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Computing base extensions of ordinary abelian varieties over finite fields

Stefano Marseglia

TL;DR

The paper develops a module-theoretic framework based on Deligne’s equivalence $AV^{ord}(q) \simeq \mathcal{M}^{ord}(q)$ to study base-field extensions of ordinary abelian varieties over finite fields. It translates base-change $A \mapsto A_r$ into explicit operations on $R_g$-modules via the functor $\mathcal{E}_2$ and characterizes isomorphism classes and twists through cases such as hr irreducible and Bass orders, with concrete algorithms. It yields methods to compute minimal fields of definition and to decide when two varieties are twists, supported by numerous examples across isogeny classes. It also develops a Galois-cohomology framework to classify twists and descent criteria for fields of definition, giving effective criteria and explicit equations in several descent scenarios. Together, these results provide practical, computational tools for understanding the arithmetic of ordinary abelian varieties over finite fields.

Abstract

We study base field extensions of ordinary abelian varieties defined over finite fields using the module theoretic description introduced by Deligne. As applications we give algorithms to determine the minimal field of definition of such a variety and to determine whether two such varieties are twists.

Computing base extensions of ordinary abelian varieties over finite fields

TL;DR

The paper develops a module-theoretic framework based on Deligne’s equivalence to study base-field extensions of ordinary abelian varieties over finite fields. It translates base-change into explicit operations on -modules via the functor and characterizes isomorphism classes and twists through cases such as hr irreducible and Bass orders, with concrete algorithms. It yields methods to compute minimal fields of definition and to decide when two varieties are twists, supported by numerous examples across isogeny classes. It also develops a Galois-cohomology framework to classify twists and descent criteria for fields of definition, giving effective criteria and explicit equations in several descent scenarios. Together, these results provide practical, computational tools for understanding the arithmetic of ordinary abelian varieties over finite fields.

Abstract

We study base field extensions of ordinary abelian varieties defined over finite fields using the module theoretic description introduced by Deligne. As applications we give algorithms to determine the minimal field of definition of such a variety and to determine whether two such varieties are twists.

Paper Structure

This paper contains 12 sections, 14 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Isogeny classes and field extensions
  4. Isomorphism classes and field extensions
  5. Computations in ${{\mathcal{B}}}({g,s})$
  6. Isomorphism classes when $h_r$ is irreducible
  7. Isomorphism classes when $R_g$ is Bass
  8. Isomorphism testing
  9. Applications to abelian varieties
  10. Twists
  11. Galois cohomology
  12. Field of definition

Key Result

Theorem 2.1

Del69 There is an equivalence of categories If ${\mathcal{F}}^{\text{ord}}(A)=(T,F)$ then $\mathop{\mathrm{rank}}\nolimits_\mathbb{Z}(T)=2\dim(A)$ and $F$ corresponds to the Frobenius endomorphism of $A$.

Theorems & Definitions (36)

  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.3
  • Proposition 3.1: chaiconradoort14
  • Example 3.2
  • Proposition 3.3
  • proof
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • ...and 26 more