On matrices of endomorphisms of abelian varieties
Zarhin, Yuri G.1
1 Pennsylvania State University, Department of Mathematics, University Park, PA 16802, USA
Mathematics Research Reports, Tome 1 (2020), pp. 55-68.

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

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DOI : 10.5802/mrr.5
Classification : 14K05, 16K20
Mots clés : Abelian varieties, Tate modules, Semisimple algebras
Zarhin, Yuri G. 1

1 Pennsylvania State University, Department of Mathematics, University Park, PA 16802, USA 
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Zarhin, Yuri G. On matrices of endomorphisms of abelian varieties. Mathematics Research Reports, Tome 1 (2020), pp. 55-68. doi : 10.5802/mrr.5. http://www.numdam.org/articles/10.5802/mrr.5/

[1] Bourbaki, N. 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 | Zbl

[2] Bourbaki, N. 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 | Zbl

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

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

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

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

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

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

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

[10] Zarhin, Yuri G. 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.) | DOI | MR

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