Question

Suppose f is a weight 2 level N cusp form. When can we realize the mod- representation of f in a form of weight 2 and level Np³, where p is some prime not dividing N? I assume that, if a simple criterion exists at all, it is a condition on the mod- representation of f restricted to inertia at p, but I'm not sure what it would say...

Answer 1

Presumably you want the form (let me call it g) of level Np^3 to be new at p, otherwise it's trivial.

Let me also assume ell isn't p.

If the form g is new at p, and has level Gamma0(p^3) at p, then the ell-adic representation attached to g will have conductor p^3. But this is a bit of a problem, because the conductor of the mod ell representation can't be that much lower than the conductor of the ell-adic representation. Indeed a theorem of Carayol and, independently, Livne, says that the p-conductor of the mod ell representation will be at least p if the p-conductor of the ell-adic representation is p^3 (the exponent can drop by at most 2). So if you're looking for Gamma_0(p^3) then you're in trouble. This is just a local calculation and isn't too deep.

Diamond and Taylor, in their second paper on the subject, give a list of the conductors of the newforms that can give rise to a given irreducible modular mod ell representation. You can see that Gamma0(p^3) is too much from the main theorem there. Of course the work in that theorem is realising everything that is possible, not ruling out everything that isn't.