Suppose $\rho:G \_{\mathbb{Q}} \rightarrow GL\_n(\mathbb{Q}\_p)$ is a Galois rep. It has a uniquely defined (up to semisimplification residual rep $\bar{\rho}$. $\bar{\rho}$ is unramified where $\rho$ is. I have examples where it becomes good at a prime where $\rho$ is bad. Can it become more ramified then $\rho$? What constraints on the degree of the ramification of $\bar{\rho}$ do we know in terms of ramification of $\rho$.


Suppose now we have an $L$ function, and know the bad factors of the $L$ function. Does this help us further? What if $p$ is itself a bad prime: can we still say anything? Note that I cannot assume that this Galois rep is motivic.



How much extra ramification in a residual representation
All this is completely worked out in an old paper of Carayol whose name escapes me -- just a few pages, probably mid to late 80s or early 90s. In short: assuming you're not talking about ramification at $p$ then the Swan conductors of $\rho$ and its reduction coincide, so the only change in conductor can be "tame". No, the reduction can't be more ramified than $\rho$; in general if the conductor of $\rho$ at $\ell$ is $\ell^a$ and the conductor of the reduction is $\ell^b$ then $b\leq a\leq b+2$ and one can explicitly classify when $b\not=a$ can occur. This was all worked out when people were trying to prove that the various forms of Serre's conjecture (a mod $p$ representation is modular, vs a mod p representation is modular of this precise level) are equivalent. Probably Ribet's Inventiones 100 paper will reference Carayol's paper? Ribet level-lowered in the only case that Carayol left open; Carayol's methods were for the most part local.


