Dirichlet and Hecke Characters
- Introduction
- Creation Functions
- Functions on Groups and Group Elements
Modulus(G): GrpDrchNF → RngOrdIdl, SeqEnumModulus(G): GrpHecke → RngOrdIdl, SeqEnumModulus(chi): GrpDrchNFElt → RngOrdIdl, SeqEnumModulus(chi): GrpHeckeElt → RngOrdIdl, SeqEnumOrder(chi): GrpDrchNFElt → RngIntEltOrder(psi): GrpHeckeElt → RngIntEltRandom(G): GrpDrchNF → GrpDrchNFEltRandom(G): GrpHecke → GrpHeckeEltDomain(G): GrpDrchNF → FldNumDomain(G): GrpDrchNFElt → FldNumDomain(G): GrpHecke → PowIdlDomain(G): GrpHeckeElt → PowIdlDecomposition(chi): GrpDrchNFElt → ListComponents(chi): GrpDrchNFElt → AssocComponents(psi): GrpHeckeElt → AssocComponent(chi, P): GrpDrchNFElt, RngOrdIdl → GrpDrchNFElComponent(psi, P): GrpHeckeElt, RngOrdIdl → GrpDrchNFEltComponent(chi, oo): GrpDrchNFElt, RngIntElt → GrpDrchNFElComponent(psi, oo): GrpHeckeElt, RngIntElt → GrpDrchNFEltComponent(chi, P): GrpDrchNFElt, PlcNumElt → GrpDrchNFElComponent(psi, P): GrpHeckeElt, PlcNumElt → GrpDrchNFEltConductor(chi): GrpDrchNFElt → RngOrdIdl, SeqEnumConductor(psi): GrpHeckeElt → RngOrdIdl, SeqEnumAssociatedPrimitiveCharacter(chi): GrpDrchNFElt → GrpDrchNFEltAssociatedPrimitiveCharacter(psi): GrpHeckeElt → GrpHeckeEltRestrict(chi, D): GrpDrchNFElt, GrpDrchNF → GrpDrchNFEltRestrict(psi, H): GrpHeckeElt, GrpHecke → GrpHeckeEltRestrict(chi, I): GrpDrchNFElt, RngOrdIdl → GrpDrchNFEltRestrict(psi, I): GrpHeckeElt, RngOrdIdl → GrpHeckeEltRestrict(chi, I, oo): GrpDrchNFElt, RngOrdIdl, SeqEnum → GrpDrchNFEltRestrict(psi, I, oo): GrpHeckeElt, RngOrdIdl, SeqEnum → GrpHeckeEltRestrict(G, D): GrpDrchNF, GrpDrchNF → GrpDrchNFRestrict(G, H): GrpHecke, GrpHecke → GrpHeckeRestrict(G, I): GrpDrchNF, RngOrdIdl → GrpDrchNFRestrict(G, I): GrpHecke, RngOrdIdl → GrpHeckeRestrict(G, I, oo): GrpDrchNF, RngOrdIdl, SeqEnum → GrpDrchNFRestrict(G, I, oo): GrpHecke, RngOrdIdl, SeqEnum → GrpHeckeTargetRestriction(G, C): GrpDrchNF, FldCyc → GrpDrchNFTargetRestriction(H, C): GrpHecke, FldCyc → GrpDrchNFSetTargetRing(~ chi, e): GrpDrchNFElt, RngEltSetTargetRing(~ psi, e): GrpHeckeElt, RngEltExtend(chi, D): GrpDrchNFElt, GrpDrchNF → GrpDrchNFEltExtend(psi, H): GrpHeckeElt, GrpHecke → GrpHeckeEltExtend(chi, I): GrpDrchNFElt, RngOrdIdl → GrpDrchNFEltExtend(psi, I): GrpHeckeElt, RngOrdIdl → GrpHeckeEltExtend(chi, I, oo): GrpDrchNFElt, RngOrdIdl, SeqEnum → GrpDrchNFEltExtend(psi, I, oo): GrpHeckeElt, RngOrdIdl, SeqEnum → GrpHeckeEltExtend(G, D): GrpDrchNF, GrpDrchNF → GrpDrchNFExtend(G, H): GrpHecke, GrpHecke → GrpHeckeExtend(G, I): GrpDrchNF, RngOrdIdl → GrpDrchNFExtend(G, I): GrpHecke, RngOrdIdl → GrpHeckeExtend(G, I, oo): GrpDrchNF, RngOrdIdl, SeqEnum → GrpDrchNFExtend(G, I, oo): GrpHecke, RngOrdIdl, SeqEnum → GrpHecke
- Predicates on Group Elements
- Passing between Dirichlet and Hecke Characters
HeckeLift(chi): GrpDrchNFElt → GrpHeckeElt, GrpHeckeDirichletRestriction(psi): GrpHeckeElt → GrpDrchNFEltNormInduction(K, chi): FldNum, GrpDrchElt → GrpHeckeEltQuadraticCharacter(e): FldNumElt → GrpHeckeElt- Example: Dirichlet Characters over the Rationals
- Example: Basic Character Operations
