Computing endomorphism rings and Frobenius matrices of Drinfeld modules

Let Fq[T] be the polynomial ring over a finite field Fq. We study the endomorphism rings of Drinfeld Fq[T]-modules of arbitrary rank over finite fields. We compare the endomorphism rings to their subrings generated by the Frobenius endomorphism and deduce from this a refinement of a reciprocity law...

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Bibliographic Details
Published in:Journal of number theory 2022-08, Vol.237, p.145-164
Main Authors: Garai, Sumita, Papikian, Mihran
Format: Article
Language:English
Subjects:
Drinfeld modules
Endomorphism rings
Gorenstein rings
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Summary:Let Fq[T] be the polynomial ring over a finite field Fq. We study the endomorphism rings of Drinfeld Fq[T]-modules of arbitrary rank over finite fields. We compare the endomorphism rings to their subrings generated by the Frobenius endomorphism and deduce from this a refinement of a reciprocity law for division fields of Drinfeld modules proved in our earlier paper. We then use these results to give an efficient algorithm for computing the endomorphism rings and discuss some interesting examples produced by our algorithm.
ISSN:0022-314X
1096-1658
DOI:10.1016/j.jnt.2019.11.018