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$\newcommand{\End}{\operatorname{End}}$For an elliptic curve $E$, I understand that the notation $\End(E)$ denotes the ring of endomorphisms of $E$. Since $\End(E)$ is torsion free, it's possible to take $\alpha, \beta \in \End(E)$ and construct objects of the form $\frac{\alpha}{\beta} \cong \End(E)\otimes_{\mathbb{Z}} \mathbb{Q}$. Many authors denote this division ring by $\End^0(E)$. (Example: Ben Smith's slide 9 related complex multiplication. I have seen it at numerous other places.)

My question is, what does the $0$ in $\End^0(E)$ indicate? Also, are there other objects like $\End^1(E)$, etc.?

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    $\begingroup$ I doubt it means anything beyond "the thing tensored with $\mathbb Q$". To the best of my knowledge, there is no such thing as $End^1$. $\endgroup$
    – Wojowu
    Commented Nov 2, 2018 at 22:49
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    $\begingroup$ Not sure if this is the earliest usage, but it's the notation that Mumford uses in his classic book on Abelian Varieties. Indeed, he defines $\text{Hom}^0(X,Y)=\mathbb{Q}\otimes_{\mathbb{Z}}\text{Hom}(X,Y)$, as well as the analogous notation for End. This gives a new category in which the objects are abelian varieties and the morphisms are in Hom$^0$, which Mumford says is "the so-called category of 'abelian varieties up to isogeny'." $\endgroup$
    – Joe Silverman
    Commented Nov 3, 2018 at 2:15
  • $\begingroup$ For motives there is the notion of correspondences of degree $r$. One has $\mathrm{Corr}^0(X,Y) = \mathrm{Hom}(h(X),h(Y))$. See e.g. Scholl's article in the Motives volume I. The Hom between $h^1$ is the group of isogenies, $\mathrm{End}_{\mathcal{M}_\mathbf{Q}}(h^1(E)) = \mathrm{End}(E) \otimes_\mathbf{Z} \mathbf{Q}$ (Proposition 4.5). $\endgroup$
    – user19475
    Commented Nov 3, 2018 at 5:21

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