Creation of Class Fields

Since the beginning of class field theory one of the core problems has been to find defining equations for the fields. Although most of the original proofs are essentially constructive, they rely on complicated and involved computations and thus were not suited for hand computations.

The method used here to compute defining equations is essentially the same as the one used for number fields. The main differences are due to the problem of p-extensions in characteristic p where Artin-Schreier-Witt theory is used, and the fact that the divisor class group is infinite.

AbelianExtension(D, U) : DivFunElt, GrpAb -> FldFunAb

Given an effective divisor D and a subgroup U of the ray class group, (see RayClassGroup), Cl_D of D such that the quotient Cl_D/U is finite, create the extension defined by this data. Note that, at this point, no defining equations are computed.

MaximalAbelianSubfield(K) : FldFunG -> FldFunAb

For a relative extension K/k of global function fields, compute the maximal abelian subfield K/A/k of K/k as an abelian extension of k. In particular, this function compute the norm group of K/k as a subgroup of a suitable ray class group.

HilbertClassField(K, p) : FldFun, PlcFunElt -> FldFunAb

For a global function field K and a place p of K, compute the (a) Hilbert class field of K as an abelian extension of K. This field is characterised by being the maximal abelian unramified extension of K where p is totally split.

FunctionField(A) : FldFunAb -> FldFun

    WithAut: BoolElt                    Default: false

    SetVerbose("ClassField", n):        Maximum: 3

Given an abelian extension of function fields as created by AbelianExtension, compute defining equations for the corresponding (ray) class field.

More precisely: Let Cl/U = ∏Cl/U_i be a decomposition of the norm group of A such that the Cl/U_i are cyclic of prime power order. Then for each U_i the function will compute a defining equation, thus A is represented by the compositum.

If WithAut is true, the second sequence returned contains a generating automorphism for each of the fields returned as the first return value. If WithAut is false, the first (and only) return value is a function field in non-simple representation.

MaximalOrderFinite(A) : FldFunAb -> RngFunOrd

MaximalOrderInfinite(A) : FldFunAb -> RngFunOrd

Compute a finite or an infinite maximal order of the function field of the abelian extension A.

This computes the maximal orders of each function field defined by each of the defining polynomials of the function field of A then combines them and finishes off the computation by using the algorithm of [BL94] on the result. As the cyclic components are essentially Kummer, Artin--Schreier or Artin--Schreier---Witt extensions the algorithms in [Sut12], [Sut13] and [Sut16] are used to compute the maximal orders of the component function fields.

> k<w> := GF(4);
> kt<t> := PolynomialRing(k);
> ktx<x> := PolynomialRing(kt);
> K := FunctionField(x^3-w*t*x^2+x+t);
> lp := Places(K, 2);
> D := 4*lp[2]+2*lp[6];
> R, mR := RayClassGroup(D);

> inf := InfinitePlaces(K);
> U1 := sub<R | [x@@ mR : x in inf]>;
> U2 := sub<R | [5*R.i : i in [1..Ngens(R)]]>;
> U3 := sub<R | U1, U2>;
> A := AbelianExtension(D, U3);
> A;
Abelian extension of type [ 5 ]
Defined modulo 4*(t^2 + t + w, $.1 + w^2*t + 1) +
2*(t^2 + w^2*t + w^2, $.1 + w*t + w)
over Algebraic function field defined over
Univariate rational function field over GF(2^2) by
x^3 + w*t*x^2 + x + t
> FunctionField(A);
Algebraic function field defined over K by
$.1^5 + (w*K.1^2 + (w*t + w^2)*K.1 + (w^2*t^2 + t
    + w^2))*$.1^3 + ((t^2 + 1)*K.1^2 + w*t*K.1 +
    (w*t^4 + w))*$.1 + (t^3 + w*t^2 + w^2*t)*K.1^2
    + (w*t^4 + w^2*t^2 + w^2)*K.1 + t^4 + w^2*t^3
    + w^2*t + w
> E := $1;
> #InfinitePlaces(E); 
10

> MaximalOrderFinite(A);
Maximal Order of Algebraic function field defined over K by
$.1^5 + (t*K.1^2 + (w*t^2 + w^2)*K.1 + (w*t^2 + t + w))*$.1^3 + ((t^2 + w)*K.1^2
    + w^2*t^3*K.1 + (t^4 + t^2 + w^2))*$.1 + (t^4 + w^2*t^2 + w*t + w)*K.1^2 +
    (w*t^5 + w*t^4 + w*t^3)*K.1 + w^2*t^5 + t^4 + w*t^3 + w^2*t + 1 over Maximal
    Equation Order of K over Univariate Polynomial Ring in t over GF(2^2)
> MaximalOrderInfinite(A);
Maximal Order of Algebraic function field defined over K by
$.1^5 + (w^2*K.1^2 + (w^2*t + 1)*K.1 + (t^2 + w*t + 1))*$.1^3 + ((w^2*t^2 +
    w^2)*K.1^2 + t*K.1 + (t^4 + 1))*$.1 + (w*t^3 + w^2*t^2 + t)*K.1^2 + (w^2*t^4
    + t^2 + 1)*K.1 + w*t^4 + t^3 + t + w^2 over Maximal Order of K over
    Valuation ring of Univariate rational function field over GF(2^2) with 
    generator 1/t
> DiscriminantDivisor(A);
4*(t^2 + t + w, K.1 + w^2*t + 1) + 4*(t^2 + w^2*t + w^2, K.1 + w*t + w)
> $1 eq Divisor(Discriminant($3)) + Divisor(Discriminant($2));
true

> q, mq := quo<R | U1>;
> l4 := [ x`subgroup : x in Subgroups(q : Quot := [4])];
> #l4;
7168
> A := [AbelianExtension(D, x@@mq) : x in l4];
> s4 := [<Genus(a), NumberOfPlacesOfDegreeOne(a), a> : a in A];
> Maximum([x[2]/x[1] : x in s4]);
16/5 15
> E := FunctionField(s4[15][3]);
> l22 := [ x`subgroup : x in Subgroups(q : Quot := [2,2])];
> #l22;
43435

> MaximalOrderFinite(s4[15][3]);
Maximal Order of E over Maximal Equation Order of K over Univariate Polynomial
Ring in t over GF(2^2)
> MaximalOrderInfinite(s4[15][3]);
Maximal Order of E over Maximal Order of K over Valuation ring of Univariate
rational function field over GF(2^2) with generator 1/t