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

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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Published 1 January 2010