Create the group of divisors of the algebraic function field F/k.
Given a place P in a function field, return the prime divisor 1 * P.
Given an algebraic function a, return the principal divisor (a).
The divisor corresponding to the ideal factorization of I.
The divisor corresponding to the ideal factorization of the ideals I and J belonging to the `finite' and `infinite' maximal order.
Given the group G of divisors of a function field, return the zero divisor.
A canonical divisor of the function field F/k.
The different divisor of the underlying extension of the function field F/k(x).
Change the print name employed when displaying D to be the contents of s which must have length 1 in this case.
The group of divisors form the Magma category DivFun. The notional power structure exists as parent but allows no operations.
Given the group G of divisors of a function field F/k, return F.
The set of places of the algebraic function field F/k.
The number of effective divisors of degree less equal n who consist of places of degree less equal m only. P[i] contains the (generic) number of places of degree 1 <= i <= min { n, m }. The formula used is described in [Heß99].
A divisor of degree one over the exact constant field of the global function field F/k.
Returns divisors D_1, D_2 such that D = kD_1 + D_2 and the exponents in D_2 are of absolute value less than |k|. The operations div and mod yield D_1 resp. D_2.
The greatest common divisor of the divisors D1 and D2.
The least common multiple of the divisors D1 and D2.
Returns true iff the divisor D is canonical and a differential having D as its divisor.
Given a divisor D, return the function field.
The degree of the divisor D over k, the constant field of definition.
A sequence containing the places occurring in the divisor D.
The numerator of D.
The denominator of D.
Create two ideals of the `finite' and `infinite' maximal order respectively corresponding to the divisor D.
The dimension of the Riemann-Roch space ( L)(D) over k, the constant field of definition.
The index of speciality of the divisor D, which equals the dimension of ( L)(W - D) where W is a canonical divisor.
Reduction: BoolElt Default: true
Simplification: "Full", "None" Default: "Full"
Compute a basis for the Riemann-Roch space of D in short form:Let F = k(x, y) be an algebraic function field defined by f(x, y)=0 over k. Given a divisor D of F/k this function returns a basis of the k-vector space ( L)(D) = { a in F^ x | (a) >= - D } union { 0 } in the short form B = [ b_1 ..., b_n ], [ d_1, ..., d_n ] with b_i in F^ x and d_i in Z for all 1 <= i <= n, where n denotes the degree in y of the defining equation f of F, such that ( L)(D) = { sum_(i=1)^n lambda_i b_i | lambda_i in k[x] ( with deg ) lambda_i <= d_i ( for ) 1 <= i <= n }.
The optional argument Reduction controls whether to use divisor reduction internally or not; it defaults to true. For small divisors this is sometimes faster.
The optional argument Simplification controls whether the resulting basis is simplified or not; it defaults to "Full". Simplification sometimes is not insignificantly expensive.
The algorithm is described in [Heß99].
Reduction: BoolElt Default: true
Simplification: "Full", "None" Default: "Full"
A sequence containing a basis of the Riemann-Roch space ( L)(D), for the divisor D.The optional argument Reduction controls whether to use divisor reduction internally or not; it defaults to true. For small divisors this is sometimes faster.
The optional argument Simplification controls whether the resulting basis is simplified or not; it defaults to "Full". Simplification sometimes is not insignificantly expensive.
Given a function field F/k and a divisor D belonging to F/k, return a vector space V and a k-linear mapping h: V longrightarrow F such that V is isomorphic to the Riemann-Roch space ( L)(D) subset F under h.
The exponent of the place P in the divisor D.
Let D be a divisor. Denote the result of both functions by tilde(D), r, A and a (for the second function the input A always equals the output A). A has (must have) positive degree and the following holds:
- D = tilde(D) + rA - (a),
- tilde(D) >= 0 and deg(tilde(D)) < g + deg(A) (over the exact constant field),
- tilde(D) has minimal degree among all such divisors satisfying (i), (ii).
