CRM: Centro De Giorgi
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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.

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