Abelian varieties isogenous to a power of an elliptic curve

2016/02/19 by Bruce W. Jordan, Allan G. Keeton, Bjorn Poonen +3 · 1 voice · 1 citation
Mathematics · #math.AG #math.NT

paper · pdf · doi:10.1112/s0010437x17007990

published as Compositio Math. 154 (2018) 934-959 · 21 pages, comments welcome

arxiv published 2016/02/19 · arxiv created 2017/03/20 · arxiv updated 2019/02/20

Abstract

Let E be an elliptic curve over a field k. Let R:= End E. There is a functor \mathscrH 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 Hom(-,E) in the opposite direction. We prove necessary and sufficient conditions on E for these functors to be equivalences of categories.

Cited by

• Algebraic Modelings of the Supersingular Isogeny Problem

Discussions

• This module point of view is due to Serre and Tate, and was more recently revisited in arxiv.org/abs/1602.06237. More precisely they show that $A \mapsto M=\Hom_{F_p}(A, E_0)$ is an anti-equivalence o [bsky, 0 points, 1 comments]

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