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.
|Commitee:||Eykholt, Richard, Hulpke, Alexander, Penttila, Tim|
|School:||Colorado State University|
|School Location:||United States -- Colorado|
|Source:||DAI-B 74/01(E), Dissertation Abstracts International|
|Keywords:||Abelian varieties, Complex multiplication, Conjugacy class|
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