# Galois representations attached to newforms

Suppose that $f$ is a weight $k$ newform for $\Gamma_1(N)$ with attached $p$-adic Galois representation $\rho_f$. Denote by $\rho_{f,p}$ the restriction of $\rho_f$ to a decomposition group at $p$. When is $\rho_{f,p}$ semistable (as a representation of $\mathrm{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$?

To make things really concrete, I'm happy to assume that $k=2$ and that the $q$-expansion of $f$ lies in $\mathbf{Z}[[q]]$.

Certainly if $N$ is prime to $p$ then $\rho_{f,p}$ is in fact crystalline, while if $p$ divides $N$ exactly once then $\rho_{f,p}$ is semistable (just thinking about the Shimura construction in weight 2 here, and the corresponding reduction properties of $X_1(N)$ over $\mathbf{Q}$ at $p$). For $N$ divisible by higher powers of $p$, we know that these representations are de Rham, hence potentially semistable. Can we say more? For example, are there conditions on "numerical data" attached to $f$ (e.g. slope, $p$-adic valuation of $N$, etc.) which guarantee semistability or crystallinity over a specific extension? Can we bound the degree and ramification of the minimal extension over which $\rho_{f,p}$ becomes semistable in terms of numerical data attached to $f$? Can it happen that $N$ is highly divisible by $p$ and yet $\rho_{f,p}$ is semistable over $\mathbf{Q}_p$?

I feel like there is probably a local-Langlands way of thinking about/ rephrasing this question, which may be of use...

As a possible example of the sort of thing I have in mind: if $N$ is divisible by $p$ and $f$ is ordinary at $p$ then $\rho_{f,p}$ becomes semistable over an abelian extension of $\mathbf{Q}p$ and even becomes crystalline over such an extension provided that the Hecke eigenvalues of $f$ for the action of $\mu_{p-1}\subseteq (\mathbf{Z}/N\mathbf{Z})^{\times}$ via the diamond operators are not all 1.

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can you give some references of the known results (for a learner) such as the pst of the attached Galois reps? Thank you! –  natura Feb 10 '10 at 18:17

The right way to do this sort of question is to apply Saito's local-global theorem, which says that the (semisimplification of the) Weil-Deligne representation built from $D_{pst}(\rho_{f,p})$ by forgetting the filtration is precisely the one attached to $\pi_p$, the representation of $GL_2(\mathbf{Q}_p)$ attached to the form via local Langlands. Your suggestions about the $p$-adic valuation of $N$ and so on are rather "coarse" invariants---$\pi_p$ tells you everything and is the invariant you really need to study.

So now you can just list everything that's going on. If $\pi_p$ is principal series, then $\rho$ will become crystalline after an abelian extension---the one killing the ramification of the characters involved in the principal series. If $\pi_p$ is a twist of Steinberg by a character, $\rho_{f,p}$ will become semistable non-crystalline after you've made an abelian extension making the character unramified. And if $\pi_p$ is supercuspidal, $\rho_{f,p}$ will become crystalline after a finite non-trivial extension that could be either abelian or non-abelian, and figuring out which is a question about $\pi_p$ (it will be a base change from a quadratic extension if $p>2$ and you have to bash out the possibilities).

Seems to me then that semistable $\rho$s will show up precisely when $\pi_p$ is either unramified principal series or Steinberg, so the answer to your question is (if I've got everything right) that $\rho_{f,p}$ will be semistable iff either $N$ (the level of the newform) is prime to $p$, or $p$ divides $N$ exactly once and the component at $p$ of the character of $f$ is trivial. Any other observations you need should also be readable from this sort of data in the same way.

One consequence of this I guess is that $\rho_{f,p}$ is semi-stable iff the $\ell$-adic representation attached to $f$ is semistable at $p$.

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Thanks for your thoughts, Rob. I was thinking of ordinary as $p$ not dividing $a_p$ and was concentrating on weight 2. My statement about crystalline/ semistable over an abelian extension comes from analyzing the ordinary parts of the p-divisible groups of modular abelian varieties, as in the work of Mazur-Wiles and Tilouine. I agree that the filtered module contains all the information I'm asking about, but this seems like more of a rephrasing of the question, as knowledge of the filtered module requires understanding $L$ first... –  B. Cais Feb 7 '10 at 18:11
If $p$ doesn't divided $a_p$, then $\rho=\rho_{f,p}$ is ordinary (in the sense of Greenberg) by a theorem of Wiles and hence is semistable. If, in addition, $p$ divides $N$ and $f$ is new, then $\rho$ is not cristalline and hence can't become cristalline over any extension since it is already semistable. Am I missing something? –  Rob Harron Feb 7 '10 at 19:42
Rob: What Wiles shows is that $p$ not dividing $a_p$ implies potentially ordinary, as he only works with his Galois representations up to $\overline{\mathbf{Q}}_p$-equivalence. The point is that the ordinary filtration on the Galois side may not be defined over $\mathbf{Q}_p$. Any weight 2 neworm of level $p^r$ primitive nebentypus has associated Galois representation that is potentially crystalline but non-crystalline over $\mathbf{Q}_p$ if $r>0$. Wiles + Perrin-Riou as you indicate only gives $p$ not dividing $a_p$ implies potentially semistable, which one has anyway... –  B. Cais Feb 8 '10 at 0:49