Definition unramified local representation

Question

We can define unramified representation of $GL(2,\mathbb Q_p$) as a representation $(\pi , V) $ such that $V^K\neq \{0\}$ for $K$ the maximal compact subgroup i.e. $GL(2,\mathbb Z_p)$.

However, I do not understand why this is called unramified.

What I want to see (hopefully) is some sort of connection with unramified extension. For example, a Galois representation is called unramified if its factors through the Galois group of the maximal unramified extension or equivalently, the quotient of the absolute Galois group with the inertia subgroup.

Answer 1

Let $f$ be a modular form (normalised new eigenform), and $p$ a prime. Then the following are equivalent:

1. The $GL_2(\mathbf{Q}_p)$-representation $\pi_p$ is unramified, where $\pi = \sideset{}{'}\bigotimes_\ell \pi_v$ is the automorphic representation associated to $f$.

2. For some (equivalently every) prime $\ell \ne p$, the restriction of the $\ell$-adic Galois representation $\rho_{f, \ell}$ to a decomposition group at $p$ is an unramified representation of $Gal(\overline{\mathbf{Q}}_p / \mathbf{Q}_p)$.

The reason why (1) and (2) are equivalent is because each of them is equivalent to the much simpler statement:

3. The level of $f$ is not divisible by $p$.

But the equivalence of (1) and (2) should give you some feel for why "unramified" is used for both concepts.

Answer 2

Local class field theory provides a natural isomorphism $GL_1(\mathbb Q_p) = \mathbb Q_p^\times \cong W_{\mathbb Q_p}^{ab}$ between the multiplicative group of $\mathbb Q_p$ and the abelianisation of the Weil group of $\mathbb Q_p$. Under this isomorphism, unramified characters of the abelianised Weil group correspond to unramified representations of $GL_1(\mathbb Q_p)$.