On CM abelian varieties over imaginary quadratic fields

Abstract

In this paper, we associate canonically to every imaginary quadratic field $K=\Bbb Q(\sqrt{-D})$ one or two isogenous classes of CM abelian varieties over $K$, depending on whether $D$ is odd or even ($D \ne 4$). These abelian varieties are characterized as of smallest dimension and smallest conductor, and such that the abelian varieties themselves descend to $\Bbb Q$. When $D$ is odd or divisible by 8, they are the `canonical' ones first studied by Gross and Rohrlich. We prove that these abelian varieties have the striking property that the vanishing order of their $L$-function at the center is dictated by the root number of the associated Hecke character. We also prove that the smallest dimension of a CM abelian variety over $K$ is exactly the ideal class number of $K$ and classify when a CM abelian variety over $K$ has the smallest dimension.