That's the one, @EvanJenkins
ok, so I now have my affine groupoid scheme
(cont.) ... So as the stack is so concrete, we can try to calculate this automorphism 2-group.
My first question is, has this been done?
At least, away from the primes 2 and 3 there shouldn't be massive problems.
Then, if we have this 2-group, one can ask whether the action lifts to objects related to Ell.
For instance, we have the universal elliptic curve over Ell, which we can view as the total space of a stack over Ell
[I think? Does this sound reasonable? Perhaps it's not a stack over Ell, but over the base site, but with a map to Ell...]
Then we also have the sheaf of ring spectra from which we get elliptic cohomology theories: is this an equivariant sheaf? (in the appropriately weak way)