Let $k$ be a finite field of char $p \geq 3$. Given an absolutely irreducible, continuous, odd representation $\overline{\rho}: G\_\mathbb{Q} \longrightarrow GL\_2(k)$ and a deformation condition $D$ for $\overline{\rho}$, let $S(D)$ be the collection of all newforms with associated $p$-adic representation in $D$. If $f \in S(D)$ then is its level bounded? I remember reading somewhere that one might work out the level using local Langlands but do not recall the reference or the argument.



Bounding the level for eigenforms satisfying a deformation condition
Let's bound the level of such an $f$ in two stages. Firstly, let's look at a prime $\ell \ne p$. Here there is a theorem of Livne and (independently) Carayol which says that if $\rho$ is a lifting of $\bar\rho$, the exponent of $\ell$ dividing the Artin conductor of $\rho$ is bounded (it's at most 2 more than the $\ell$-conductor of $\bar\rho$). That leaves just the power of $p$ dividing the level to be controlled; and it's easy to see that if you require $\rho|\_{D\_p}$ to be upper-triangular, with fixed determinant and Hodge--Tate weights, then the conductors of the characters occuring along the diagonal are bounded above and this gives you a bound on the level of $f$ at $p$.


