An L-series or an L-function is an infinite sum L(s)=∑n=1^∞an/ns in the complex variable s with complex coefficients an. Such functions arise in many places in mathematics and they are usually naturally associated with some kind of mathematical object, for instance a character, a number field, a curve, a modular form or a cohomology group of an algebraic variety. The coefficients an are certain invariants associated with that object. For example, in the case of a character χ: (Z/mZ) * to C * they are simply its values an=χ(n) when gcd(n, m)=1 and 0 otherwise.
Magma is able to associate an L-series to various types of object. The intrinsic which provides access to such pre-defined L-series is
{LSeries(object: optional parameters)}
Every such function returns a variable of type LSer. A range of functions may now be applied to this L-series object as described in the following sections (and in fact more are available than listed), and these are independent of the object to which the L-series was originally associated. In fact, an object of type LSer only "remembers" its origin for printing purposes.
Precision: RngIntElt Default:
The Riemann zeta function ζ(s) is returned.The number of digits of precision to which the values ζ(s) are to be computed may be specified using the Precision parameter. If it is omitted, the precision of the default real field will be used.
Check that ζ(2) agrees numerically with π2/6.
> L := RiemannZeta( : Precision:=40); > Evaluate(L,2); 1.644934066848226436472415166646025189219 > Pi(RealField(40))^2/6; 1.644934066848226436472415166646025189219
Method: MonStgElt Default: em "Default"
ClassNumberFormula: BoolElt Default: false
Precision: RngIntElt Default:
Create the Dedekind zeta function ζ(K, s) of a number field K. The series is defined by ∑I (Norm)K/Q(I) - s, where the sum is taken over the non-zero ideals I of the maximal order of K. For K=Q, the series coincides with the Riemann zeta function.The optional parameter Method may be "Artin", "Direct" or "Default" and specifies whether the zeta function should be computed as a product of L-series of Artin representations or directly, by counting prime ideals. (The default behaviour depends upon the field.)
For the "Direct" method, the Dedekind zeta function has a simple pole at s=1 whose residue must be known in order to compute the L-values. The class number formula gives an expression for this residue in terms of the number of real/complex embeddings of K, the regulator, the class number and the number of roots of unity in K. If the optional parameter ClassNumberFormula is set to true, then these quantities are computed on initialization (using Magma's functions Signature(K), Regulator(K), #ClassGroup(MaximalOrder(K)) and #TorsionSubgroup(UnitGroup(K))) and it might take some time if the discriminant of K is large. If ClassNumberFormula is false (default) then the residue is computed numerically from the functional equation. This is generally faster, unless the discriminant of K is small and the precision is set to be very high.
The number of digits of precision to which the values ζ(K, s) are to be computed may be specified using the Precision parameter. If it is omitted the precision is taken to be that of the default real field.
