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Abelian varieties isogenous to a power of an elliptic curve

Jordan, Bruce W. and Keeton, Allan G. and Poonen, Bjorn and Rains, Eric M. and Shepherd-Barron, Nicholas and Tate, John T. (2018) Abelian varieties isogenous to a power of an elliptic curve. Compositio Mathematica, 154 (5). pp. 934-959. ISSN 0010-437X. https://resolver.caltech.edu/CaltechAUTHORS:20170922-140322875

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Abstract

Let E be an elliptic curve over a field k. Let R:=End E. There is a functor Hom_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 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 F_p.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1112/S0010437X17007990DOIArticle
https://arxiv.org/abs/1602.06237arXivDiscussion Paper
Additional Information:© The Authors 2018. Published online: 21 March 2018. B.P. was supported in part by National Science Foundation grant DMS-1069236 and DMS-1601946 and grants from the Simons Foundation (#340694 and #402472 to Bjorn Poonen). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation or the Simons Foundation. It is a pleasure to thank Everett Howe, Tony Scholl, and Christopher Skinner for helpful discussions. We thank also the referees for valuable suggestions on the exposition.
Funders:
Funding AgencyGrant Number
NSFDMS-1069236
NSFDMS-1601946
Simons Foundation340694
Simons Foundation402472
Issue or Number:5
Classification Code:MSC: 14K15 (primary); 11G10; 14K02; 14K05 (secondary)
Record Number:CaltechAUTHORS:20170922-140322875
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170922-140322875
Official Citation:Jordan, B., Keeton, A., Poonen, B., Rains, E., Shepherd-Barron, N., & Tate, J. (2018). Abelian varieties isogenous to a power of an elliptic curve. Compositio Mathematica, 154(5), 934-959. doi:10.1112/S0010437X17007990
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:81760
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:22 Sep 2017 21:35
Last Modified:03 Oct 2019 18:46

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