On CM abelian varieties over imaginary quadratic fields

In this paper, we associate canonically to every imaginary quadratic field K=[inline-graphic not available: see fulltext] one or two isogenous classes of CM (complex multiplication) abelian varieties over K, depending on whether D is odd or even (D≠4). These abelian varieties are characterized as of smallest dimension and smallest conductor, and such that the abelian varieties themselves descend to [inline-graphic not available: see fulltext]. When D is odd or divisible by 8, they are the scalar restriction of ‘canonical’ elliptic curves 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.