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. doi:10.1112/S0010437X17007990. 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:

URL	URL Type	Description
https://doi.org/10.1112/S0010437X17007990	DOI	Article
https://arxiv.org/abs/1602.06237	arXiv	Discussion 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 Agency	Grant Number
NSF	DMS-1069236
NSF	DMS-1601946
Simons Foundation	340694
Simons Foundation	402472

Issue or Number:

Classification Code:

MSC: 14K15 (primary); 11G10; 14K02; 14K05 (secondary)

DOI:

10.1112/S0010437X17007990

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

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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:

15 Nov 2021 19:45

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