Abstract
The category of abelian varieties over \({\mathbb{F}_q}\) is shown to be anti-equivalent to a category of ℤ-lattices that are modules for a non-commutative pro-ring of endomorphisms of a suitably chosen direct system of abelian varieties over \({\mathbb{F}_q}\). On full subcategories cut out by a finite set w of conjugacy classes of Weil q-numbers, the anti-equivalence is represented by what we call w-locally projective abelian varieties.
Article PDF
Avoid common mistakes on your manuscript.
References
J. Bergström, V. Karemaker and S. Marseglia, Polarizations of abelian varieties over finite fields via canonical liftings, International Mathematics Research Notices 2023 (2023), 3194–3248.
R. Brauer, H. Hasse and E. Noether, Beweis eines Hauptsatzes in der Theorie der Algebren, Journal für die reine und angewandte Mathematik 167 (1932), 399–404.
T. Centeleghe and J. Stix, Categories of abelian varieties over unite fields I: Abelian varieties over \({\mathbb{F}_p}\), Algebra & Number Theory 9 (2015), 225–265.
P. Deligne, Varietes abeliennes ordinaire sur un corps fini, Inventiones Mathematicae 8 (1969), 238–243.
P. Freyd, Abelian Categories, Harper’s Series in Modern Mathematics, Harper and Row, New York, 1964.
J. Giraud, Remarque sur une formule de Shimura-Taniyama, Inventiones Mathematicae 5 (1968), 231–236.
T. Honda, Isogeny classes of abelian varieties over finite fields, Journal of the Mathematical Society of Japan 20 (1968), 83–95.
E. W. Howe, Principally polarized ordinary abelian varieties over finite fields, Transactions of the American Mathematical Society 347 (1995), 2361–2401.
B. W. Jordan, A. G. Keeton, B. Poonen, E. M. Rains, N. Shepherd-Barron and J. T. Tate, Abelian varieties isogenous to a power of an elliptic curve, Compositio Mathematica 154 (2018), 934–959.
E. Kani, Products of CM elliptic curves, Collectanea Mathematica 62 (2011), 297–339.
K. Lauter, The maximum or minimum number of rational points on genus three curves over finite fields, Compositio Mathematica 134 (2002), 87–111.
J. S. Milne, Etale Cohomology, Princeton Mathematical Series, Vol. 33, Princeton University Press, Princeton, NJ, 1980.
A. Oswal and A. N. Shankar, Almost ordinary abelian varieties over unite fields, Journal of the London Mathematical Society 101 (2020), 923–937.
I. Reiner, Maximal Orders, London Mathematical Society Monographs, New Series, Vol. 28, The Clarendon press, Oxford University Press, Oxford, 2003.
M. Demazure and A. Grothendieck (eds.), Schémas en groupes. I: Propriétés générales des schémas en groupes. Lecture Notes in Mathematics, Vol. 151, Springer, Berlin-New York, 1970.
J. Tate, Endomorphisms of Abelian varieties over unite fields, Inventiones Mathematicae 2 (1966), 134–144.
J. Tate, Classes d’isogenie des variétés abéliennes sur un corps fini, in Sém. Bourbaki, Vol. 1968/69, Lecture Notes in Mathematics, Vol. 175, Springer, Berlin, pp. 95–110.
W. C. Waterhouse, Abelian varieties over finite fields, Annales Scientifiques de l’Ecole Normale Supérieure 2 (1969), 521–560.
W. C. Waterhouse, J. S. Milne, Abelian varieties over finite fields, in 1969 Number Theory Institute, Proceedings of Symposia in Pure Mathematics, Vol. 20, American Mathematical Society, Providence, RI, 1971, pp. 53–64.
C.-F. Yu, Superspecial abelian varieties over finite prime fields, Journal of Pure and Applied Algebra 216 (2012), 1418–1427.
Yu. G. Zarhin, Endomorphisms of abelian varieties and points of finite order in characteristic p, Matematicheskie Zametki 21 (1977), 737–744; English translatiion in Mathematical Notes 21 (1978), 415–419.
Yu. G. Zarhin, Almost isomorphic abelian varieties, European Journal of Mathematics 3 (2017), 22–33.
Acknowledgements
We are grateful to the anonymous referee for numerous suggestions that helped improve the presentation of our results.
Funding
Open Access funding enabled and organized by Projekt DEAL.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Moshe Jarden on the occasion of his 80th birthday
The second author acknowledges support by Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Centre TRR 326 “Geometry and Arithmetic of Uniformized Structures”, project number 444845124.
Rights and permissions
Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution and reproduction in any medium, provided the appropriate credit is given to the original authors and the source, and a link is provided to the Creative Commons license, indicating if changes were made (https://creativecommons.org/licenses/by/4.0/).
About this article
Cite this article
Centeleghe, T.G., Stix, J. Categories of abelian varieties over finite fields II: Abelian varieties over \({\mathbb{F}_q}\) and Morita equivalence. Isr. J. Math. 257, 103–170 (2023). https://doi.org/10.1007/s11856-023-2536-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-023-2536-2