Duke Mathematical Journal

Central values of Hecke $L$-functions of CM number fields

Article information

Source
Duke Math. J. Volume 98, Number 3 (1999), 541-564.

Dates
First available in Project Euclid: 19 February 2004

http://projecteuclid.org/euclid.dmj/1077228359

Digital Object Identifier
doi:10.1215/S0012-7094-99-09817-4

Mathematical Reviews number (MathSciNet)
MR1695801

Zentralblatt MATH identifier
0965.11045

Citation

Rodriguez Villegas, Fernando; Yang, Tonghai. Central values of Hecke L -functions of CM number fields. Duke Math. J. 98 (1999), no. 3, 541--564. doi:10.1215/S0012-7094-99-09817-4. http://projecteuclid.org/euclid.dmj/1077228359.

References

• [G] Paul B. Garrett, Holomorphic Hilbert modular forms, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990.
• [Gr] Benedict H. Gross, Arithmetic on elliptic curves with complex multiplication, Lecture Notes in Mathematics, vol. 776, Springer, Berlin, 1980.
• [Gr2] Benedict H. Gross, Heegner points on $X\sb 0(N)$, Modular forms (Durham, 1983) ed. R. Rankin, Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984, pp. 87–105.
• [HKS] Michael Harris, Stephen S. Kudla, and William J. Sweet, Theta dichotomy for unitary groups, J. Amer. Math. Soc. 9 (1996), no. 4, 941–1004.
• [K] Stephen S. Kudla, Splitting metaplectic covers of dual reductive pairs, Israel J. Math. 87 (1994), no. 1-3, 361–401.
• [MRoh] Hugh L. Montgomery and David E. Rohrlich, On the $L$-functions of canonical Hecke characters of imaginary quadratic fields. II, Duke Math. J. 49 (1982), no. 4, 937–942.
• [RR] R. Ranga Rao, On some explicit formulas in the theory of Weil representation, Pacific J. Math. 157 (1993), no. 2, 335–371.
• [RV] Fernando Rodríguez Villegas, On the square root of special values of certain $L$-series, Invent. Math. 106 (1991), no. 3, 549–573.
• [RV2] Fernando Rodríguez Villegas, Square root formulas for central values of Hecke $L$-series. II, Duke Math. J. 72 (1993), no. 2, 431–440.
• [RVZ] Fernando Rodriguez Villegas and Don Zagier, Square roots of central values of Hecke $L$-series, Advances in number theory (Kingston, ON, 1991), Oxford Sci. Publ., Oxford Univ. Press, New York, 1993, pp. 81–99.
• [RVZ2] Fernando Rodríguez Villegas and Don Zagier, Which primes are sums of two cubes? Number theory (Halifax, NS, 1994), CMS Conf. Proc., vol. 15, Amer. Math. Soc., Providence, RI, 1995, pp. 295–306.
• [Ro] Jonathan D. Rogawski, The multiplicity formula for $A$-packets, The zeta functions of Picard modular surfaces eds. R. P. Langlands and D. Ramakrishnan, Univ. Montréal, Montreal, QC, 1992, pp. 395–419.
• [Roh] David E. Rohrlich, Galois conjugacy of unramified twists of Hecke characters, Duke Math. J. 47 (1980), no. 3, 695–703.
• [Roh2] David E. Rohrlich, The nonvanishing of certain Hecke $L$-functions at the center of the critical strip, Duke Math. J. 47 (1980), no. 1, 223–232.
• [Roh3] David E. Rohrlich, On the $L$-functions of canonical Hecke characters of imaginary quadratic fields, Duke Math. J. 47 (1980), no. 3, 547–557.
• [Roh4] David E. Rohrlich, Root numbers of Hecke $L$-functions of CM fields, Amer. J. Math. 104 (1982), no. 3, 517–543.
• [Roh5] David E. Rohrlich, Galois theory, elliptic curves, and root numbers, Compositio Math. 100 (1996), no. 3, 311–349.
• [Ru] Karl Rubin, Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer, Invent. Math. 64 (1981), no. 3, 455–470.
• [S] Goro Shimura, On the factors of the jacobian variety of a modular function field, J. Math. Soc. Japan 25 (1973), 523–544.
• [S2] Goro Shimura, The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math. 29 (1976), no. 6, 783–804.
• [S3] Goro Shimura, On the periods of modular forms, Math. Ann. 229 (1977), no. 3, 211–221.
• [W] Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982.
• [We] André Weil, Sur la formule de Siegel dans la théorie des groupes classiques, Acta Math. 113 (1965), 1–87.
• [Y] Tonghai Yang, Theta liftings and Hecke $L$-functions, J. Reine Angew. Math. 485 (1997), 25–53.
• [Y2] Tonghai Yang, Common zeros of theta functions and central Hecke $L$-values of CM number fields of degree $4$, Proc. Amer. Math. Soc. 126 (1998), no. 4, 999–1004.
• [Y3] Tonghai Yang, Eigenfunctions of the Weil representation of unitary groups of one variable, Trans. Amer. Math. Soc. 350 (1998), no. 6, 2393–2407.
• [Y4] T. H. Yang, Nonvanishing of central value of Hecke characters and the rank of their associated elliptic curves, to appear in Compositio Math.