Abelian varieties over finite fields with commutative endomorphism algebra: theory and algorithms
We give a categorical description of all abelian varieties with commutative endomorphism ring over a finite field with $q=p^a$ elements in a fixed isogeny class in terms of pairs consisting of a fractional $\mathbb Z[\pi,q/\pi]$-ideal and a fractional $W\otimes_{\mathbb Z_p} \mathbb Z_p[\pi,q/\pi]$-...
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Main Authors | , , |
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Format | Journal Article |
Language | English |
Published |
13.09.2024
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Subjects | |
Online Access | Get full text |
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Summary: | We give a categorical description of all abelian varieties with commutative
endomorphism ring over a finite field with $q=p^a$ elements in a fixed isogeny
class in terms of pairs consisting of a fractional $\mathbb Z[\pi,q/\pi]$-ideal
and a fractional $W\otimes_{\mathbb Z_p} \mathbb Z_p[\pi,q/\pi]$-ideal, with
$\pi$ the Frobenius endomorphism and $W$ the ring of integers in an unramified
extension of $\mathbb Q_p$ of degree $a$. The latter ideal should be compatible
at $p$ with the former and stable under the action of a semilinear Frobenius
(and Verschiebung) operator; it will be the Dieudonn\'e module of the
corresponding abelian variety. Using this categorical description we create
effective algorithms to compute isomorphism classes of these objects and we
produce many new examples exhibiting exotic patterns. |
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DOI: | 10.48550/arxiv.2409.08865 |