Level raising by prime powers

2009-11-05T16:35:46Z

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 by Kevin Buzzard for Level raising by prime powers

2009-11-05T18:38:28Z

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.

Level raising by prime powers - MathOverflow

Level raising by prime powers

Answer by Kevin Buzzard for Level raising by prime powers