> P<x> := PolynomialRing(Integers()); > K := NumberField(x^2+1); > L := LSeries(K); > Evaluate(L, 2); 1.50670300992298503088656504818
> R<x> := PolynomialRing(Rationals()); > F := NumberField(x^12-3); > L := LSeries(F: Method:="Direct"); > Conductor(L), LCfRequired(L); 1579460446107205632 92968955438The set-up time for the direct method is negligible, but the L-value computation will take days for this number of coefficients. On the other hand, the normal closure of F is not too large and has only representations of small dimension:
> G := GaloisGroup(F); > #G, [Degree(ch): ch in CharacterTable(G)]; 24 [ 1, 1, 1, 1, 2, 2, 2, 2, 2 ] > time L := LSeries(F : Method:="Artin"); Time: 0.340It took longer to define the L-series, but the advantage is that it is a product of L-series with very small conductors, and the L-value calculations are almost instant:
> [Conductor(f[1]) : f in Factorisation(L)]; [ 1, 12, 3888, 243, 576, 15552, 15552 ] > time Evaluate(L, 2); 1.63925427193646882835990708820 Time: 0.840
> _<x> := PolynomialRing(Rationals()); > K<s5> := NumberField( x^2-5 ); > L<s205> := NumberField( x^2-205 ); > C := Compositum(K,L); > e1 := C!(5+s5); > e2 := C!(41+s205); > E := ext<C | Polynomial( [ -e1*e2, 0, 1] )>; > A := AbsoluteField(E); > DefiningPolynomial(A); x^8 - 820*x^6 + 223040*x^4 - 24206400*x^2 + 871430400 > Signature(A); // totally real 8 0 > L := LSeries(A); > LCfRequired(L); // approx value, with old CFE 2739 > CFENew(L); // new vers of CheckFunctionalEquation 0.000000000000000000000000000000 > Evaluate(L, 1/2); // zero as expected 0.000000000000000000000000000000
So the evaluation of L at 1/2 is zero as expected. In fact, L is a product, and one factor has odd sign:
> L`prod; [ <L-series of Riemann zeta function, 1>, <L-series of Artin representation Q8: (1,1,-1,1,-1) of ext<Q|x^8-820*x^6+223040* x^4-24206400*x^2+871430400>, conductor 5, 1>, <L-series of Artin representation Q8: (1,1,1,-1,-1) of ext<Q|x^8-820*x^6+223040* x^4-24206400*x^2+871430400>, conductor 41, 1>, <L-series of Artin representation Q8: (1,1,-1,-1,1) of ext<Q|x^8-820*x^6+223040* x^4-24206400*x^2+871430400>, conductor 205, 1>, <L-series of Artin representation Q8: (2,-2,0,0,0) of ext<Q|x^8-820*x^6+223040*x ^4-24206400*x^2+871430400>, conductor 42025, 2> ] > [ ComplexField(9)!Sign(x[1]) : x in $1 ]; [ 1.00000000, 1.00000000, 1.00000000, 1.00000000, -1.00000000 ] > Sign(L`prod[5][1]); -1.00000000000000000000000000000
This phenomenon happens fairly generally with symplectic representations, and quaternionic Galois groups give the most direct examples. The example of smallest conductor 2832 appears to be for the field x8 + 12x6 + 36x4 + 36x2 + 9, while x8 - x7 + x6 - 4x5 + 5x4 - 8x3 + 4x2 - 8x + 16 has Galois group (SL)2((F)3) and conductor 1632.
Precision: RngIntElt Default:
Creates the L-series of an Artin representation A. (For information about Artin representations see Chapter ARTIN REPRESENTATIONS.)
> K := QuadraticField(-1); > triv,sign := Explode(ArtinRepresentations(K)); > Evaluate(LSeries(triv), 2); // zeta(2)=pi^2/6 1.64493406684822643647241516665 > Evaluate(LSeries(sign), 2); 0.915965594177219015054603514933
> load galpols; > f:=PolynomialWithGaloisGroup(7,6); // Alt(7) > K:=NumberField(f); > A:=ArtinRepresentations(K); > a:=A[2];a; Artin representation A7: (6,2,3,0,0,1,-1,-1,-1) of ext<Q|x^7-2*x^6-7*x^5+11*x^4+ 16*x^3-14*x^2-11*x+2>Its L-series with the default precision of 30 digits needs a lot of coefficients to compute with.
> L:=LSeries(a); > LCfRequired(L); // approx value 1830809We decrease the precision to 8 digits
> L:=LSeries(a: Precision:=8); > LCfRequired(L); // approx value with CheckFunctionalEquation 52584Now it takes under 15 seconds to verify the functional equation of L(a, s) and to compute its value at s=2.
> CFENew(L); // new vers of CheckFunctionalEquation 1.4901161E-8 > Evaluate(L,2); 1.1105972
Precision: RngIntElt Default:
Create the L-series L(E, s) of an elliptic curve E defined over Q or over a number field.The number of digits of precision to which the values L(E, s) are to be computed may be specified using the Precision parameter. If it is omitted the precision is taken to be that of the default real field.
