Precision: RngIntElt Default: 10
Given a minimal representation π of GL2(Qp), this returns the representation ρπ of the Weil-Deligne group associated to π under the local Langlands correspondence, as a local Galois representation. (See Section The Local Langlands Correspondence and Chapter LOCAL GALOIS REPRESENTATIONS.)For supercuspidal π with admissible pair (E, χ) the Frobenius data of ρπ are recovered as follows. When E/Qp is unramified the Loeffler-Weinstein rectifier is applied. When E/Qp is ramified the value of χ at a uniformizer of E is obtained from the Atkin-Lehner eigenvalue together with the tame and wild eps-factors, giving the full ρπ when χ has cyclic prime-order image and π arises from a modular form. In the remaining ramified cases an error is raised.
Given an ordinary minimal supercuspidal representation π of GL2(Qp), this returns the associated admissible pair (E, χ). (See Section The Local Langlands Correspondence.) Two objects are returned: a quadratic field extension E/Qp, and a map χ which is a character of the unit group of E.