10
$\begingroup$

Suppose $f$ is a weight $2$ level $N$ cusp form. When can we realize the mod-$\ell$ representation of $f$ in a form of weight $2$ and level $Np^3$, 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-$\ell$ representation of $f$ restricted to inertia at $p$, but I'm not sure what it would say...

$\endgroup$

1 Answer 1

10
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ Sorry Kevin (and FC), I simply forgot to do so! No judgment was implied, only my own foolishness. $\endgroup$ Nov 6, 2009 at 0:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.