2003/01/26 by Tonghai Yang, Yang, Tonghai · 1 citation
Computer Science · Mathematics · #11G05 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.math/0301306
openalex publication_date 2003/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we associate canonically to every imaginary quadratic field K=\Bbb Q(√(-D)) one or two isogenous classes of CM 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 \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.