The sequence of gap numbers of the divisor D at P where P must be a place of degree one:Let F/k be an algebraic function field, D a divisor and P a place of degree one. An integer m >= 1 is a gap number of D at P if dim bigl( D + (m - 1)P bigr) = dim(D + mP) holds. The gap numbers m of D satisfy 1 <= m <= 2g - 1 - deg(D) and their cardinality equals the index of speciality i(D). GapNumbers(D, P) returns such a particular sequence. The sequences of gap numbers of D at various P are independent of constant field extensions for perfect k and are the same for all but a finite number of places P of degree one (consider e.g. k algebraically closed). If P is omitted in the function call, this uniform sequence is returned by GapNumbers(D). The places P where D has different sequences of gap numbers are called Weierstraß places of D and are returned by WeierstrassPlaces(D). In the above mentioned functions it is equivalent to replace D by either F or the zero divisor.
SeparatingElement: FldFunGElt Default:
separating element used internally for the computation can be specified, it defaults to SeparatingElement(F). See the description of GapNumbers.
We construct the function field F/GF(9) and compute the Riemann-Roch space corresponding to a certain divisor.
> k<w> := GF(9);
> R<x> := FunctionField(k);
> P<y> := PolynomialRing(R);
> f := y^4 + (2*x^5 + x^4 + 2*x^3 + x^2)*y^2 + x^8
> + 2*x^6 + x^5 +x^4 + x^3 + x^2;
> F<a> := FunctionField(f);
> Genus(F);
7
> _, P1 := RandomPlace(F, 1);
> _, P2 := RandomPlace(F, 1);
> D := P1 - P2;
> D;
(1/x, w^7/x^7*a^3 + w^5/x^5*a^2 + w^3/x^2*a + w) - (x, 2/(x^4 + x^2 + 2*x)*a^3 +
w^3/x*a^2 + (w^5*x^3 + w^3*x + w^7)/(x^3 + x + 2)*a + w^5)
> IsPrincipal(336*D);
true
> infty := Poles(F!x)[1];
> V, h := RiemannRochSpace(11*infty);
> V;
KModule V of dimension 5 over GF(3^2)
> h;
Mapping from: ModFld: V to FldFun: F
> B := h(Basis(V));
> B;
[ x/(x^3 + x + 2)*a^3 + (2*x^4 + 2*x^3 + x)/(x^3 + x + 2)*a, 1/(x^3 + x + 2)*a^3
+ (2*x^3 + 2*x^2 + 1)/(x^3 + x + 2)*a, a^2 + 2*x^3 + 2*x^2, 1/x*a^2 + 2*x^2
+ 2*x, 1 ]
> (B[2] + 2*B[3])@@h;
( 0 1 2 0 0)
> Qc<c> := PolynomialRing(RationalField()); > Qcs<s> := PolynomialRing(Qc); > F<s> := FunctionField(s^2 + c^2 - 1); > c := F!c; > Genus(F); 0 > Zeros(s); [ (c - 1, s), (c + 1, s) ] > Zeros(c-1); [ (c - 1, s) ] > P := Zeros(c-1)[1]; > Degree(P); 1 > Dimension(1*P); 2 > Basis(1*P); [ 1/(c - 1)*s, 1 ] > a := Basis(1*P)[1]; > Degree(a); 1 > MinimalPolynomial(a); .1^2 + (c + 1)/(c - 1) > (a^2 - 1)/(a^2 + 1); c > a * ((a^2 - 1)/(a^2 + 1) - 1); s.1^2 + (c + 1)/(c - 1) > (a^2 - 1)/(a^2 + 1); c > a * ((a^2 - 1)/(a^2 + 1) - 1); s
> Qx<x> := PolynomialRing(RationalField()); > k<i> := NumberField(x^2 + 1); > kc<c> := PolynomialRing(k); > kcs<s> := PolynomialRing(kc); > F<s> := FunctionField(s^2 + c^2 - 1); > c := F!c; > Genus(F); 0 > e := c + i*s; > ebar := c - i*s; > Degree(e); 1 > c eq (e + ebar) / 2; true > s eq (e - ebar) / (2*i); true
SeparatingElement: FldFunGElt Default:
The ramification divisor of D (using SeparatingElement for the computation which defaults to SeparatingElement(F) for F/k the function field of D):Let F/k be an algebraic function field, x a separating variable and D a divisor. The ramification divisor of D is defined to be i(D) ( W - D ) + bigl( W_x(D) bigr) + nu (dx), where W is a canonical divisor of F/k, W_x(D) is the determinant of the Wronksian matrix of D with respect to x and nu is the sum of the Wronskian orders of D with respect to x. It is effective and consists of the Weierstraß places of D. The constant field k is required to be exact.
