Abelian varieties over finite fields with commutative endomorphism algebra: theory and algorithms

2024/09/13 by Jonas Bergström, Bergström, Jonas, Valentijn Karemaker +3 · 1 citation
Computer Science · Mathematics · #11G10 #11G25 #14-04 (Secondary) #14K15 (Primary) 14G15 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2409.08865

openalex publication_date 2024/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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[π,q/π]-ideal and a fractional W⊗_{\mathbb Zp} \mathbb Z_p[π,q/π]-ideal, with π 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 Dieudonné 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.

Cited by

• Isogeny graphs of abelian varieties and singular ideals in orders

Related

• Isogeny classes and endomorphisms algebras of abelian varieties over\n finite fields
• Isomorphism classes of Drinfeld modules over finite fields
• Labeling abelian varieties over finite fields
• Isogenies of certain abelian varieties over finite fields with p-ranks\n zero
• Categories of abelian varieties over finite fields I. Abelian varieties over \mathbbF_p