Local Galois Representations

GaloisRepresentation(pi): RepLoc -> GalRep

WeilRepresentation(pi): RepLoc -> GalRep

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 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.) 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 \(E.\)