Note that the computation time for evaluating an L-series grows roughly like the square root of the conductor (or its norm to Q if the base field is a number field). Therefore an evaluation might take an unreasonable amount of time if the conductor of E is much larger than, say, 1010 or so. If only the leading term at s=1 is required, over Q it is faster to use AnalyticRank or ConjecturalRegulator.
Note also for general number fields it is only conjectured that L(E/K, s) has a meromorphic continuation to C and also possesses a functional equation. This conjecture is implicitly used in the computations.
> E := EllipticCurve([0,1,1,0,0]); > Conductor(E); 43 > L:=LSeries(E); > Evaluate(L, 1); 0.000000000000000000000000000000 > Evaluate(L, 1 : Derivative:=1); 0.343523974618478230618071163922Now base change E to K=Q(i). The Mordell-Weil rank of E over K is 2:
> Rank(E) + Rank(QuadraticTwist(E,-1)); 2So we expect L(E/K, s) to have a zero of order 2:
> K := QuadraticField(-1); > EK := BaseChange(E, K); > L := LSeries(EK); > Evaluate(L, 1); 0.000000000000000000000000000000 > Evaluate(L, 1 : Derivative:=1) lt 10^-20; true > Evaluate(L, 1 : Derivative:=2); 1.62399545025600030722546910344
Method: MonStgElt Default: em "Default"
Precision: RngIntElt Default:
Given an elliptic curve E defined over the rationals and a number field K, create the L-series L(E/K, s) associated with E/K. Note that in general it is only conjectured that L(E/K, s) has an analytic continuation to C and possesses a functional equation. This conjecture is implicitly used in the computations.Technically, the resulting L-series is the tensor product of two l-adic representations, the one associated to E/Q and the one associated to K/Q. Method specifies how LSeries(K) should be defined. It is the same parameter as for LSeries(FldNum). Note that the conductor of the L-series L(E/K, s) usually increases very rapidly with the discriminant of K. Consequently, if the used method is "Direct" or the irreducible constituents of PermutationCharacter(K) have large dimension, the computation time may be quite substantial.
The number of digits of precision to which the values L(E/K, s) are to be computed may be specified using the Precision parameter. If it is omitted the precision is taken to be that of the default real field.
> E := EllipticCurve([0, 0, 0, 1, 0]); > F := QuadraticTwist(E, 5); > L := LSeries(E, QuadraticField(5)); > Evaluate(L, 1); 1.53733828470360522458966069195 > Evaluate(LSeries(E),1) * Evaluate(LSeries(F),1); 1.53733828470360522458966069195Here is another example over a cyclotomic field. The L-value computations are fast, as here they only involve one-dimensional twists:
> E := EllipticCurve([0, 0, 0, 1, 0]); > L := LSeries(E, CyclotomicField(11)); > time Evaluate(L, 1); 0.000000000000000000000000000000 Time: 1.560
Precision: RngIntElt Default:
Twisted L-series of an elliptic curve E/Q by an Artin representation A.
> E := EllipticCurve(CremonaDatabase(),"11A3"); > K := CyclotomicField(5); > art := ArtinRepresentations(K); > for A in art do Evaluate(LSeries(E,A),1); end for; 0.253841860855910684337758923351 0.685976714588516438169889514223 + 1.10993363969520543571381847366*$.1 2.83803828204429619496466743332 0.685976714588516438169889514223 - 1.10993363969520543571381847366*$.1All the L-values are non-zero, so according to the Birch-Swinnerton-Dyer conjecture E has rank 0 over Q(ζ5). Indeed:
> #TwoSelmerGroup(BaseChange(E,K)); 1
> load galpols;
> E:=EllipticCurve("11a3"); // X_1(11)
> f:=PolynomialWithGaloisGroup(8,5); // Quaternion Galois group
> K:=NumberField(f);
> A:=ArtinRepresentations(K);
> assert exists(a){a: a in A | Degree(a) eq 2};a;
Artin representation Q8: (2,-2,0,0,0) of ext<Q|x^8-12*x^6+36*x^4-36*x^2+9>
> L:=LSeries(E,a: Precision:=10);
> LCfRequired(L);
208818
> time Evaluate(L,1);
1.678012769
Time: 7.470
> Sign(L);
1.000000000
Precision: RngIntElt Default:
LocalData: List Default: [* *]
Returns the L-series of a hyperelliptic curve C/Q.The number of digits of precision to which the values L(C, s) are to be computed may be specified using the Precision parameter. If it is omitted the precision is taken to be that of the default real field.
