Global Function Fields

In this section \(F/k\) denotes a global function field.

Functions relative to the Exact Constant Field

NumberOfPlacesOfDegreeOverExactConstantField(F, m): FldFun, RngIntElt -> RngIntElt

NumberOfPlacesDegECF(F, m): FldFun, RngIntElt -> RngIntElt

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.

NumberOfPlacesOfDegreeOneOverExactConstantField(F): FldFunG -> RngIntElt

NumberOfPlacesOfDegreeOneECF(F): FldFunG -> RngIntElt

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.

NumberOfPlacesOfDegreeOneOverExactConstantField(F, m): FldFunG, RngIntElt -> RngIntElt

NumberOfPlacesOfDegreeOneECF(F, m): FldFun, RngIntElt -> RngIntElt

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.

SerreBound(F): FldFunG -> RngIntElt

SerreBound(F, m): FldFunG, RngIntElt -> RngIntElt

SerreBound(q, g): RngIntElt, RngIntElt -> RngIntElt

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 the global function field \(F\), 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.

IharaBound(F): FldFunG -> RngIntElt

IharaBound(F, m): FldFunG, RngIntElt -> RngIntElt

IharaBound(q, g): RngIntElt, RngIntElt -> RngIntElt

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 the global function field \(F\), 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.

NumberOfPlacesOfDegreeOneECFBound(F): FldFunG -> RngIntElt

NumberOfPlacesOfDegreeOneOverExactConstantFieldBound(F): FldFunG -> RngIntElt

NumberOfPlacesOfDegreeOneECFBound(F, m): FldFun, RngIntElt -> RngIntElt

NumberOfPlacesOfDegreeOneOverExactConstantFieldBound(F, m): FldFunG, RngIntElt -> RngIntElt

NumberOfPlacesOfDegreeOneECFBound(q, g): RngIntElt, RngIntElt -> RngIntElt

NumberOfPlacesOfDegreeOneOverExactConstantFieldBound(q, g): RngIntElt, RngIntElt -> RngIntElt

The minimum of the Serre and Ihara bound. Contrary to the Degree function the degree is here taken over the respective exact constant fields.

LPolynomial(F): FldFunG -> RngUPolElt

The \(L\)-polynomial of the global function field \(F/k\) (with respect to the exact constant field).

LPolynomial(F, m): FldFunG, RngIntElt -> RngUPolElt

The \(L\)-polynomial of the constant field extension of degree \(m\) of the global function field \(F/k\) (with respect to the exact constant field).

ZetaFunction(F): FldFunG -> FldFunRatUElt

The Zeta function of the global function field \(F/k\) (with respect to the exact constant field).

ZetaFunction(F, m): FldFunG, RngIntElt -> FldFunRatUElt

The Zeta function of the constant field extension of degree \(m\) of the global function field \(F/k\) (with respect to the exact constant field).

Functions Relative to the Constant Field

Places(F, m): FldFunG, RngIntElt -> SeqEnum[PlcFunElt]

A sequence containing the places of degree \(m\) of the global function field \(F/k\).

HasPlace(F, m): FldFunG, RngIntElt -> BoolElt, PlcFunElt

Returns true and a place of degree \(m\) if and only if there exists such a place in the global function field; false otherwise.

HasRandomPlace(F, m): FldFunG, RngIntElt -> BoolElt, PlcFunElt

Returns true and a random place of degree \(m\) in the global function field (false if there are none).

RandomPlace(F, m): FldFunG, RngIntElt -> PlcFunElt

Returns a random place of degree \(m\) in the global function field or throws an error if there is none.

Example: Invariants of a Global Function Field (ex-8f580e)

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> 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
> NumberOfPlacesDegECF(F, 1);
22
> NumberOfPlacesOfDegreeOneECFBound(F);
32
> HasRandomPlace(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

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Example: Invariants of a Global Relative Field (ex-9f3184)

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Some of the above functions are demonstrated for a global relative field.

> 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);
> NumberOfPlacesOfDegreeOneECF(FF2) eq NumberOfPlacesOfDegreeOneECF(RER_FF2);
true
> SerreBound(FF2);
170
> NumberOfPlacesDegECF(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

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Functions related to Class Group

UnitRank(O): RngFunOrd -> RngIntElt

Given a maximal ‘finite’ order \(O\) in a global function field, return the unit rank of \(O\).

UnitGroup(O): RngFunOrd -> GrpAb, Map

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.

Regulator(O): RngFunOrd -> RngIntElt

The regulator of the unit group of the ‘finite’ maximal order \(O\).

PrincipalIdealMap(O): RngFunOrd -> Map

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.

Example: Unit Group of an Equation Order (ex-1591a6)

Following on from the last example,

> 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

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ClassGroup(F : parameters): FldFunG -> GrpAb, Map, Map

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.

ClassGroup(O): RngFunOrd -> GrpAb, Map, Map

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.

ClassGroupExactSequence(O): RngFunOrd -> Map, Map, Map

Returns the maps in the center of the exact sequence \(0 \rightarrow U \rightarrow F^\times \rightarrow Id \rightarrow Cl \rightarrow 0\) where \(U\) is the unit group of \(O\), \(F^\times\) 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 \(O\).

ClassGroupAbelianInvariants(F : parameters): FldFunG -> SeqEnum

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.

ClassGroupAbelianInvariants(O): RngFunOrd -> SeqEnum

Computes a sequence of integers containing the Abelian invariants of the ideal class group of the ‘finite’ maximal order \(O\).

ClassNumber(F): FldFunG -> RngIntElt

The order of the group of divisor classes of degree zero of \(F/k\).

ClassNumber(O): RngFunOrd -> RngIntElt

The order of the ideal class group of the ‘finite’ maximal order \(O\).

Example: Class Groups of Relative Fields (ex-dd2e5e)

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An example of class groups of relative fields is shown.

> 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

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GlobalUnitGroup(F): FldFunG -> GrpAb, Map

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.

ClassGroupPRank(F): FldFunG -> RngIntElt

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.

HasseWittInvariant(F): FldFunG -> RngIntElt

Return the Hasse–Witt invariant of \(F/k\). See HasseWittInvariant for a detailed description.

IndependentUnits(O): RngFunOrd -> SeqEnum[RngFunOrdElt]

A sequence of independent units of the ‘finite’ maximal order \(O\).

FundamentalUnits(O): RngFunOrd -> SeqEnum[RngFunOrdElt]

A sequence of fundamental units of the ‘finite’ maximal order \(O\).

Example: Basis and Fundamental Units of a Maximal Order (ex-065c59)

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> 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

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