SeparatingElement: FldFunGElt Default:
The Weierstrass places of the divisor D (using SeparatingElement for the computation which defaults to SeparatingElement(F) for F/k the function field of D):Let F/k be an algebraic function field, D a divisor and P a place of degree one. An integer m >= 1 is a gap number of D at P if dim bigl( D + (m - 1)P bigr) = dim(D + mP) holds. The gap numbers m of D at P satisfy 1 <= m <= 2g - 1 - deg(D) and their cardinality equals the index of speciality i(D). The sequences of gap numbers of D are independent of constant field extensions for perfect k and are the same for all but a finite number of places P of degree one (consider e.g. k algebraically closed). The places P of degree one at which D has different sequences of gap numbers are called Weierstraß places of D. This function returns a list of all places of F/k (having not necessarily degree one) which are lying below Weierstraß places of D viewed in F bar(k) / bar(k) (k perfect). The constant field k is required to be exact. Note that if the characteristic of F is positive this function is currently quite slow for large genus because of Differentiation().
SeparatingElement: FldFunGElt Default:
Let F/k be an algebraic function field with separating element x and let v_1, ... v_l be a basis of ( L)(D). For the differentiation D_x with respect to x consider the successively smallest nu_1 <= ... <= nu_l in Z^( >= 0) such that the rows D_x^((nu_i))(v_1), ..., D_x^((nu_i))(v_l), 1 <= i <= l are F-linearly independent. The numbers nu_1, ..., nu_l are the Wronskian orders of D with respect to x and are returned. If D has dimension zero, the empty list is returned. The constant field k is required to be exact.The separating element can be given by setting the SeparatingElement parameter appropriately.
Return the complementary divisor D^# of D. The function field F/k of D must be a finite extension of a rational function field k(x). D^# equals Diff(F/k(x)) - D for F the function field of D and Diff(F/k(x)) the different divisor of F/k(x).
A basis of the space of differentials of the divisor D. See DifferentialBasis for details.
A vector space and the isomorphism from this space to the differential space of the divisor D.
An element x in F and a sequence of elements L in the rational function field are returned such that x generates F over the constant field and L contains the images of the generators of F over its constant field in the rational function field.
Let F/k be a global function field. The group of divisor classes is isomorphic to the product of a copy of Z and the group of divisors classes of degree zero which is a finite abelian group. Magma features an algorithm to compute the divisor class group by computing an abelian group G in the form Z/c_1Z x ... x Z/c_(2g)Z x Z with integers c_1 | ... | c_(2g) and a surjective homomorphism f: Div(F) -> G from the divisor group to G whose kernel consists precisely of the principal divisors.
The algorithm employed is a randomized index calculus style method of expected subexponential running time for "small" constant field size and "large" genus. A description of this and other algorithms of this section can be found in [Heß99].
A bound B such that the places of degree (over the exact constant field) less than or equal to B, taken together with the places of a divisor of degree one, generate the whole divisor class group of any global function field of genus g over the exact constant field of q elements.
A bound B such that all places of degree (over the exact constant field) less than or equal to B, taken together with the places of a divisor of degree one, generate the whole divisor class group. Particular properties of the function field are taken into account.
An approximation of the class number of the global function field F/k with multiplicative error less than 1 + e for e > 0. The formula biggl| log bigl( h / q^g bigr) - sum_(r=1)^(b) q^(-r)/r bigl( N_r - (q^r + 1) bigl) biggr| <= 2g q^(-b/2) / bigl( (q^(1/2) - 1)(b + 1) bigr) is used where N_r denotes the number of places of degree one in the constant field extension of degree r of F/k.
