Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Effective nonvanishing of canonical Hecke $ L$-functions


Authors: George Boxer and Peter Diao
Journal: Proc. Amer. Math. Soc. 138 (2010), 3891-3897
MSC (2010): Primary 11M99
DOI: https://doi.org/10.1090/S0002-9939-2010-10430-6
Published electronically: June 4, 2010
MathSciNet review: 2679611
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Motivated by work of Gross, Rohrlich, and more recently Kim, Masri, and Yang, we investigate the nonvanishing of central values of $ L$-functions of ``canonical'' weight $ 2k-1$ Hecke characters for $ \mathbb{Q}(\sqrt{-p})$, where $ 3 < p \equiv 3 \pmod 4$ is prime. Using the work of Rodriguez-Villegas and Zagier, we show that there are nonvanishing central values provided that $ p \geq 6.5(k-1)^2$ and $ (-1)^{k+1} \left(\frac{2}{p}\right) = 1$. Moreover, we show that the number of such $ \psi \in \Psi_{p,k}$ satisfies

$\displaystyle \char93 \{\psi\in\Psi_{p,k}\mid L(\psi,k)\not=0\}\geq \frac{h(-p)}{\char93 \operatorname{Cl}(K)[2k-1]}. $


References [Enhancements On Off] (What's this?)

  • [DFI] Duke, W., J. Friedlander, and H. Iwaniec, Bounds for Automorphic L-functions. II. Inventiones Mathematicae 115 (1994), 219-239. MR 1258904 (95a:11044)
  • [Gr] B. Gross, Arithmetic on elliptic curves with complex multiplication. With an appendix by B. Mazur. Lecture Notes in Mathematics, 776. Springer, Berlin, 1980. iii+95 pp. MR 563921 (81f:10041)
  • [KMY] B. D. Kim, R. Masri, and Tonghai Yang, Nonvanishing of Hecke $ L$-functions and the Bloch-Kato conjecture, preprint.
  • [K] Krasikov, Ilia, Nonnegative Quadratic Forms and Bounds on Orthogonal Polynomials. Journal of Approximation Theory 111 (2001), 31-49. MR 1840019 (2002j:42034)
  • [LX] C. Liu and L. Xu, The vanishing order of certain Hecke $ L$-functions of imaginary quadratic fields. J. Number Theory 108 (2004), 76-89. MR 2078658 (2005g:11221)
  • [Ma] R. Masri, Asymptotics for sums of central values of canonical Hecke $ L$-series. Int. Math. Res. Not. IMRN 2007, no. 19, Art. ID rnm065, 27 pp. MR 2359540 (2009b:11090)
  • [Ma2] R. Masri, Quantitative nonvanishing of $ L$-series associated to canonical Hecke characters. Int. Math. Res. Not. IMRN 2007, no. 19, Art. ID rnm070, 16 pp. MR 2359543 (2008j:11170)
  • [MV] P. Michel and A. Venkatesh, Heegner points and nonvanishing of Rankin-Selberg $ L$-functions. Proceedings of the Gauss-Dirichlet conference, Clay Math. Proc., 7, Amer. Math. Soc., Providence, RI, 2007. MR 2362200 (2009a:11116)
  • [MiY] S. D. Miller and T. H. Yang, Nonvanishing of the central derivative of canonical Hecke $ L$-functions. Math. Res. Lett. 7 (2000), 263-277. MR 1764321 (2001i:11058)
  • [MR] H. Montgomery and D. Rohrlich, On the $ L$-functions of canonical Hecke characters of imaginary quadratic fields. II. Duke Math. J. 49 (1982), 937-942. MR 683009 (84e:12014)
  • [RV] F. Rodriguez-Villegas, Square root formulas for central values of Hecke $ L$-series II. Duke Math. J. 72 (1993), 431-440. MR 1248679 (95d:11156)
  • [RVY] F. Rodriguez-Villegas and T. Yang, Central values of Hecke $ L$-functions of CM number fields, Duke Math. J. 98 (1999), 541-564. MR 1695801 (2000j:11074)
  • [RVZ] F. Rodriguez-Villegas and D. Zagier, Square roots of central values of Hecke $ L$-series. Advances in number theory (Kingston, ON, 1991), 81-99, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993. MR 1368412 (96j:11069)
  • [R] D. Rohrlich, The nonvanishing of certain Hecke $ L$-functions at the center of the critical strip. Duke Math. J. 47 (1980), 223-232. MR 563377 (81k:12017)
  • [R2] D. Rohrlich, On the $ L$-functions of canonical Hecke characters of imaginary quadratic fields. Duke Math. J. 47 (1980), 547-557. MR 587165 (81m:12020)
  • [R3] D. Rohrlich, Galois conjugacy of unramified twists of Hecke characters. Duke Math. J. 47 (1980), 695-703. MR 587174 (82a:12009)
  • [Sh] G. Shimura, On periods of modular forms. Math. Ann. 229 (1977), 211-221. MR 0463119 (57:3080)
  • [Y] T. H. Yang, Nonvanishing of central Hecke $ L$-values and rank of certain elliptic curves. Compos. Math. 117 (1999), 337-359. MR 1702416 (2001a:11093)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11M99

Retrieve articles in all journals with MSC (2010): 11M99


Additional Information

George Boxer
Affiliation: Frist Center, Princeton University, Mailbox 2704, Princeton, New Jersey 08544

Peter Diao
Affiliation: Frist Center, Princeton University, Mailbox 2868, Princeton, New Jersey 08544

DOI: https://doi.org/10.1090/S0002-9939-2010-10430-6
Received by editor(s): November 3, 2009
Received by editor(s) in revised form: February 4, 2010
Published electronically: June 4, 2010
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society