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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