Apologies for any basic mistakes in this question; I'm a beginner to this theory and don't have anyone at my institution to consult for advice.

What I mean is, if you take an elliptic curve $E$ over $\mathbb{Q}$ say, you should get associated an cohomology theory by taking some deformation of it/putting it in a family and completing. (At least as far as I know, the deformation is necessary, since otherwise LEFT fails at supersingular primes. Is this correct?)

It seems to me that you could pretty much do this with any formal group over $\mathbb{Z}$; just find some deformation so that the heights are all generically $1$, then apply LEFT. (Or is this maybe harder to do than I think? Seems it wouldn't be, though, just from stacky considerations.)

My question is, is there some intrinsic interest in the cohomology theories actually arising from elliptic curves, as opposed to these ones? Or is the reason for the interest entirely because of the geometry of the universal tmf, and its speculative connections to loop groups, the Witten genus, etc.?

Or maybe is the principal interest in the direction of searching for a good equivariant theory which encodes all the elliptic curve's geometry, instead of just its formal Neron model?

10more comments