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Proceedings of the American Mathematical Society
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
Published electronically: June 4, 2010
MathSciNet review: 2679611
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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]}. $


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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: http://dx.doi.org/10.1090/S0002-9939-2010-10430-6
PII: S 0002-9939(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.