Dissertation/Thesis Abstract

Conjugacy classes of matrix groups over local rings and an application to the enumeration of abelian varieties
by Williams, Cassandra L., Ph.D., Colorado State University, 2012, 121; 3523830
Abstract (Summary)

The Frobenius endomorphism of an abelian variety over a finite field [special characters omitted] of dimension g can be considered as an element of the finite matrix group GSp2g([special characters omitted]). The characteristic polynomial of such a matrix defines a union of conjugacy classes in the group, as well as a totally imaginary number field K of degree 2g over [special characters omitted]. Suppose g = 1 or 2. We compute the proportion of matrices with a fixed characteristic polynomial by first computing the sizes of conjugacy classes in GL2([special characters omitted]) and GSp4([special characters omitted]). Then we use an equidistribution assumption to show that this proportion is related to the number of abelian varieties over a finite field with complex multiplication by the maximal order of K via a theorem of Everett Howe.

Indexing (document details)
Advisor: Achter, Jeffrey
Commitee: Eykholt, Richard, Hulpke, Alexander, Penttila, Tim
School: Colorado State University
Department: Mathematics
School Location: United States -- Colorado
Source: DAI-B 74/01(E), Dissertation Abstracts International
Subjects: Mathematics
Keywords: Abelian varieties, Complex multiplication, Conjugacy class
Publication Number: 3523830
ISBN: 9781267571700