This section list some related functions that are either useful in the context of class fields for function fields or are necessary for their computation. They will most certainly change their appearance.
Strict: BoolElt Default: false
Exception: DivFunElt Default: false
Raw: BoolElt Default: false
Given an effective divisor m and a sequence S of pairs (Qi, ei) of places and elements, find an element a and a place Q0 such that vQi(a - ei) ≥vQi(m), and a is integral everywhere outside Qi (0≤i≤n).If Exception is not false, it has to be a place that will be used for Q0.
If Strict is true, the element a will be chosen such vQi(a - ei) = vQi(m)
If Raw is true, different rather technical return values are computed that are used internally.
Strict: BoolElt Default: false
Given a sequence S of either finite or infinite places of a function field, a sequence Z of elements of a function field and a sequence V of integers, return an element z such that z - Z[i] has valuation at least V[i] at S[i] and positive valuation at all other places of same finiteness as those in S which do not appear in S. If the parameter Strict is set to true then z will be computed such that vS[i](z - Z[i]) = V[i].
IntegralOutside: BoolElt Default: false
Given a sequence S of either finite or infinite places of a function field, a sequence Z of elements of a function field and a sequence V of integers, return an element z such that z - Z[i] has valuation at least V[i] at S[i]. If the parameter IntegralOutside is set to true then z will be computed such that it is integral at all other places of same finiteness as those in S which do not appear in S.
> 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);We will now try to find an element x in K such that vpi(x - ei) ≥mi for pi = lp[i], mi = i and random elements ei:
> e := [Random(K, 3) : i in lp]; > m := [i : i in [1..#lp]]; > D := &+ [ m[i]*lp[i] : i in [1..#lp]]; > x := StrongApproximation(D, [<lp[i], e[i]> : i in [1..#lp]]); > [Valuation(x-e[i], lp[i]) : i in [1..#lp]]; [ 1, 2, 3, 4, 5, 6 ]Note, that we only required ≥ for the valuations, to enforce = we would need to pass the Strict option. This will double the running time.
Exception: DivFunElt Default:
Given an effective divisor m, find a place P coprime to m and an integer r≥0 such that rP - m is a non special divisor and return r and P.If Exception is specified, it must be an effective divisor n coprime to m. In this case the function finds r>0 such that rn - m is non special and returns r and n.
Cond: DivFunElt Default:
AS: RngWittElt Default:
Extra: RngIntElt Default: 5
Given a global function field, try to compute its norm group. The norm group is defined to be the group generated by norms of unramified divisors. This group can be related to a subgroup of some ray class group.Provided F is abelian, this function will compute a divisor m and a sub group U of the ray class group modulo m such that F is isomorphic to the ray class field thus defined.
This function uses a heuristic algorithm. It will terminate after the size of the quotient by the norm group is less or equal than the degree for Extra many places.
If Cond is given, it must be an effective divisor that will be used as the potential conductor of F. Note: if Cond is too small, ie. a proper divisor of the true conductor, the result of this function will be wrong. However, if the conductor is not passed in, the discriminant divisor is used as a starting point. As this is in general far too large, the function will be much quicker if a better (smaller) starting point is passed in.
If AS is given, it must be a Witt vector e of appropriate length and F should be the corresponding function field. This allows a much better initial guess for the conductor than using the discriminant.
Given a function a in some global function field and a place p such that a is integral at p (has non-negative valuation) return the sign of a, ie. the first non-zero coefficient if the expansion of a at p. The sign function is not unique. Magma choses a sign function when creating the residue class field map.
Given a global function field F and a place p, return a new function field G that is Fq-isomorphic to F and has p as the only infinite place.
Return the valuation of u - (zp - z) at P and an element z such that this valuation is either positive or not congruent to 0 mod p where p is the characteristic of the field of u.