Consider the stack $Ell$ (of groupoids) of elliptic curves. I'm interested in the autoequivalence 2-group of $Ell$, the objects of which consists of transformations $Ell \Rightarrow Ell: Ring \to Gpd$ valued in equivalences of groupoids. The arrows are isomorphisms of such transformations.

In a chat opinions ranged from optimistic that it would be large to hunches it would be small (in fact $\mathbb{Z}/2 \rightrightarrows \ast$)

Even if this 2-group is very small, I would also be interested in knowing if it's possible to calculate much of the *endomorphism* monoidal groupoid of $Ell$, given that we know a very explicit presentation of it (using a Hopf algebroid built from finitely generated polynomial rings). Here we would take all transformations, not just those valued in equivalences.

EDIT, Will Jagy: to get fuller context, you can scroll arbitrarily, and conveniently, back in the transcript http://chat.stackexchange.com/transcript/9417/2013/6/28/5-16 where David's announcement of this question occurs pretty late in this segment. I will check later today, the terminal hour marker '16' may change, that is how the system works.

group... – David Roberts Jun 28 '13 at 10:06