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Abelian varieties isogenous to a power of an elliptic curve

Part of: Abelian varieties and schemes Arithmetic algebraic geometry

Published online by Cambridge University Press:  21 March 2018

Bruce W. Jordan ,
Allan G. Keeton ,
Bjorn Poonen ,
Eric M. Rains ,
Nicholas Shepherd-Barron  and
John T. Tate
Bruce W. Jordan
Affiliation:
Department of Mathematics, Baruch College, The City University of New York, One Bernard Baruch Way, New York, NY 10010-5526, USA email [email protected]
Allan G. Keeton
Affiliation:
Center for Communications Research, 805 Bunn Drive, Princeton, NJ 08540-1966, USA email [email protected]
Bjorn Poonen
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA email [email protected]
Eric M. Rains
Affiliation:
Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA email [email protected]
Nicholas Shepherd-Barron
Affiliation:
Mathematics Department, King’s College London, Strand, London WC2R 2LS, UK email [email protected]
John T. Tate
Affiliation:
Mathematics Department, Harvard University, 1 Oxford Street, Cambridge MA 02138-2901, USA email [email protected]
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Abstract

Let $E$ be an elliptic curve over a field $k$. Let $R:=\operatorname{End}E$. There is a functor $\mathscr{H}\!\mathit{om}_{R}(-,E)$ from the category of finitely presented torsion-free left $R$-modules to the category of abelian varieties isogenous to a power of $E$, and a functor $\operatorname{Hom}(-,E)$ in the opposite direction. We prove necessary and sufficient conditions on $E$ for these functors to be equivalences of categories. We also prove a partial generalization in which $E$ is replaced by a suitable higher-dimensional abelian variety over $\mathbb{F}_{p}$.

Keywords

Abelian varietyelliptic curveisogeny

MSC classification

Primary: 14K15: Arithmetic ground fields
Secondary: 11G10: Abelian varieties of dimension $> 1$ 14K02: Isogeny 14K05: Algebraic theory
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 5 , May 2018 , pp. 934 - 959
Copyright
© The Authors 2018 

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