#### Vol. 9, No. 1, 2015

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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
##### Abstract

We assign functorially a $ℤ$-lattice with semisimple Frobenius action to each abelian variety over ${\mathbb{F}}_{\phantom{\rule{0.3em}{0ex}}p}$. This establishes an equivalence of categories that describes abelian varieties over ${\mathbb{F}}_{\phantom{\rule{0.3em}{0ex}}p}$ 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
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