Kyoto Journal of Mathematics

Central critical values of modular Hecke L-functions

Haruzo Hida

Full-text: Open access


We give an explicit formula for the central critical value L(1/2,π̂χ) of the base-change lift π̂ to an imaginary quadratic field K of an automorphic representation π as the square of a finite sum of the values of a nearly holomorphic cusp form in π at elliptic curves with complex multiplication by K. As long as the transcendental factor of the value is a CM period, χ is basically any unitary arithmetic Hecke character of K inducing the inverse of the central character of π.

Article information

Kyoto J. Math., Volume 50, Number 4 (2010), 777-826.

First available in Project Euclid: 29 November 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F11: Holomorphic modular forms of integral weight 11F25: Hecke-Petersson operators, differential operators (one variable) 11F27: Theta series; Weil representation; theta correspondences 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols


Hida, Haruzo. Central critical values of modular Hecke $L$ -functions. Kyoto J. Math. 50 (2010), no. 4, 777--826. doi:10.1215/0023608X-2010-014.

Export citation


  • [H] E. Hecke, Zur Theorie der elliptischen Modulfunktionen, Math. Ann. 97 (1927), 210–242.
  • [Hi1] H. Hida, A p-adic measure attached to the zeta functions associated with two elliptic modular forms, I, Invent. Math. 79 (1985), 159–195.
  • [Hi2] H. Hida, Elementary Theory of L-functions and Eisenstein Series, London Math. Soc. Stud. Texts 26, Cambridge Univ. Press, Cambridge, 1993.
  • [Hi3] H. Hida, Modular Forms and Galois Cohomology, Cambridge Stud. Adv. Math. 69, Cambridge Univ. Press, Cambridge, 2000.
  • [Hi4] H. Hida, Anticyclotomic main conjectures, Doc. Math. 2006, extra vol., 465–532.
  • [Hi5] H. Hida, Hilbert Modular Forms and Iwasawa Theory, Oxford Math. Monogr., Oxford Univ. Press, Oxford, 2006.
  • [JL] H. Jacquet and R.-P. Langlands, Automorphic Forms on GL (2), Lecture Notes in Math. 114, Springer, Berlin, 1970.
  • [K] N. M. Katz, p-adic L-functions for CM fields, Invent. Math. 49 (1978), 199–297.
  • [KRY] S. S. Kudla, M. Rapoport, and T. Yang, Modular Forms and Special Cycles on Shimura Curves, Ann. of Math. Stud. 161, Princeton Univ. Press, Princeton, 2006.
  • [P] K. Prasanna, Integrality of a ratio of Petersson norms and level-lowering congruences, Ann. of Math. (2) 163 (2006), 901–967.
  • [Sh1] G. Shimura, On some arithmetic properties of modular forms of one and several variables, Ann. of Math. (2) 102 (1975), 491–515.
  • [Sh2] G. Shimura, The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math. 29 (1976), 783–804.
  • [Sh3] G. Shimura, On certain zeta functions attached to two Hilbert modular forms, I: The case of Hecke characters, Ann. of Math. (2) 114 (1981), 127–164; II: The case of automorphic forms on a quaternion algebra, 569–607.
  • [Wa] J.-L. Waldspurger, Sur les valeurs de certaines fonctions L-automorphes en leur centre de symétrie, Compositio Math. 54 (1985), 173–242.
  • [We1] A. Weil, Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964), 143–211.
  • [We2] A. Weil, Sur la formule de Siegel dans la théorie des groupes classiques, Acta Math. 113 (1965), 1–87.
  • [We3] A. Weil, Basic Number Theory, 3rd ed., Grundlehren Math. Wiss. 144, Springer, Berlin, 1974.
  • [YZZ] X. Yuan, S.-W. Zhang, and W. Zhang, Heights of CM points I Gross–Zagier formula, preprint, 2008.