[Next][Prev] [_____] [Left] [Up] [Index] [Root]

L-FUNCTIONS

 
Acknowledgements
 
Overview
 
Built-in L-series
 
Computing L-values
 
Arithmetic with L-series
 
General L-series
      Terminology
      Constructing a General L-Series
      Setting the Coefficients
      Specifying the Coefficients Later
      Generating the Coefficients from Local Factors
 
Accessing the Invariants
 
Precision
      L-series with Unusual Coefficient Growth
      Computing L(s) when Im(s) is Large (ImS Parameter)
      Implementation of L-series Computations (Asymptotics Parameter)
 
Verbose Printing
 
Advanced Examples
      Handmade L-series of an Elliptic Curve
      Self-made Dedekind Zeta Function
      L-series of a Genus 2 Hyperelliptic Curve
      Experimental Mathematics for Small Conductor
      Tensor Product of L-series Coming from l-adic Representations
      Non-abelian Twist of an Elliptic Curve
      Other Tensor Products
      Symmetric Powers
 
Weil Polynomials
 
Bibliography







DETAILS

 
Overview

 
Built-in L-series
      RiemannZeta() : -> LSer
      Example Lseries_lseries-sig-riemann (H127E1)
      LSeries(K) : FldNum -> LSer
      Example Lseries_lseries-sig-dedekind (H127E2)
      Example Lseries_lseries-sig-dedekind2 (H127E3)
      Example Lseries_armitage (H127E4)
      LSeries(A) : ArtRep -> LSer
      Example Lseries_lseries-artin (H127E5)
      Example Lseries_lseries-a7 (H127E6)
      LSeries(E) : CrvEll -> LSer
      Example Lseries_lseries-sig-elliptic (H127E7)
      LSeries(E, K) : CrvEll, FldNum -> LSer
      Example Lseries_lseries-sig-ellnf (H127E8)
      LSeries(E, A) : CrvEll, ArtRep -> LSer
      Example Lseries_lseries-sig-ellartintwist (H127E9)
      Example Lseries_lseries-etw-quaternion (H127E10)
      LSeries(C) : CrvHyp -> LSer
      Example Lseries_lseries-sig-crvhyp (H127E11)
      LSeries(Chi) : GrpDrchElt -> LSer
      Example Lseries_lseries-sig-character (H127E12)
      LSeries(hmf) : ModFrmHilElt -> LSer
      Example Lseries_lseries-hilbert-modfom (H127E13)
      LSeries(psi) : GrpHeckeElt -> LSer
      LSeries(f) : ModFrmElt -> LSer
      Example Lseries_lseries-sig-modfrm (H127E14)

 
Computing L-values
      Evaluate(L, s0) : LSer, FldComElt -> FldComElt
      CentralValue(L) : LSer -> FldComElt
      LStar(L, s0) : LSer, FldComElt -> FldComElt
      LTaylor(L,s0,n) : LSer, FldComElt, RngIntElt -> FldComElt
      Example Lseries_lseries-evaluate (H127E15)

 
Arithmetic with L-series
      L1 * L2 : LSer, LSer -> LSer
      L1 / L2 : LSer, LSer -> LSer
      TensorProduct(L1, L2, ExcFactors) : LSer, LSer, [<>] -> LSer

 
General L-series

      Terminology

      Constructing a General L-Series
            LSeries(weight, gamma, conductor, cffun) : FldReElt,[FldRatElt],FldReElt,Any -> LSer
            Example Lseries_lseries-checkfun (H127E16)

      Setting the Coefficients
            LSetCoefficients(L,cffun) : LSer, Any ->

      Specifying the Coefficients Later
            Example Lseries_lseries-lcfrequired (H127E17)

      Generating the Coefficients from Local Factors

 
Accessing the Invariants
      LCfRequired(L) : LSer -> RngIntElt
      LGetCoefficients(L, N) : LSer, RngIntElt -> List
      EulerFactor(L, p) : LSer, RngIntElt -> .var Degree : RngIntElt : var Precision: RngIntElt Default: desGiven an L-series and a prime p, this computes thepth Euler factor, either as a polynomial or a power series.The optional parameter Degree will truncate the series to that length,and the optional parameter Precision is of use when the series isdefined over the complex numbers.
      Sign(L) : LSer -> .
      GammaFactors(L) : LSer -> Seqenum
      LSeriesData(L) : LSer -> Info
      Example Lseries_lseries-invariants (H127E18)
      Factorization(L) : LSer -> SeqEnum[Tup]
      Example Lseries_lseries-invariants (H127E19)

 
Precision
      LSetPrecision(L,precision) : LSer, RngIntElt ->

      L-series with Unusual Coefficient Growth

      Computing L(s) when Im(s) is Large (ImS Parameter)

      Implementation of L-series Computations (Asymptotics Parameter)

 
Verbose Printing

 
Advanced Examples

      Handmade L-series of an Elliptic Curve
            Example Lseries_lseries-elliptic-selfmade (H127E20)

      Self-made Dedekind Zeta Function
            Example Lseries_lseries-dedekind-selfmade (H127E21)

      L-series of a Genus 2 Hyperelliptic Curve
            Example Lseries_lseries-genus2 (H127E22)

      Experimental Mathematics for Small Conductor
            Example Lseries_lseries-experimental (H127E23)

      Tensor Product of L-series Coming from l-adic Representations
            Example Lseries_lseries-tensor (H127E24)

      Non-abelian Twist of an Elliptic Curve
            Example Lseries_lseries-nonabtwist (H127E25)

      Other Tensor Products
            Example Lseries_ec-tensorprod (H127E26)
            Example Lseries_level1-modform (H127E27)
            Example Lseries_siegel-modular-form (H127E28)
            Example Lseries_tensprod-overK (H127E29)

      Symmetric Powers
            SymmetricPower(L, m) : LSer, RngIntElt -> LSer
            Example Lseries_lseries-sympow (H127E30)
            Example Lseries_sympow-ec (H127E31)
            Example Lseries_sympow-ec (H127E32)
            Example Lseries_sympow-ec (H127E33)
            Example Lseries_sympow-ec2 (H127E34)

 
Weil Polynomials
      SetVerbose("WeilPolynomials", v) : MonStgElt, RngIntElt ->
      HasAllRootsOnUnitCircle(f) : RngUPolElt -> BoolElt
      FrobeniusTracesToWeilPolynomials(tr, q, i, deg) : SeqEnum, RngIntElt, RngIntElt, RngIntElt -> SeqEnum
      WeilPolynomialToRankBound(f, q) : RngUPolElt, RngIntElt -> RngIntElt
      ArtinTateFormula(f, q, h20) : RngUPolElt, RngIntElt, RngIntElt -> RngIntElt, RngIntElt
      WeilPolynomialOverFieldExtension(f, deg) : RngUPolElt, RngIntElt -> RngUPolElt
      CheckWeilPolynomial(f, q, h20) : RngUPolElt, RngIntElt, RngIntElt -> BoolElt
      Example Lseries_weil (H127E35)

 
Bibliography

[Next][Prev] [Right] [____] [Up] [Index] [Root]
Version: V2.19 of Wed Apr 24 15:09:57 EST 2013