F/k denotes a global function field in this section.
The number of places of degree one in the global function field F/k. Contrary to the Degree() function the degree is here taken over the exact constant field.
The number of places of degree one in the constant field extension of degree m of the global function field F/k. Contrary to the Degree() function the degree is here taken over the respective exact constant fields.
The Serre bound on the number of places of degree one in a global function field of genus g over the exact constant field of q elements (of F, of the constant field extension of degree m of F). Contrary to the Degree() function the degree is here taken over the respective exact constant fields.
The Ihara bound on the number of places of degree one in a global function field F/k of genus g over the exact constant field of q elements (of F, of the constant field extension of degree m of F). Contrary to the Degree function the degree is here taken over the respective exact constant fields.
The minimum of the Serre and Ihara bound. Contrary to the Degree function the degree is here taken over the respective exact constant fields.
The number of places of degree m of the global function field F/k. Contrary to the Degree function the degree is here taken over the respective exact constant fields.
A sequence containing the places of degree m of the global function field F/k.
Returns true and a place of degree m if and only if there exists such a place in the global function field; false otherwise.
Returns true and a random place of degree m in the global function field (false if there are none).
The L-polynomial of the global function field F/k.
The L-polynomial of the constant field extension of degree m of the global function field F/k.
The Zeta function of the global function field F/k.
The Zeta function of the constant field extension of degree m of the global function field F/k.
> Y<t> := PolynomialRing(Integers());
> R<x> := FunctionField(GF(9));
> P<y> := PolynomialRing(R);
> f := y^3 + y + x^5 + x + 1;
> F<alpha> := FunctionField(f);
> Genus(F);
4
> NumberOfPlaces(F, 1);
22
> NumberOfPlacesOfDegreeOneBound(F);
32
> RandomPlace(F, 2);
true (x^2 + $.1*x + 2, alpha + $.1^2*x + $.1^5)
> LPolynomial(F);
6561*t^8 + 8748*t^7 + 7290*t^6 + 3888*t^5 + 1539*t^4 + 432*t^3 + 90*t^2
+ 12*t + 1
Given a maximal `finite' order O in a global function field, return the unit rank of O.
The unit group of a `finite' maximal order O as an Abelian group and the map from the unit group into O. Also see IsUnitWithPreimage.
The regulator of the unit group of the `finite' maximal order O.
The map from the multiplicative group of the field of fractions of O to the group of fractional ideals of O where O is a `finite' maximal order.
> PF<x> := PolynomialRing(GF(13, 2));
> P<y> := PolynomialRing(PF);
> FF1<b> := ext<FieldOfFractions(PF) | y^2 - x>;
> P<y> := PolynomialRing(FF1);
> FF2<d> := ext<FF1 | y^3 - b>;
> RER_FF2 := RationalExtensionRepresentation(FF2);
> NumberOfPlacesOfDegreeOne(FF2) eq NumberOfPlacesOfDegreeOne(RER_FF2);
true
> SerreBound(FF2);
170
> NumberOfPlaces(FF2, 1);
170
> _, P := HasPlace(FF2, 1);
> P;
(x, ((.1^44 * x + .1^100)*b + (.1^82 * x + .1^10))*d^2 + ((.1^85 * x + .1^67)*b
+ (.1^107 * x + .1^130))*d + (.1^26 * x + .1^69)*b + .1^149 * x)
> Degree(P) eq 1;
true
> LPolynomial(FF2, 2) eq LPolynomial(RER_FF2, 2);
true
> EFF2F := EquationOrderFinite(FF2);
> G, m := UnitGroup(EFF2F);
> G;
Abelian Group isomorphic to Z/168
Defined on 1 generator
Relations:
168 * G.1 = 0
> m(Random(G));
[ [ .1^120, 0 ], [ 0, 0 ], [ 0, 0 ] ]
> IsUnit(1);
true
> Regulator(EFF2F);
1
1);
true
> Regulator(EFF2F);
1
DegreeBound: RngIntElt Default:
SizeBound: RngIntElt Default:
ReductionDivisor: DivFunElt Default:
Proof: BoolElt Default:
The divisor class group of F/k as an Abelian group, a map of representatives from the class group to the divisor group and the homomorphism from the divisor group onto the divisor class group. For a detailed description see ClassGroup.
The ideal class group of the `finite' maximal order O as an Abelian group, a map of representatives from the ideal class group to the group of fractional ideals and the homomorphism from the group of fractional ideals onto the ideal class group.
