Categories of abelian varieties over finite fields II: Abelian varieties over finite fields and Morita equivalence

TL;DR

This paper establishes a deep categorical equivalence between abelian varieties over finite fields and certain modules over non-commutative rings, revealing new structural insights and classifications.

Contribution

It introduces a novel anti-equivalence between abelian varieties over finite fields and modules over a non-commutative pro-ring, expanding the understanding of their structure.

Findings

Categorical anti-equivalence between abelian varieties and $Z$-lattices with endomorphism modules

Introduction of $w$-locally projective abelian varieties for specific subcategories

Structural classification of abelian varieties via non-commutative ring modules

Abstract

The category of abelian varieties over $\mathbb{F}_q$ is shown to be anti-equivalent to a category of $\mathbb{Z}$-lattices that are modules for a non-commutative pro-ring of endomorphisms of a suitably chosen direct system of abelian varieties over $\mathbb{F}_q$. On full subcategories cut out by a finite set $w$ of conjugacy classes of Weil $q$-numbers, the anti-equivalence is represented by what we call $w$-locally projective abelian varieties.