If the conductor exponents and the local factors at (some of) the bad primes are known in advance, they can be passed as a list of tuples <prime,conductor exponent,local factor>, e.g. LocalData:=[ * < 2, 11, 1 - x > * ].
Current implementation can only compute the conductor at 2 when v2(Δ)<12, using Ogg's formula. If this is not the case, either the local factor at 2 must be supplied in LocalData, or the use of Ogg's formula at 2 forced with LocalData:="Ogg" or [* <2,"Ogg"> *].
> R<x> := PolynomialRing(Rationals()); > C := HyperellipticCurve(x^5+1); > L := LSeries(C: Precision:=18); > LCfRequired(L); // need this number of coefficients 1809 > Evaluate(L,1); // L(C,1) 1.03140710417331776 > Sign(L); // sign in the functional equation 1.00000000000000000The L-value is non-zero, indicating that the Jacobian should have rank 0. In fact, it does:
> RankBound(Jacobian(C)); 0
Precision: RngIntElt Default:
LocalData: List Default: [* *]
The L-series associated to a hyperelliptic curve C/Q base changed to a number field K.The number of digits of precision to which the values L(C, s) are to be computed may be specified using the Precision parameter. If it is omitted the precision is taken to be that of the default real field.
Current implementation can only compute the conductor at 2 when v2(Δ)<12, using Ogg's formula. If this is not the case, the use of Ogg's formula at 2 can be forced with LocalData:="Ogg". Also, the local Galois representation machinery is used in the computations, and it is currently not implemented for some of the reduction types.
> R<x> := PolynomialRing(Rationals()); > C := HyperellipticCurve(x^5-x+1); > K := CyclotomicField(4); > L := LSeries(C,K: Precision:=12); > CheckFunctionalEquation(L); 0.000000000000
Precision: RngIntElt Default:
LocalData: List Default: [* *]
The L-series of a hyperelliptic curve C defined over a number field K.The number of digits of precision to which the values L(C/K, s) are to be computed may be specified using the Precision parameter. If it is omitted the precision is taken to be that of the default real field.
If the conductor exponents and the local factors at (some of) the bad primes are known in advance, they can be passed as a list of tuples <prime ideal,conductor exponent,local factor>, e.g. LocalData:=[ * < P, 11, 1 - x > * ].
Current implementation can only compute the conductor at primes P|2 when vP(Δ)<12, using Ogg's formula. If this is not the case, either the local factor at 2 must be supplied in LocalData, or the use of Ogg's formula at 2 forced with LocalData:="Ogg" or [* <P,"Ogg"> *].
A word of warning: as is it the case for most L-functions, the number of Dirichlet coefficients necessary to compute with L(C/K, s) behaves roughly like the square root of the conductor N of the L-function. For a curve C/K of genus g and conductor NC (an ideal in OK), this conductor is given by the formula
N = |(Norm)K/Q(NC)|.|ΔK/Q|2g,
It grows very fast, and so in practice the use of these L-series is limited to small genus and small degree [K:Q], even for curves of small conductor.
C/Q(i): y2 + (x3 + x2 + x + 1)y = - (i + 1)x3 - (i + 1)x2 - ix,
define its L-series L(C, s) and compute L(C, 1) and L'(C, 1).
