Categories of abelian varieties over finite fields, I: Abelian varieties over 𝔽p

Centeleghe, Tommaso; Stix, Jakob

doi:10.2140/ant.2015.9.225

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Categories of abelian varieties over finite fields, I: Abelian varieties over $\mathbb{F}_{p}$

Tommaso Giorgio Centeleghe and Jakob Stix

Vol. 9 (2015), No. 1, 225–265

DOI: 10.2140/ant.2015.9.225

Abstract

We assign functorially a $ℤ$ -lattice with semisimple Frobenius action to each abelian variety over $F_{p}$ . This establishes an equivalence of categories that describes abelian varieties over $F_{p}$ avoiding $\sqrt{p}$ as an eigenvalue of the Frobenius in terms of simple commutative algebra. This result extends the isomorphism classification of Waterhouse and Deligne’s equivalence for ordinary abelian varieties.

Keywords

abelian varieties, finite fields, Gorenstein rings, reflexive modules

Mathematical Subject Classification 2010

Primary: 11-02

Milestones

Received: 21 July 2014

Revised: 5 December 2014

Accepted: 6 January 2015

Published: 18 February 2015

Authors

Tommaso Giorgio Centeleghe
IWR Universität Heidelberg Im Neuenheimer Feld 368 D-69120 Heidelberg Germany

Jakob Stix
Institut für Mathematik Goethe-Universität Frankfurt Robert-Mayer-Straße 6–8 D-60325 Frankfurt am Main Germany

Vol. 9, No. 1, 2015