Local Galois Representations
============================

.. magma:function:: GaloisRepresentation(pi)
   :input_types: RepLoc
   :output_types: GalRep
   :label: GaloisRepresentation_RepLoc

.. magma:function:: WeilRepresentation(pi)
   :input_types: RepLoc
   :output_types: GalRep
   :label: WeilRepresentation_RepLoc
   :parameters: Precision : RngIntElt : 10

   Given a minimal representation :math:`\pi` of
   :math:`{\operatorname{GL}}_2({\mathbb{Q}}_p),` this returns the representation
   :math:`\rho_\pi` of the Weil-Deligne group associated to :math:`\pi` under the
   local Langlands correspondence, as a local Galois representation. (See Section
   :ref:`sec-local-langlands` and Chapter
   ``galrep``.) In the current implementation, the representation only
   agrees with :math:`\rho_\pi` on inertia.

.. magma:function:: AdmissiblePair(pi)
   :input_types: RepLoc
   :output_types: RngPad, Map
   :label: AdmissiblePair_RepLoc

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