> K<i>:=CyclotomicField(4); > R<x>:=PolynomialRing(K); > C:=HyperellipticCurve((-i-1)*x^3+(-i-1)*x^2-i*x,x^3+x^2+x+1); > L:=LSeries(C: Precision:=8); > LCfRequired(L); // need this many coefficients 7855 > CheckFunctionalEquation(L); // zero to correct precision -7.4505806E-9 > Evaluate(L,1); // L(C,1) is zero 6.4853393E-10 > Evaluate(L,1: Derivative:=1); // L'(C,1) is non-zero 0.31311835According to the Birch--Swinnerton-Dyer conjecture, the Jacobian of C/Q(i) should have rank 1. We can verify that it has indeed trivial 2-torsion, 2-Selmer group of size 2, and that there are non-trivial rational points on C:
> f:=HyperellipticPolynomials(SimplifiedModel(C));
> J:=Jacobian(HyperellipticCurve(f));
> #TwoTorsionSubgroup(J),#TwoSelmerGroup(J);
1 2
> Points(C: Bound:=10);
{@ (1 : 0 : 0), (1 : -1 : 0), (0 : -1 : 1), (0 : 0 : 1), (i - 1 : -1 : 1),
(i - 1 : -i - 1 : 1), (-i : 1 : 1), (-i : -1 : 1) @}
Precision: RngIntElt Default:
Given a primitive dirichlet character χ: (Z/mZ) * to C * , create the associated Dirichlet L-series L(χ, s)=∑n=1^∞χ(n)/ns. The character χ must be defined so that its values fall in either the ring of integers, the rational field or a cyclotomic field.The number of digits of precision to which the values L(χ, s) are to be computed may be specified using the Precision parameter. If it is omitted the precision is taken to be that of the default real field.
For information on Dirichlet characters, see Section Dirichlet Characters.
> G<Chi> := DirichletGroup(37, CyclotomicField(36)); > L := LSeries(Chi); > Evaluate(L,1); // depends on the chosen generator of G 1.65325576836885655776002342451 - 0.551607898922910805875537715935*$.1
Given a cuspidal newform in a space of Hilbert modular forms, this creates the associated L-series. Note that this is not implemented for Bianchi modular forms, as the computations of Hecke eigenvalues (particularly at bad primes) is not always feasible in that case.
> K := NumberField(x^2 - 5) where x is PolynomialRing(Rationals()).1; > H := HilbertCuspForms(K,7*Integers(K),[2,2]); > f := NewformDecomposition(NewSubspace(H))[1]; > L := LSeries(Eigenform(f)); > LSetPrecision(L,9); > LCfRequired(L); // approximate value, with old CFE 198 > time CFENew(L); // new vers of CheckFunctionalEquation 0.000000000 Time: 9.690
Precision: RngIntElt Default:
Given a primitive Hecke (Grössen)character on ideals, construct the associated L-series.For more information on these see Section Hecke Grössencharacters and their L-functions.
Embedding: Map/UserProgram Default:
Precision: RngIntElt Default:
Given a modular form f, construct the L-series L(f, s)=∑n=1^∞an/ns, where f has the q-expansion ∑n=0^∞an qn. It is assumed that L(f, s) satisfies a functional equation of the standard kind (see Section Terminology for the precise form of the functional equation).The optional parameter embedding specifies a map which embeds the coefficients of f into the complex field. By default this is the identity map, so that the coefficients of f must be coercible into C. Otherwise, the value of the parameter must either be an object of type Map or a user-defined function e(x) each having domain the base ring of f and codomain the complex field (or values than can be coerced into the complex field).
The number of digits of precision to which the values L(f, s) are to be computed may be specified using the Precision parameter. If it is omitted the precision is taken to be that of the default real field.
> f := Newforms("G1N16k2A")[1]; f;
q + (-a - 1)*q^2 + (a - 1)*q^3 + 2*a*q^4 + (-a - 1)*q^5 + 2*q^6 - 2*a*q^7 +
(-2*a + 2)*q^8 + a*q^9 + 2*a*q^10 + (a + 1)*q^11 + O(q^12)
> B:=BaseRing(f); B;
Equation Order with defining polynomial x^2 + 1 over its ground order
The two distinct embeddings of B into the complex numbers give rise to
two modular forms, which can be accessed using the ComplexEmbeddings
function.
