Isomorphism classes of Drinfeld modules over finite fields

Valentijn Karemaker, Jeffrey Katen, Mihran Papikian

Research output: Contribution to journalArticlepeer-review

Abstract

We study isogeny classes of Drinfeld A-modules over finite fields k with commutative endomorphism algebra D, in order to describe the isomorphism classes in a fixed isogeny class. We study when the minimal order A[π] of D generated by the Frobenius π occurs as an endomorphism ring by proving when it is locally maximal at π, and show that this happens if and only if the isogeny class is ordinary or k is the prime field. We then describe how the monoid of fractional ideals of the endomorphism ring E of a Drinfeld module ϕ up to D-linear equivalence acts on the isomorphism classes in the isogeny class of ϕ, in the spirit of Hayes. We show that the action is free when restricted to kernel ideals, of which we give three equivalent definitions, and determine when the action is transitive. In particular, the action is free and transitive on the isomorphism classes in an isogeny class which is either ordinary or defined over the prime field, yielding a complete and explicit description in these cases, which can be implemented as a computer algorithm.

Original languageEnglish (US)
Pages (from-to)381-410
Number of pages30
JournalJournal of Algebra
Volume644
DOIs
StatePublished - Apr 15 2024

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

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