# Local Galois Representations

## `GaloisRepresentation(pi): RepLoc -> GalRep`

## `WeilRepresentation(pi): RepLoc -> GalRep`

```magma
Precision: RngIntElt                    Default: 10
```

Given a minimal representation $\pi$ of ${\operatorname{GL}}_2({\mathbb{Q}}_p),$ this returns the representation $\rho_\pi$ of the Weil-Deligne group associated to $\pi$ under the local Langlands correspondence, as a local Galois representation. (See Section [The Local Langlands Correspondence](introduction.md#sec-local-langlands) and Chapter `galrep`.)

For supercuspidal $\pi$ with admissible pair $(E,\chi)$ the Frobenius data of $\rho_\pi$ are recovered as follows. When $E/{\mathbb{Q}}_p$ is unramified the Loeffler-Weinstein rectifier is applied. When $E/{\mathbb{Q}}_p$ is ramified the value of $\chi$ at a uniformizer of $E$ is obtained from the Atkin-Lehner eigenvalue together with the tame and wild $\varepsilon$-factors, giving the full $\rho_\pi$ when $\chi$ has cyclic prime-order image and $\pi$ arises from a modular form. In the remaining ramified cases an error is raised.

## `AdmissiblePair(pi): RepLoc -> RngPad, Map`

Given an ordinary minimal supercuspidal representation $\pi$ of ${\operatorname{GL}}_2({\mathbb{Q}}_p),$ this returns the associated admissible pair $(E,\chi).$ (See Section [The Local Langlands Correspondence](introduction.md#sec-local-langlands).) Two objects are returned: a quadratic field extension $E/{\mathbb{Q}}_p$, and a map $\chi$ which is a character of the unit group of $E.$