Mathematical Research Letters

Volume 17 (2010)

Number 4

CM cycles and nonvanishing of class group $L$–functions

Pages: 749 – 760

DOI: http://dx.doi.org/10.4310/MRL.2010.v17.n4.a13

Author

Riad Masri (University of Wisconsin, Madison, Wisconsin)

Abstract

Let $K$ be a totally imaginary quadratic extension of a totally real number field $F$, and assume that $F$ has narrow ideal class number 1. Let $\chi$ be a character of the ideal class group $\Cl(K)$ of $K$, and let $L_K(\chi,s)$ be its associated $L$–function. In this paper we prove that for all $\epsilon > 0$, \begin{align*} \# \{\chi \in \textrm{CL}(K)^{\wedge}:~L_K(\chi,\frac{1}{2}) \neq 0\} \gg_{\epsilon, F} d_K^{\frac{1}{100}-\epsilon} \end{align*} as the absolute discriminant $d_K \rightarrow \infty$.

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