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




Williams, Cassandra L., author
Achter, Jeffrey, advisor
Eykholt, Richard, committee member
Hulpke, Alexander, committee member
Penttila, Tim, committee member

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The Frobenius endomorphism of an abelian variety over a finite field Fq of dimension g can be considered as an element of the finite matrix group GSp2g(Z/lr). 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 Q. 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(Z/lr) and GSp4(Z/lr. 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.


2012 Summer.
Includes bibliographical references.

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abelian variety
conjugacy class
complex multiplication


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