Thanks to a result of Deligne, the category of ordinary abelian varieties
over a fixed finite field can be described in terms of finite free Z-modules
equipped with a linear operator F (playing the role of Frobenius) satisfying
certain axioms. Deligne's result is based on Serre-Tate canonical lifting.
In a recent joint work with Tommaso Centeleghe, we prove a similar result
for the full subcategory of all abelian varieties over the prime field
supported on non-real Weil numbers, thereby obtaining a description of
non-ordinary isogeny classes in Deligne's spirit. However, we must
substitute lifting to characteristic zero by the commutative algebra of
Gorenstein orders, because thanks to work of Chai, Conrad and Oort
functorial lifting is not always possible.
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