-> U -> F^ x -> Id -> Cl -> 0 where U is the unit group of O, F^ x is the multiplicative group of the field of fractions of O, Id is the group of fractional ideals of O and Cl is the class group of O for a `finite' maximal order O.where U is the unit group of O, F^ x is the multiplicative group of the field of fractions of O, Id is the group of fractional ideals of O and Cl is the class group of O for a `finite' maximal order O.
DegreeBound: RngIntElt Default:
SizeBound: RngIntElt Default:
ReductionDivisor: DivFunElt Default:
Proof: BoolElt Default:
Computes a sequence of integers containing the Abelian invariants of the divisor class group of F/k. For a detailed description see ClassGroupAbelianInvariants.
Computes a sequence of integers containing the Abelian invariants of the ideal class group of the `finite' maximal order O.
The order of the ideal class group of the `finite' maximal order O.
The order of the group of divisor classes of degree zero of F/k.
> PF<x> := PolynomialRing(GF(13, 2));
> P<y> := PolynomialRing(PF);
> FF1<b> := ext<FieldOfFractions(PF) | y^2 - x>;
> P<y> := PolynomialRing(FF1);
> FF2<d> := ext<FF1 | y^3 - b : Check := false>;
> MFF2I := MaximalOrderInfinite(FF2);
> G, m, mi := ClassGroup(FF2);
> m(Random(G));
Divisor in reduced representation:
Divisor in ideal representation:
Fractional ideal of Maximal Equation Order of FF2 over Maximal Equation Order of
FF1 over Univariate Polynomial Ring in x over GF(13^2)
Generators:
1
(.1^60/x * b + .1^57/x)*d^2 + (.1^141/x * b + .1^4/x)*d + .1^80/x * b, Ideal of
MFF2I
Generators:
1
1,
- 2,
6 * (1/x, ((.1^132*x^2 + .1^164 * x + 12)/x^3 * b + (.1^85*x^2 + .1^155 * x +
12)/x^3) * d^2 + ((.1^75*x^2 + .1^81 * x + 12)/x^3 * b + (.1^29*x^2 + .1^155 * x
+ 12)/x^3) * d + (.1^163*x^2 + .1^29 * x + 12)/x^3 * b + (.1^141*x + 12)/x^2),
(x)^2 * (1/x)^2
> mi(&+[Divisor(Random(FF2, 3)) : i in [1 .. 3]]);
0
> ClassNumber(FF2);
1
The group of global units of F/k, i. e. the multiplicative group of the exact constant field, as an Abelian group, together with the map into F. Also see IsGlobalUnit and IsGlobalUnitWithPreimage.
Compute the p-rank of the class group of F/k where p is the characteristic of F/k. For a detailed description see ClassGroupPRank.
Return the Hasse--Witt invariant of F/k. See HasseWittInvariant for a detailed description.
A sequence of independent units of the `finite' maximal order O.
A sequence of fundamental units of the `finite' maximal order O.
> R<x> := FunctionField(GF(3));
> P<y> := PolynomialRing(R);
> f := y^4 + x*y + x^4 + x + 1;
> F<a> := FunctionField(f);
> O := MaximalOrderFinite(F);
> Basis(O);
[ 1, a, a^2, a^3 ]
> Discriminant(O);
x^12 + x^3 + 1
> UnitRank(O);
1
> U := FundamentalUnits(O);
> U;
[ [ x^33 + x^31 + 2*x^30 + 2*x^28 + 2*x^27 + x^25 + 2*x^24 + x^22 + 2*x^19 +
2*x^15 + x^10 + 2*x^9 + 2*x^7 + x^6 + 2*x + 2, x^32 + 2*x^30 + x^29 + 2*x^28
+ 2*x^27 + 2*x^26 + x^22 + x^21 + 2*x^19 + x^18 + x^17 + x^16 + x^13 + x^11
+ 2*x^10 + 2*x^9 + 2*x^3 + 1, x^29 + x^27 + 2*x^25 + 2*x^23 + x^22 + 2*x^21
+ x^20 + x^18 + 2*x^17 + x^16 + x^15 + 2*x^14 + x^11 + 2*x^10 + 2*x^4 + x,
x^30 + 2*x^27 + x^24 + x^21 + 2*x^18 + x^9 + 2*x^6 + 2 ] ]
> Norm(U[1]);
1
> Regulator(O);
33