Physics, mathematics, and technology

2016 Issue №4

Back to the list Download the article

A review of the existing generalizations of the Deuring Reduction Theorem



The article is focused on various generalizations of the Deuring Reduction Theorem. Our research proves that the most appropriate theorem for further elaboration is the one that relates the decomposition of pK into prime ideals with the decomposition of A[p] into indecomposable BT1-group schemes up to isomorphism. The article investigates basic problems of the theorem's further generalization and some ways of solving them as well as formulates tasks for further work in this direction.


1. Benger N., Charlemagne M., Freeman D. M. On the Security of Pairing-Friendly Abelian Varieties over Non-prime Fields // Pairing-Based Cryptography–Pairing 2009, Lecture Notes in Comput. Sci., vol. 5671. Springer, 2009. P. 52—65.
2. Blake C. A Deuring criterion for abelian varieties // Bulletin of the London Mathematical Society. Dec. 2014. Vol. 46, issue 6. P. 1256.
3. Bradford J. Commutative Endomorphism Rings of Simple Abelian Varieties over Finite Fields. Ph. D. Thesis, University of Maryland, 2012.
4 Clark P. L. Bounds for torsion on abelian varieties with integral moduli // arXiv:math/0407264, 2004.
5. Clark P. L., Xarles X. Local bounds for torsion points on abelian varieties // Canad. J. Math. 2008. Vol. 60. P. 532—555.
6. Demazure M. Lectures on p-divisible groups // Lecture Notes in Mathematics, vol. 302. Berlin, 1986.
7. Dodson B. The structure of galois groups of cm-fields // ITransactions of the American Mathematical Society. 1984. № 283(1). Р. 1—32.
8. Ekedahl T. On supersingular curves and abelian varieties // Math. Scand. 1987. № 60. Р. 151—178.
9. Van der Geer G. Cycles on the moduli space of abelian varieties // Moduli of curves and abelian varieties, Aspects Math, E33. 1999. P. 65—89.
10. Goren E. On certain reduction problems concerning abelian surfaces // Manuscripta Math. 1997. № 94. Р. 33—43.
11. Goren E. Lectures on Hilbert modular varieties and modular forms // CRM Monograph, vol. 14. AMS 2002.
12. Kraft H. Kommutative algebraische p-gruppen (mit anwendungen auf p-divisible gruppen und abelsche varietäten) // Manuscript, University of Bonn, September, 1985.
13. Moonen B. Group schemes with additional structures and Weyl group cosets // Moduli of Abelian varieties. 2001. Vol. 195. P. 255—298.
14. Oort F. Simple p-kernels of p-divisible groups // Advances in Mathematics. 2005. № 198. Р. 275—310.
15. Oort F. A stratification of a moduli space of abelian varieties // Moduli of abelian varieties. 2001. Vol. 195. P. 345—416.
16. Pries R. A short guide to p-torsion of abelian varieties in characteristic p // arXiv:math/0609658v1, September, 2006.
17. Ribet K. Torsion points of abelian varieties in cyclotomic extensions // L’enseignement Mathématique. 1981. Vol. 27. P. 315—319.
18. Sugiyama K.-I. On a generalization of Deuring’s results // Finite Fields  and Their Applications. March 2014. Vol. 26. P. 69—85.
19. Xarles X. Torsion points of abelian varieties over p-adic fields // preprint. URL:
20. Yu C.-F. A note on supersingular abelian varieties // arXiv:1412.7107, 2015. 
21. Zaytsev A. Generalization of Deuring reduction theorem // Journal of Algebra. 2013. Vol. 392. P. 97—114.