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
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)
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
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Version: V2.19 of
Wed Apr 24 15:09:57 EST 2013