Returns an integer B such that all places of degree less than or equal to B of a global function field of genus g over the exact constant field of q elements have to be considered in order to approximate the class number with multiplicative error less than 1 + e for e > 0.
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.
The optional parameter DegreeBound allows to control the size of the factor basis which consists of all places of degree less equal DegreeBound (plus a small additional amount; the degree is taken over the exact constant field). If not provided the algorithm tries to choose an appropriate value.
The optional parameter SizeBound bounds the size of the factor basis to not exceed SizeBound places. Every time the factor basis has to be enlarged during the computation it will be by no more than SizeBound additional places. If not provided there is no bound on the size of the factor basis. Every enlargement of the factor basis will append all places of the next degree.
The optional parameter ReductionDivisor contains the reduction divisor used in the relation search stage. Reasonable choices are divisors of small positive degree. If not provided the algorithm tries to choose an appropriate reduction divisor.
The optional parameter Proof indicates whether the computed result should be proven in a proof step. If a small degree bound for the factor basis is used and the divisor class group happens to be a product of a large number of cyclic groups the proof step can be very time consuming and Proof := false might be helpful. Once a value is given for Proof it remains the default value until set differently. The initial value of Proof for every function field is true.
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.
The optional parameters are the same as for ClassGroup.
The order of the group of divisor classes of degree zero of F/k.
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.
Whether a is a global unit, i.e. a constant (equivalent to IsConstant)
Returns true and the preimage of a in the global unit group, false otherwise.
The map from the multiplicative group of the function field to the group of divisors.
Returns the three maps in the center of the exact sequence 0 -> k^ x -> F^ x -> Div -> Cl -> 0 where k^ x is the global unit group of the function field, F^ x is the multiplicative group of the function field, Div is the divisor group and Cl is the divisor class group.
The group of S-units as an Abelian group and the map into the function field
Returns true if a is an S-unit, false otherwise.
Returns true and the preimage of a in the S-unit group if a is an S-unit, false otherwise.
The S-Regulator.
The map from the multiplicative group of the function field to the group of divisors (mod places in S).
Returns true and a generator if D is principal modulo places in S, false otherwise
The S-class group as an Abelian group, a map of representatives from the S-class group to the group of divisors (mod places in S) and the homomorphism from the group of divisors (mod places in S) onto the S-class group.
Returns the three maps in the center of the exact sequence 0 -> U(S) -> F^ x -> Div(S) -> Cl(S) -> 0 where U(S) is the S-unit group, F^ x is the multiplicative group of the function field, Div(S) is the group of divisors (mod places in S) and Cl(S) is the S-class group.
Computes a sequence of integers containing the Abelian invariants of the S-class group.
The order of the torsion part of the S-class group.
> 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)
> ClassNumberApproximation(F, 1.3);
24890.25505701632912193514
> L := LPolynomial(F);
> L;
6561*t^8 + 8748*t^7 + 7290*t^6 + 3888*t^5 + 1539*t^4 + 432*t^3 + 90*t^2
+ 12*t + 1
> Evaluate(L, 1);
28561
Compute the p-rank of the class group of F/k where p is the characteristic of F/k. More precisely: Let F/k be a function field of characteristic p. Consider the subgroup Cl^0(F/k)[p] of p-torsion elements of the group of divisor classes of degree zero. This function returns its dimension as an GF(p)-vector space. Possible values range from 0 to g, where g is the genus of F/k. k is currently required to be a finite field.
Return the Hasse--Witt invariant of F/k. More precisely: Let F/k be a function field of characteristic p. Let F bar(k) / bar(k) be the constant field extension by the algebraic closure bar(k) of k within an algebraic closure bar(F) of F/k. Consider the subgroup Cl^0(F bar(k) / bar(k))[p] of p-torsion elements of the group of divisor classes of degree zero. This function returns its dimension as an GF(p)-vector space. Possible values range from 0 to g, where g is the genus of F/k. k is required to be perfect.
The Tate--Lichtenbaum pairing Cl_0[m] x Cl_0/mCl_0 -> k.[Next][Prev] [Right] [Left] [Up] [Index] [Root]