# Galois representation being unramified is Galois local

Let $$K$$ be number field and $$\rho:G_K\rightarrow \text{Gl}(V)$$ a Galois representation. Let $$\nu$$ be a place of $$K$$ (non-archimedean if it helps/is necessary). We say that $$\rho$$ is unramified at $$\nu$$ if $$\rho(I_\nu)$$ is trivial. My question is if this can be tested galois-locally, i.e. if $$L$$ is a finite Galois extension of $$K$$ and $$\rho\vert_{G_L}$$ is unramified at all primes $$\omega\vert \nu$$, does it follow that $$\rho$$ is unramified at $$\nu$$?

• No. Consider any Galois representation $\rho$ with finite image (e.g. a Dirichlet character). Its image is isomorphic to $\mathrm{Gal}(L/K)$ for some $L$. But $\rho|_L$ is trivial and hence unramified for all primes. A similar idea works whenever the image of inertia is finite. – Mathmo123 Oct 3 '20 at 18:45

For example, take an elliptic curve $$E/K$$ which has potential good reduction at $$\nu$$ but not good reduction. Let $$L$$ be some finite extension over which $$E$$ achieves good reduction at all $$\omega|\nu$$. Then Ogg-Neron-Shafarevic tells us that the action on the Tate module of $$E$$ at some prime not divisible by $$\nu$$ is not unramified at $$\nu$$ (since we don't have good reduction) but is unramified at all those $$\omega$$ (since we get good reduction at all those places).
I think whenever $$I_\nu$$ has finite image you can come up with examples like that - the issue is that if the action factors through a finite quotient then you can find some finite extension $$L$$ which eats up'' that image and so the restriction to $$L$$ will always look unramified. Probably (?) the only way you can always guarantee that what you want holds is if you require that $$L/K$$ is unramified (in which case $$I_\nu = I_\omega$$).
• The last sentence is correct (so you can delete the "probably"). If $L/K$ ramifies at $v$, then the regular representation of $Gal(L / K)$ is ramified at $v$, but its restriction to $G_L$ is trivial, so certainly unramified at the primes above $v$. – David Loeffler Oct 2 '20 at 7:15