Conjugacy classes of matrix groups over local rings and an application to the enumeration of abelian varieties

Abstract

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 GSp_2g([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.