Third Italian Number Theory Meeting

# Abelian varieties over finite fields (based on joint work with J. Stix)

speaker: Tommaso Centeleghe (Universitaet Heidelberg)

abstract: Let $$p$$ be a prime number, $$\mathbf{F}_q$$ "the" finite field of characteristic $$p$$ and order $$q=p^e$$, and $${\sf AV}_q$$ the category of abelian varieties over $$\mathbf{F}_q$$. In an ongoing project the authors study the problem of concretely describing $${\sf AV}_q$$ and certain full subcategories in terms of linear algebra data. The general method is inspired by Waterhouse's thesis and builds on the construction of a lattice $$T(A)$$ functorially attached to any object $$A$$ of $${\sf AV}_q$$ and equipped with the extra structure given by the action of a certain pro-ring $$\mathcal{R}_q$$. The first result obtained pertains only the case $$q=p$$ and asserts the existence of an equivalence between the full subcategory $${\sf AV}_p^{\rm com}\subset {\sf AV}_p$$ of varieties whose Frobenius operator avoids the eigenvalues $\pm\sqrt{p}$, and the category of pairs $$(T, F)$$ consisting of a finite free $$\mathbf{Z}$$-module $$T$$ and a linear operator $$F:T\to T$$ satisfying two axioms easy to state. This generalizes the $$q=p$$ case of theorem of Deligne from '69, who gave an analogous description of the subcategory $${\sf AV}_q^{\rm ord}\subset{\sf AV}_q$$ of ordinary abelian varieties. While the main ingredient in Deligne's proof is Serre-Tate theory of canonical liftings of ordinary abelian varieties, the method employed by the authors relies on the Gorenstein property of the orders generated by a Weil $$p$$-numbers and it complex conjugate, and avoids lifting objects to characteristic zero.

timetable:
Tue 22 Sep, 16:30 - 16:55, Aula Contini
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