Categories of abelian varieties over finite fields II: Abelian varieties over $${\mathbb{F}_q}$$ and Morita equivalence

Abstract

AbstractThe category of abelian varieties over $${\mathbb{F}_q}$$ F q is shown to be anti-equivalent to a category of ℤ-lattices that are modules for a non-commutative pro-ring of endomorphisms of a suitably chosen direct system of abelian varieties over $${\mathbb{F}_q}$$ 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.