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For a scheme $X$, denote by $\mathcal{Ell}_X[\text{isog}^{-1}]$ the category of elliptic curves on $X$ localized at isogenies. Consider the functor $$ \mathcal{Ell}^{isog}:Sch/S^{op}\rightarrow \text{Gpd}, \quad X \rightarrow \mathcal{Ell}_X[\text{isog}^{-1}]. $$ It was asked in this M.SE question if this was an algebraic stack. It is not, because

"forget isomorphism class and remember only isogeny class" map from the usual moduli space ought to be algebraic, but an algebraic map of curves has finite fibers but (over C, say) isogeny classes of non-isomorphic curves are infinite. (https://math.stackexchange.com/q/675207)

I'm wondering if the functor $\mathcal{Ell}^{isog}$ if not represented by an algebraic stack is a $2$-sheaf. I don't have any intuition about that, so I'd be glad for any hint.

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  • $\begingroup$ You say the target is groupoids but you specified a set (isogeny classes); do you mean to specify the groupoid whose objects are elliptic curves and whose morphisms are... isogenies-up-to-scale or something like that? (I'm not sure what exactly you need to do to get a groupoid whose $\pi_0$ is isogeny classes.) $\endgroup$
    – Qiaochu Yuan
    Sep 28, 2020 at 2:10
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    $\begingroup$ Then specify a groupoid! "Isogeny classes of elliptic curves" is a set. $\endgroup$
    – Qiaochu Yuan
    Sep 28, 2020 at 2:40
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    $\begingroup$ Fair enough, apologies... :-( $\endgroup$
    – curious math guy
    Sep 28, 2020 at 2:41
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    $\begingroup$ Sorry - I don't mean to sound aggressive but there's an important point here about the difference between sheaves and stacks. $\endgroup$
    – Qiaochu Yuan
    Sep 28, 2020 at 2:58
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    $\begingroup$ Do you want the groupoid $Ell_X[isog^{-1}]$, where $Ell_X$ is the category of elliptic curves over $X$ and isogenies between them? That is, the universal way to get a groupoid from the category of elliptic curves and isogenies? $\endgroup$
    – David Roberts ♦
    Sep 28, 2020 at 4:26

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This does not satisfy the sheaf condition.

Consider a curve that is the union of two $\mathbb P^1$s, glued at $0$ and $\infty$. We can form an open cover consisting of the complement of $0$ and the complement of $\infty$, each two $\mathbb A^1$s glued at a point. The intersection of the cover is two disjoint $\mathbb G_m$s.

If you take two constant elliptic curves over the two open sets, and glue them by an isogeny which is two different isogenies of different degrees on the two copies of $\mathbb G_m$, there won't be a descent to the base curve.

The reason is that it's impossible to have a family of elliptic curves which is locally constant but when you travel around a loop in the base, has "monodromy" in the form of a self-isogeny of non-1 degree.

Alternately we could say that it's not possible to adjust this descent data by an isogeny on each open set to get the descent data for an honest family of elliptic curves.

However, maybe you could sheafify it.

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