Admissible Representations of \({\operatorname{GL}}_2({\mathbb{Q}}_p)\)

• Introduction
  • Motivation
  • Definitions
  • The Principal Series
  • Supercuspidal Representations
  • The Local Langlands Correspondence
  • Connection with Modular Forms
  • Category
  • Verbose Output
• Creation of Admissible Representations
  • LocalComponent(M, p): ModSym, RngIntElt → RepLoc
  • Example: Creation Example
• Attributes of Admissible Representations
  • CentralCharacter(pi): RepLoc → GrpDrchElt
  • Conductor(pi): RepLoc → RngIntElt
  • DefiningModularSymbolsSpace(pi): RepLoc → ModSym
  • IsMinimal(pi): RepLoc → BoolElt, GrpDrchElt, RepLoc
  • Example: Attributes Example
• Structure of Admissible Representations
  • IsPrincipalSeries(pi): RepLoc → BoolElt
  • IsSupercuspidal(pi): RepLoc → BoolElt
  • PrincipalSeriesParameters(pi): RepLoc → GrpDrchElt, GrpDrchElt
  • CuspidalInducingDatum(pi): RepLoc → ModGrp
• Local Galois Representations
  • GaloisRepresentation(pi): RepLoc → GalRep
  • WeilRepresentation(pi): RepLoc → GalRep
  • AdmissiblePair(pi): RepLoc → RngPad, Map
• Examples
  • Example: example1
  • Example: example2
  • Example: example3