L-Functions

• Overview
• Built-in \(L\)-series
  • RiemannZeta() → LSer
  • Example: Riemann Zeta Function at 2
  • LSeries(K): FldNum → LSer
  • Example: Dedekind Zeta Function of Q(i)
  • Example: Dedekind Zeta of a Degree 12 Field
  • Example: Dedekind Zeta Vanishing at the Centre
  • LSeries(A): ArtRep → LSer
  • Example: Artin L-Series over Q(i)
  • Example: Artin Representation of A7
  • LSeries(E): CrvEll → LSer
  • Example: L-Series of a Rank 1 Curve
  • LSeries(E, K): CrvEll, FldNum → LSer
  • Example: Base Change of an Elliptic Curve
  • LSeries(E, A): CrvEll, ArtRep → LSer
  • Example: Twisting by Cyclotomic Characters
  • Example: Twist by a Quaternion Representation
  • LSeries(C): CrvHyp[FldRat] → LSer
  • Example: L-Series of a Hyperelliptic Curve
  • LSeries(C,K): CrvHyp[FldRat], FldNum → LSer
  • Example: Base Change of a Hyperelliptic Curve
  • LSeries(C): CrvHyp[FldNum] → LSer
  • Example: Genus 2 Curve over Q(i)
  • LSeries(Chi): GrpDrchElt → LSer
  • Example: Dirichlet L-Function Modulo 37
  • LSeries(hmf): ModFrmHilElt → LSer
  • Example: L-Series of a Hilbert Modular Form
  • LSeries(omf): ModFrmAlgElt → LSer
  • Example: L-Series of an Orthogonal Modular Form
  • LSeries(psi): GrpHeckeElt → LSer
  • LSeries(psi): GrossenChar → LSer
  • LSeries(f): ModFrmElt → LSer
  • Example: Newform over the Gaussian Integers
  • LSeries(S): ModSym → LSer
  • Example: L-Series from Modular Symbols
• 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: Derivatives of a Rank 3 L-Series
  • DedekindZetaExact(K, z): Fld, RngIntElt → FldRatElt
• General \(L\)-series
  • Terminology
  • Constructing a General \(L\)-Series
    • LSeries(weight, gamma, conductor, cffun): FldReElt, [FldRatElt], FldReElt, Any → LSer
    • LSeries(HS, N, cffun): HodgeStruc, RngIntElt, Any → LSer
    • CheckFunctionalEquation(L): LSer → FldComElt
    • CFENew(L): LSer → FldReElt
    • Example: Checking a Functional Equation
  • Setting the Coefficients
    • LSetCoefficients(L,cffun): LSer, Any → {}
  • Specifying the Coefficients Later
    • Example: Riemann Zeta with Deferred Coefficients
  • Generating the Coefficients from Local Factors
• Accessing the Invariants
  • LCfRequired(L): LSer → RngIntElt
  • LGetCoefficients(L, N): LSer, RngIntElt → List
  • EulerFactor(L, p): LSer, RngIntElt → RngElt
  • Degree(L): LSer → RngIntElt
  • Conductor(L): LSer → RngElt
  • Sign(L): LSer → RngElt
  • MotivicWeight(L): LSer → RngIntElt
  • GammaFactors(L): LSer → SeqEnum
  • LSeriesData(L): LSer → Info
  • BadPrimeData(L): LSer → SeqEnum
  • Example: Coefficients of a Modular Form L-Series
  • Factorization(L): LSer → SeqEnum[Tup]
  • Factorisation(L): LSer → SeqEnum[Tup]
  • Example: Factorization of L-Series
• Modifying the \(L\)-function
  • ChangeEulerFactor(L,p,f): LSer, RngIntElt, RngUPolElt → LSer
  • ChangeLocalInformation(L,p,d,f): LSer, RngIntElt, RngIntElt, RngUPolElt → LSer
  • ChangeLocalInformation(L,bp): LSer, List → LSer
  • CopyCoefficients(L,M): LSer, LSer
  • Example: Changing Local Information
• 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
• Arithmetic with \(L\)-series
  • L1 * L2: LSer, LSer → LSer
  • L1 / L2: LSer, LSer → LSer
  • Hodge Structure
