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Proceedings of the American Mathematical Society

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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
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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 \begin{equation*} \#\{\psi \in \Psi _{p,k}\mid L(\psi ,k)\not =0\}\geq \frac {h(-p)}{\#\operatorname {Cl}(K)[2k-1]}. \end{equation*}


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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

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.