Abelian varieties over finite fields and cyclic algebras over number fields
According to a result of Tate and Honda, a simple abelian variety A over a finite field is determined up to isogeny by an eigenvalue of the Frobenius endomorphism on the first étale cohomology group of A, the Weil number. The category of abelian varieties over finite fields
is much more complicated. Deligne proved that the category of ordinary abelian varieties over a finite field is equivalent to the category of ordinary Deligne modules. Centeleghe and Stix proved a more general result about the category of all abelian varieties. They fix a set of Weil numbers and construct an equivalence from the category of abelian varieties with Frobenius eigenvalues from this set to a subcategory of modules over the endomorphism algebra of a balanced abelian variety. Over a prime field, the target category is the category of Deligne modules, but in general, this category is rather inexplicit.
We give a more direct generalization of the Deligne theorem. Namely, we show that the category of abelian varieties over a finite field with a given set of Frobenius eigenvalues is equivalent to a category of modules very similar to Deligne modules. First, we prove that the
endomorphism algebra of a simple abelian variety is a cyclic algebra of a very special kind. We then investigate how the category of torsion-free modules over an explicit order in a sum of such algebras is related to the given category of abelian varieties.
תאריך עדכון אחרון : 10/06/2025