    • HasHodgeStructure(L): LSer → BoolElt, HodgeStruc
    • HodgeStructure(L): LSer → HodgeStruc
    • HodgeStructure(X): SeqEnum → HodgeStruc
    • HodgeStructure(w, G): RngIntElt, SeqEnum → HodgeStruc
    • Dual(HS): HodgeStruc → HodgeStruc
    • TateTwist(HS, k): HodgeStruc, RngIntElt → HodgeStruc
    • Translate(L, z): LSer, RngIntElt → LSer
    • Translate(L, z): LSer, FldRatElt → LSer
    • GammaFactors(HS): HodgeStruc → SeqEnum
    • GammaShifts(HS): HodgeStruc → SeqEnum
    • Degree(HS): HodgeStruc → RngIntElt
    • Weight(HS): HodgeStruc → RngIntElt
    • EffectiveHodgeStructure(HS): HodgeStruc → HodgeStruc, RngIntElt
    • RootNumber(HS): HodgeStruc → FldCycElt
    • TensorProduct(H1, H2): HodgeStruc, HodgeStruc → HodgeStruc
    • SymmetricPower(HS, m): HodgeStruc, RngIntElt → HodgeStruc
    • Determinant(HS): HodgeStruc → HodgeStruc
    • AlternatingSquare(HS): HodgeStruc → HodgeStruc
    • HodgeVector(HS): HodgeStruc → SeqEnum, RngIntElt
    • CriticalPoints(HS): HodgeStruc → SeqEnum
    • CriticalPoints(L): LSer → SeqEnum
    • ImaginaryTwist(HS): HodgeStruc → HodgeStruc
    • Example: Hodge Structures of L-Series
  • Tensor Products
    • TensorProduct(L1, L2, ExcFactors): LSer, LSer, [<>] → LSer
    • TensorProduct(L1, L2, ExcFactors, K): LSer, LSer, [<>], FldNum → LSer
    • TensorProduct(L1, L2): LSer, LSer → LSer
    • TensorProduct(L1, L2, K): LSer, LSer, FldNum → LSer
    • Example: Tensor Product of Two Elliptic Curves
    • Example: Tensor Product of Level 1 Modular Forms
    • Example: Tensor Product and a Siegel Modular Form
    • Example: Tensor Product over a Quadratic Field
    • Example: Consani-Scholten Tensor Product
  • Symmetric Powers
    • Determinant(L): LSer → LSer
    • SymmetricPower(L, m): LSer, RngIntElt → LSer
    • Example: Symmetric Powers of a Dirichlet Character
    • Example: Symmetric Powers of a Grossencharacter
    • Example: Symmetric Powers of an Elliptic Curve
    • Example: Symmetric Power with Central Vanishing
    • Symmetrization(L, p): LSer, SeqEnum → LSer
    • IsOrthogonal(L): LSer → BoolElt
    • IsSymplectic(L): LSer → BoolElt
    • OrthogonalSymmetrization(L, p): LSer, SeqEnum → LSer
    • SymplecticSymmetrization(L, p): LSer, SeqEnum → LSer
    • Example: General Symmetrizations of a Modular Form
    • Example: Orthogonal Symmetric Cube
    • Example: Symplectic Symmetrizations in Degree 4
    • Example: Higher Orthogonal Symmetrizations
    • Example: Imprimitive Orthogonal Symmetrization
• Advanced Examples
  • Handmade \(L\)-series of an Elliptic Curve
    • Example: Elliptic Curve L-Series from Scratch
  • Self-made Dedekind Zeta Function
    • Example: Dedekind Zeta Function from Scratch
  • Handmade \(L\)-series of a Hyperelliptic Curve
    • Example: Genus 2 L-Series from Scratch
  • Experimental Mathematics for Small Conductor
    • Example: Recovering a Conductor 11 L-Series
  • Tensor Product of \(L\)-series Coming from \(l\)-adic Representations
    • Example: Tensor Product of l-Adic L-Series
  • Non-abelian Twist of an Elliptic Curve
    • Example: BSD in a Non-Abelian Extension
  • Unitary Twist with Central Vanishing
    • Example: Rohrlich's Central Vanishing Twist
• Bibliography