In the lists below K usually refers to a function field, O to an order.

Subsections

• Category and Parent
• Related Structures
• Other Structure Functions

Function fields form the Magma category FldFun, orders form RngFunOrd. The notional power structures exist as parents of function fields and their orders, they allow no operations.

Category(K) : FldFun -> Cat

Parent(K) : FldFun -> Pow

Category(O) : RngFunOrd -> Cat

Parent(O) : RngFunOrd -> Pow

Given an order O, this returns the order over which O is defined. This will be F_q[x] (for finite orders) or F_q(x) (for infinite orders).

Degree(K) : FldFun -> RngIntElt

Given a function field K, return the degree [K:G] of K over its ground field G. For an order O it returns the relative degree of O over its ground order.

The discriminant of the order O, up to a sign.

The regulator of a function field K. M should be an integer matrix giving the image of a subgroup of the unit group of K in the logarithm space.

Regulator(O) : RngFunOrd -> RngIntElt

The regulator of the finite maximal order O.

The signature of K; i.e. the (ramification index, relative degree) pairs for the places lying over the infinite place.

The defining polynomial of the order O.

Roots of the defining polynomial of K, as Puiseux series expansions in z = x^(1/e).

The unit rank of K.

The dimension of the exact constant field of K.

The genus of the function field K.

A sequence of s sequences, where the i-th sequence contains certain Dirichlet elements up to bound b w.r.t direction i (maximum of m elements per direction).

The sequence of (ramification index, relative degree) pairs of the pairwise distinct places lying over the infinite place of K.

Split(K, p) : FldFun, RngUPolElt -> SeqEnum

The sequence of (ramification index, relative degree) pairs of the pairwise distinct places lying over the place of K associated to the prime polynomial p.

The integral closure of the order O.

The result of the Dedekind test on the finite equation order O of a function field.

DedekindTest(O, p) : RngFunOrd, RngUPolElt -> BoolElt

The result of the Dedekind test on the infinite equation order O. with respect to the prime polynomial p.

A 0-reduced basis for the finite order O, together with the B^ * values of the basis elements.

Returns [eB * (b_1), ..., eB * (b_n)], max(B * (b_i)), sum(B * (b_i), and e for the 'finite' function field order O, where the b_i are basis elements.

Given an F_q[x]-order O, return the order with a 0-reduced basis.

The function field K containing the order O.

FundamentalUnits(K) : FldFun -> SeqEnum

A sequence of fundamental units of the maximal finite order O of the global function field K.