 (Paris, 1981/1982), Progr. Math., vol. 38, Birkhäuser Boston, Boston, MA, 1983, pp. 73–88.
  • [Hs] M. Harris, Period invariants of Hilbert modular forms. I. Trilinear differential operators and $L$-functions, Cohomology of arithmetic groups and automorphic forms (Luminy-Marseille, 1989), Lecture Notes in Math., vol. 1447, Springer, Berlin, 1990, pp. 155–202.
  • [H] H. Hida, Elementary theory of $L$-functions and Eisenstein series, London Mathematical Society Student Texts, vol. 26, Cambridge University Press, Cambridge, 1993.
  • [H1] H. Hida, $p$-ordinary cohomology groups for $\rm SL(2)$ over number fields, Duke Math. J. 69 (1993), no. 2, 259–314.
  • [H2] H. Hida, On $p$-adic $L$-functions of $\rm GL(2)\times \rm GL(2)$ over totally real fields, Ann. Inst. Fourier (Grenoble) 41 (1991), no. 2, 311–391.
  • [H3] H. Hida, On $p$-adic Hecke algebras for $\rm GL\sb 2$ over totally real fields, Ann. of Math. (2) 128 (1988), no. 2, 295–384.
  • [H4] H. Hida, On abelian varieties with complex multiplication as factors of the Jacobians of Shimura curves, Amer. J. Math. 103 (1981), no. 4, 726–776.
  • [H5] H. Hida, On abelian varieties with complex multiplication as factors of the abelian variety attached to Hilbert modular forms, Japan. J. Math. (N.S.) 5 (1979), no. 1, 157–208.
  • [J] H. Jacquet, Automorphic forms on $\rm GL(2)$. Part II, Lecture Notes in Mathematics, vol. 278, Springer-Verlag, Berlin, 1972.
  • [JL] H. Jacquet and R. P. Langlands, Automorphic Forms on $\rm GL(2)$, Lecture Notes in Math., vol. 114, Springer-Verlag, Berlin, 1970.
  • [Ku] P. F. Kurchanov, Dirichlet series of Jacquet-Langlands cusp forms over fields of CM type, Mth. USSR Izv. 14 (1980), 61–78.
  • [L] R. P. Langlands, On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Math., vol. 544, Springer-Verlag, Berlin, 1976.
  • [Ma1] Y. I. Manin, Periods of cusp forms and $p$-adic series, Math. USSR Sb. 92 (1973), 371–393.
  • [Ma2] Y. I. Manin, The Values of $p$-adic Hecke series at integer points of the critical strips, Math. USSR Sb. 93 (1974), 621–626.
  • [MM] Y. Matsushima and S. Murakami, On vector bundle valued harmonic forms and automorphic forms on symmetric riemannian manifolds, Ann. of Math. (2) 78 (1963), 365–416.
  • [MSh] Y. Matsushima and G. Shimura, On the cohomology groups attached to certain vector valued differential forms on the product of the upper half planes, Ann. of Math. (2) 78 (1963), 417–449.
  • [Mz] B. Mazur, Courbes elliptiques et symboles modulaires, Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 414, Springer, Berlin, 1973, 277–294. Lecture Notes in Math., Vol. 317.
  • [M] T. Miyake, On automorphic forms on $\rm GL\sb2$ and Hecke operators, Ann. of Math. (2) 94 (1971), 174–189.
  • [R] D. E. Rohrlich, Nonvanishing of $L$-functions for $\rm GL(2)$, Invent. Math. 97 (1989), no. 2, 381–403.
  • [Sh1] G. Shimura, Sur les intégrales attachées aux formes automorphes, J. Math. Soc. Japan 11 (1959), 291–311.
  • [Sh2] G. Shimura, The special values of the zeta functions associated with Hilbert modular forms, Duke Math. J. 45 (1978), no. 3, 637–679.
  • [Sh3] G. Shimura, On the critical values of certain Dirichlet series and the periods of automorphic forms, Invent. Math. 94 (1988), no. 2, 245–305.
  • [Sh4] G. Shimura, On some arithmetic properties of modular forms of one and several variables, Ann. of Math. (2) 102 (1975), no. 3, 491–515.
  • [Sh5] G. Shimura, On Eisenstein series, Duke Math. J. 50 (1983), no. 2, 417–476.
  • [Sh6] G. Shimura, On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. (3) 31 (1975), no. 1, 79–98.
  • [Sh7] G. Shimura, On the fundamental periods of automorphic forms of arithmetic type, Invent. Math. 102 (1990), no. 2, 399–428.
  • [Sh8] G. Shimura, The critical values of certain Dirichlet series attached to Hilbert modular forms, Duke Math. J. 63 (1991), no. 3, 557–613.
  • [St]1 J. Sturm, Addendum to: “Special values of zeta functions, and Eisenstein series of half integral weight”, Amer. J. Math. 102 (1980), no. 4, 781–783.
  • [St]2 J. Sturm, Special values of zeta functions, and Eisenstein series of half integral weight, Amer. J. Math. 102 (1980), no. 2, 219–240.
  • [W] A. Weil, Dirichlet Series and Automorphic Forms, Lecture Notes in Math., vol. 189, Springer-Verlag, Berlin, 1971.
  • [W1] A. Weil, Basic number theory, Springer-Verlag, New York, 1974.
  • [Y] H. Yoshida, On the zeta functions of Shimura varieties and periods of Hilbert modular forms, preprint.