DirichletCharacter(I, B): RngOrdIdl, Tup → GrpDrchNFElt, GrpDrchNFDirichletCharacter(I, oo, B): RngOrdIdl, SeqEnum, Tup → GrpDrchNFElt, GrpDrchNFDirichletCharacter(G, B): GrpDrchNF, Tup → GrpDrchNFElt, GrpDrchNFDirichletCharacter(I, L): RngOrdIdl, List → GrpDrchNFElt, GrpDrchNFDirichletCharacter(I, oo, L): RngOrdIdl, SeqEnum, List → GrpDrchNFElt, GrpDrchNFDirichletCharacter(G, L): GrpDrchNF, List → GrpDrchNFElt, GrpDrchNFHeckeCharacter(I, B): RngOrdIdl, Tup → GrpHeckeEltHeckeCharacter(I, oo, B): RngOrdIdl, SeqEnum, Tup → GrpHeckeEltHeckeCharacter(G, B): GrpHecke, Tup → GrpHeckeEltHeckeCharacter(I, L): RngOrdIdl, List → GrpHeckeEltHeckeCharacter(I, oo, L): RngOrdIdl, SeqEnum, List → GrpHeckeEltHeckeCharacter(G, L): GrpHecke, List → GrpHeckeElt- Example: Dirichlet Character with Prescribed Values
CentralCharacter(chi): GrpDrchNFElt → GrpDrchNFEltCentralCharacter(psi): GrpHeckeElt → GrpDrchNFElt- Example: Central Character of a Dirichlet Character
DirichletCharacterOverNF(chi): GrpDrchElt → GrpDrchNFEltDirichletCharacterOverQ(chi): GrpDrchNFElt → GrpDrchElt- Example: Dirichlet Characters over Q and Number Fields
- L-functions of Hecke Characters
- Hecke Grössencharacters and their L-functions
Grossencharacter(psi, chi, T): GrpHeckeElt, GrpDrchNFElt, SeqEnum → GrossenCharRawEval(I, GR): RngOrdFracIdl, GrossenChar → FldNumElt, FldCycElt, FldCycEltGrossencharacter(psi, T): GrpHeckeElt, SeqEnum → GrossenCharConductor(GR): GrossenChar → RngOrdIdl, SeqEnumModulus(GR): GrossenChar → RngOrdIdl, SeqEnumIsPrimitive(GR): GrossenChar → BoolEltAssociatedPrimitiveGrossencharacter(psi): GrossenChar → GrossenCharComponents(GR): GrossenChar → AssocComponent(GR, P): GrossenChar, RngOrdIdl → GrpDrchNFElComponent(GR, oo): GrossenChar, RngIntElt → GrpDrchNFElComponent(GR, P): GrossenChar, PlcNumElt → GrpDrchNFElExtend(GR, I): GrossenChar, RngOrdIdl → GrossenCharRestrict(GR, I): GrossenChar, RngOrdIdl → GrossenCharCentralCharacter(GR): GrossenChar → GrpDrchNFEltGrossenTwist(GR, D): GrossenChar, List → GrossenChar, GrpHeckeTateTwist(GR, n): GrossenChar, RngIntElt → GrossenCharTateTwist(psi, n): GrpHeckeElt, RngIntElt → GrossenChar- Example: Grossencharacters on the Gaussian Field
- Example: Canonical Grossencharacter of Q(sqrt(-23))
- Example: Grossencharacter with Vanishing Central Value
- Example: Grossencharacter Central Value and Gamma Periods
- Example: Twisting a Grossencharacter by a Hilbert Character
- Example: Grossencharacters of Trivial Conductor
- Example: Grossencharacter on the Eighth Cyclotomic Field
- Example: Grossencharacter from a Hypergeometric Motive
- Local Root Numbers
RootNumber(GR, P): GrossenChar, RngOrgIdl → FldComEltRootNumber(psi, P): GrpHeckeElt, RngOrgIdl → FldComEltRootNumber(GR, P): GrossenChar, PlcNumElt → FldComEltRootNumber(psi, P): GrpHeckeElt, PlcNumElt → FldComEltRootNumber(GR, p): GrossenChar, RngIntElt → FldComEltRootNumber(psi, p): GrpHeckeElt, RngIntElt → FldComEltRootNumbers(GR): GrossenChar → SeqEnumRootNumbers(psi): GrossenChar → SeqEnumRootNumber(GR): GrossenChar → SeqEnumRootNumber(psi): GrossenChar → SeqEnum- Example: Local Root Numbers for Curve 27a
- Example: Rohrlich's Root Number Exercise
- Grössencharacters and Elliptic Curves
- Bibliography