Nonvanishing of Hecke L–functions for Cm Fields and Ranks of Abelian Varieties

In this paper we prove a nonvanishing theorem for central values of L– functions associated to a large class of algebraic Hecke characters of CM number fields. A key ingredient in the proof is an asymptotic formula for the average of these central values. We combine the nonvanishing theorem with work of Tian and Zhang [TZ] to deduce that infinitely many of the CM abelian varieties associated to these Hecke characters have Mordell-Weil rank zero. Included among these abelian varieties are higher dimensional analogues of the elliptic Q-curves A(D) of B. Gross [Gr].