On matrices of endomorphisms of abelian varieties
Mathematics Research Reports, Volume 1 (2020) , pp. 55-68.

We study endomorphisms of abelian varieties and their action on the -adic Tate modules. We prove that for every endomorphism one may choose a basis of each -Tate module such that the corresponding matrix has rational entries and does not depend on .

Received: 2020-02-01
Accepted: 2020-05-25
Published online: 2020-10-14
DOI: https://doi.org/10.5802/mrr.5
Classification: 14K05,  16K20
Keywords: Abelian varieties, Tate modules, Semisimple algebras
@article{MRR_2020__1__55_0,
     author = {Yuri G. Zarhin},
     title = {On matrices of endomorphisms of abelian varieties},
     journal = {Mathematics Research Reports},
     publisher = {MathOA foundation},
     volume = {1},
     year = {2020},
     pages = {55-68},
     doi = {10.5802/mrr.5},
     language = {en},
     url = {mrr.centre-mersenne.org/item/MRR_2020__1__55_0/}
}
Yuri G. Zarhin. On matrices of endomorphisms of abelian varieties. Mathematics Research Reports, Volume 1 (2020) , pp. 55-68. doi : 10.5802/mrr.5. https://mrr.centre-mersenne.org/item/MRR_2020__1__55_0/

[1] N. Bourbaki Éléments de mathématique. Fasc. XXVI. Groupes et algèbres de Lie. Chapitre I: Algèbres de Lie, Seconde édition. Actualités Scientifiques et Industrielles, No. 1285, Hermann, Paris, 1971, 146 pages | MR 0271276 | Zbl 0213.04103

[2] N. Bourbaki Éléments de mathématique. Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII: Sous-algèbres de Cartan, éléments réguliers. Chapitre VIII: Algèbres de Lie semi-simples déployées, Actualités Scientifiques et Industrielles, No. 1364. Hermann, Paris, 1975, 271 pages | MR 0453824 | Zbl 0329.17002

[3] I. N. Herstein Noncommutative rings, Carus Mathematical Monographs, Volume 15, Mathematical Association of America, Washington, DC, 1994, xii+202 pages (Reprint of the 1968 original) | MR 1449137 | Zbl 0177.05801

[4] Nathan Jacobson Lie algebras, Dover Publications, Inc., New York, 1979, ix+331 pages (Republication of the 1962 original) | MR 559927 | Zbl 0121.27504

[5] Serge Lang Algebra, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984, xv+714 pages | MR 783636 | Zbl 0712.00001

[6] David Mumford Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Oxford University Press, 1974, viii+279 pages | Zbl 0326.14012

[7] I. Reiner Maximal orders, London Mathematical Society Monographs. New Series, Volume 28, The Clarendon Press, Oxford University Press, Oxford, 2003, xiv+395 pages (Corrected reprint of the 1975 original) | MR 1972204 | Zbl 1024.16008

[8] Kenneth A. Ribet Galois action on division points of Abelian varieties with real multiplications, Amer. J. Math., Volume 98 (1976) no. 3, pp. 751-804 | Article | MR 457455 | Zbl 0348.14022

[9] John Tate Endomorphisms of abelian varieties over finite fields, Invent. Math., Volume 2 (1966), pp. 134-144 | Article | MR 206004 | Zbl 0147.20303

[10] Yuri G. Zarhin Endomorphism algebras of abelian varieties with special reference to superelliptic Jacobians, Geometry, algebra, number theory, and their information technology applications (Springer Proc. Math. Stat.) Volume 251, Springer, Cham, 2018, pp. 477-528 ((A. Akbary S. Gun, eds.)) | Article | MR 3880401