Level raising by prime powers - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T11:17:30Z http://mathoverflow.net/feeds/question/4268 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/4268/level-raising-by-prime-powers Level raising by prime powers David Hansen 2009-11-05T16:35:46Z 2009-11-05T18:38:28Z <p>Suppose f is a weight 2 level N cusp form. When can we realize the mod-<img src="http://latex.mathoverflow.net/png?%5Cell" alt="\ell" title="" /> representation of f in a form of weight 2 and level Np<sup>3</sup>, 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-<img src="http://latex.mathoverflow.net/png?%5Cell" alt="\ell" title="" /> representation of f restricted to inertia at p, but I'm not sure what it would say...</p> http://mathoverflow.net/questions/4268/level-raising-by-prime-powers/4282#4282 Answer by Kevin Buzzard for Level raising by prime powers Kevin Buzzard 2009-11-05T18:38:28Z 2009-11-05T18:38:28Z <p>Presumably you want the form (let me call it g) of level Np^3 to be new at p, otherwise it's trivial.</p> <p>Let me also assume ell isn't p.</p> <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.</p> <p>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.</p>