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It is well-known that "the" stack of elliptic curves (allow me to be vague as to singular curves, compactifications etc) has a presentation by a groupoid in schemes. One of the things that needs to be proved to see this is that the sheaves of isomorphisms between two elliptic curves over a base is representable (I'm simplifying a bit, clearly), say by an algebraic space (see e.g. L&M-B Corollaire 3.13)

What happens if we take instead the presheaf of isogenies between two elliptic curves? My guess is that it is a sheaf. But is it representable? I have no intuition at all.

Clearly elliptic curves and isogenies are not an algebraic stack as usually defined: they do not form a category fibred in groupoids. I'm asking specifically about the (pre)sheaf of isogenies here, and not elliptic curves up to isogeny. This has applications to Charles Rezk's work discussed in his ICM talk, if people want motivation.

One could ask analogous questions about abelian varieties, but I'll hold off.

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In general you have a Hom scheme $\mathrm{Hom}_S(X,Y)$ for $X$ and $Y$ two schemes over $S$ whenever $S$ is noetherian, $X$ flat and projective, $Y$ quasi-projective. It decomposes into connected components depending on the Hilbert polynomial of the graph of $f$ in $X \times_S Y$. If $X$ and $Y$ are curves, this gives $\mathrm{Hom}_S(X,Y) = \coprod_d \mathrm{Hom}^d_S(X,Y)$ where $\mathrm{Hom}^d_S(X,Y)$ parametrizes maps of degree $d$. Now an isogeny of elliptic curves is just a map of positive degree which preserves the origin, so you're done.

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  • $\begingroup$ Awesome! So the sheaf $\underline{Isog}_S(E_1,E_2)$ of isogenies is represented by a subscheme of $Hom_S(E_1,E_2)$? $\endgroup$ Sep 19 '14 at 8:47
  • $\begingroup$ Yep. (blank space) $\endgroup$ Sep 19 '14 at 8:48
  • $\begingroup$ I've been pondering whether this has been possible for a long, long time, but procrastinating as I thought it something completely off the wall. So thanks! $\endgroup$ Sep 19 '14 at 8:48
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It should be representable by an ind-scheme. You should look at the papere by Mumford "On the equations defining abelian varieties II", section 9. He calls this moduli space $\mathcal{M}_{\infty}$, it parametrizes towers of abelian varieties.

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  • $\begingroup$ Mumford seems to be working up to isogeny, though I may have misread that. Also, his scheme $\mathcal{M}_\infty$ doesn't seem at all related to what I asked! $\endgroup$ Sep 19 '14 at 8:45

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