2020/05/29 by Everett W. Howe, Howe, Everett W.
Computer Science · Mathematics · Social Sciences · #11G10 (Primary) 11G15 #11G25 #14G15 #14K15 #14K22 (Secondary) #Algebraic Geometry (math.AG) #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Historical Geopolitical and Social Dynamics #Number Theory (math.NT) #math.AG #math.NT #msc:11G10 #msc:11G15 #msc:11G25 #msc:14G15 #msc:14K15 #msc:14K22
paper · pdf · doi:10.48550/arxiv.2005.14365
21 pages
arxiv created 2020/05/29 · openalex publication_date 2020/05/29 · arxiv updated 2020/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that for a large class of rings R, the number of principally polarized abelian varieties over a finite field in a given simple ordinary isogeny class and with endomorphism ring R is equal either to 0, or to a ratio of class numbers associated to R, up to some small computable factors. This class of rings includes the maximal order of the CM field K associated to the isogeny class (for which the result was already known), as well as the order R generated over Z by Frobenius and Verschiebung. For this latter order, we can use results of Louboutin to estimate the appropriate ratio of class numbers in terms of the size of the base field and the Frobenius angles of the isogeny class. The error terms in our estimates are quite large, but the trigonometric terms in the estimate are suggestive: Combined with a result of Vladut on the distribution of Frobenius angles of isogeny classes, they give a heuristic argument in support of the theorem of Katz and Sarnak on the limiting distribution of the multiset of Frobenius angles for principally polarized abelian varieties of a fixed dimension over finite fields.