> f1, f2 := Explode(ComplexEmbeddings(f)[1]); > Coefficient(f,2), Coefficient(f1,2), Coefficient(f2,2); -a - 1 -1.00000000000000000000000000000 + 1.00000000000000000000000000000*$.1 -1.00000000000000000000000000000 - 1.00000000000000000000000000000*$.1Thus, f1 and f2 have genuine complex coefficients and we can construct the associated L-series and compute their L-values, for instance at s=1.
> L1 := LSeries(f1); > L2 := LSeries(f2); > CFENew(L1); // new vers of CheckFunctionalEquation 4.73316543132607083247037139170E-30 > CFENew(L2); 4.73316543132607083247037139170E-30 > v1 := Evaluate(L1,1); v2 := Evaluate(L2,1); v1,v2; 0.359306437003505684066327207778 + 0.0714704939991172686588458066909*$.1 0.359306437003505684066327207778 - 0.0714704939991172686588458066909*$.1If instead we invoke LSeries(f), Magma will note that f is defined over a number field and complain that the coefficients of f are not well-defined complex numbers.
> L := LSeries(f); Runtime error: For f over a number field, you have to specify a complex embeddingInstead of using ComplexEmbeddings, one can instead explicitly specify an embedding of the coefficients of B into the complex numbers using the parameter Embedding with the function LSeries The following statements define the same L-function as L2 above.
> C<i> := ComplexField(); > L2A := LSeries(f: Embedding:=hom< B -> C | i > ); > L2B := LSeries(f: Embedding:=func< x | Conjugates(x)[1] > ); > L2C := LSeries(f1: Embedding:=func< x | ComplexConjugate(x) > );
Finally, we illustrate the very important fact that Magma expects, but does not check that the L-function associated to a modular form satisfies a functional equation.
> L := LSeries(f1+f2); // or L:=LSeries(f: Embedding:=func<x|Trace(B!x)>);
Although Magma is happy with this definition, it is in fact illegal. The modular form f has a character whose values lie in the field of the 4-th roots of unity.
> Order(DirichletCharacter(f)); 4
The two embeddings f1 and f2 of f have different (complex conjugate) characters and f1 + f2 does not satisfy a functional equation of the standard kind. Magma will suspect this when it tries to determine the sign in the functional equation and thereby print a warning:
> Evaluate(L,1); |Sign| is nowhere near 1, wrong functional equation? 0.363706143196151750493063407843 > CFENew(L); // new vers of CheckFunctionalEquation 0.293560852738072340956684556739
The function CheckFunctionalEquation or CFENew should return 0 (to current precision), so the functional equation is not satisfied, and the result of evaluating L will be a random number. So it is the user's responsibility to ensure that the modular form does satisfy a functional equation as described in Section Terminology.
Precision: RngIntElt Default:
Given a 1-dimensional cuspidal modular symbol space, construct the L-series determined by its Hecke operators (for the Euler factors). In fact, the dual Hecke operators are used for efficiency purposes.
> S := CuspidalSubspace(ModularSymbols(864,4,+1)); // wt 4 > x := PolynomialRing(Integers()).1; > K := Kernel([<5,x-19>,<7,x-13>],S); // 19 at p=5, 13 at p=7 > assert Dimension(K) eq 1; > L := LSeries(K); > LCfRequired(L); // approx for old CheckFunctionalEquation 467 > CFENew(L); 0.000000000000000000000000000000 > time DualHeckeOperator(K,997); [31536] Time: 0.040
> chi := DirichletGroup(4).1; // nontrivial character modulo 4 > S := CuspidalSubspace(ModularSymbols(chi,5,+1)); // weight 5 > L := LSeries(S); > CFENew(L); 0.000000000000000000000000000000 > EulerFactor(L,2); 4*x + 1 > DualHeckeOperator(S,2); [-4]