Divisors of function fields¶
Sage allows extensive computations with divisors on function fields.
EXAMPLES:
The divisor of an element of the function field is the formal sum of poles and zeros of the element with multiplicities:
sage: K.<x> = FunctionField(GF(2)); R.<t> = K[]
sage: L.<y> = K.extension(t^3 + x^3*t + x)
sage: f = x/(y+1)
sage: f.divisor()
- Place (1/x, 1/x^3*y^2 + 1/x)
+ Place (1/x, 1/x^3*y^2 + 1/x^2*y + 1)
+ 3*Place (x, y)
- Place (x^3 + x + 1, y + 1)
>>> from sage.all import *
>>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); R = K['t']; (t,) = R._first_ngens(1)
>>> L = K.extension(t**Integer(3) + x**Integer(3)*t + x, names=('y',)); (y,) = L._first_ngens(1)
>>> f = x/(y+Integer(1))
>>> f.divisor()
- Place (1/x, 1/x^3*y^2 + 1/x)
+ Place (1/x, 1/x^3*y^2 + 1/x^2*y + 1)
+ 3*Place (x, y)
- Place (x^3 + x + 1, y + 1)
The Riemann-Roch space of a divisor can be computed. We can get a basis of the space as a vector space over the constant field:
sage: p = L.places_finite()[0]
sage: q = L.places_infinite()[0]
sage: (3*p + 2*q).basis_function_space()
[1/x*y^2 + x^2, 1, 1/x]
>>> from sage.all import *
>>> p = L.places_finite()[Integer(0)]
>>> q = L.places_infinite()[Integer(0)]
>>> (Integer(3)*p + Integer(2)*q).basis_function_space()
[1/x*y^2 + x^2, 1, 1/x]
We verify the Riemann-Roch theorem:
sage: D = 3*p - q
sage: index_of_speciality = len(D.basis_differential_space())
sage: D.dimension() == D.degree() - L.genus() + 1 + index_of_speciality
True
>>> from sage.all import *
>>> D = Integer(3)*p - q
>>> index_of_speciality = len(D.basis_differential_space())
>>> D.dimension() == D.degree() - L.genus() + Integer(1) + index_of_speciality
True
AUTHORS:
Kwankyu Lee (2017-04-30): initial version
- class sage.rings.function_field.divisor.DivisorGroup(field)[source]¶
Bases:
UniqueRepresentation
,Parent
Groups of divisors of function fields.
INPUT:
field
– function field
EXAMPLES:
sage: K.<x> = FunctionField(GF(5)); _.<Y> = K[] sage: F.<y> = K.extension(Y^2 - x^3 - 1) sage: F.divisor_group() Divisor group of Function field in y defined by y^2 + 4*x^3 + 4
>>> from sage.all import * >>> K = FunctionField(GF(Integer(5)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1) >>> F = K.extension(Y**Integer(2) - x**Integer(3) - Integer(1), names=('y',)); (y,) = F._first_ngens(1) >>> F.divisor_group() Divisor group of Function field in y defined by y^2 + 4*x^3 + 4
- Element[source]¶
alias of
FunctionFieldDivisor
- function_field()[source]¶
Return the function field to which the divisor group is attached.
EXAMPLES:
sage: K.<x> = FunctionField(GF(5)); _.<Y> = K[] sage: F.<y> = K.extension(Y^2 - x^3 - 1) sage: G = F.divisor_group() sage: G.function_field() Function field in y defined by y^2 + 4*x^3 + 4
>>> from sage.all import * >>> K = FunctionField(GF(Integer(5)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1) >>> F = K.extension(Y**Integer(2) - x**Integer(3) - Integer(1), names=('y',)); (y,) = F._first_ngens(1) >>> G = F.divisor_group() >>> G.function_field() Function field in y defined by y^2 + 4*x^3 + 4
- class sage.rings.function_field.divisor.FunctionFieldDivisor(parent, data)[source]¶
Bases:
ModuleElement
Divisors of function fields.
INPUT:
parent
– divisor groupdata
– dictionary of place and multiplicity pairs
EXAMPLES:
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[] sage: F.<y> = K.extension(Y^3 - x^2*(x^2 + x + 1)^2) sage: f = x/(y + 1) sage: f.divisor() Place (1/x, 1/x^4*y^2 + 1/x^2*y + 1) + Place (1/x, 1/x^2*y + 1) + 3*Place (x, (1/(x^3 + x^2 + x))*y^2) - 6*Place (x + 1, y + 1)
>>> from sage.all import * >>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1) >>> F = K.extension(Y**Integer(3) - x**Integer(2)*(x**Integer(2) + x + Integer(1))**Integer(2), names=('y',)); (y,) = F._first_ngens(1) >>> f = x/(y + Integer(1)) >>> f.divisor() Place (1/x, 1/x^4*y^2 + 1/x^2*y + 1) + Place (1/x, 1/x^2*y + 1) + 3*Place (x, (1/(x^3 + x^2 + x))*y^2) - 6*Place (x + 1, y + 1)
- basis_differential_space()[source]¶
Return a basis of the space of differentials \(\Omega(D)\) for the divisor \(D\).
EXAMPLES:
We check the Riemann-Roch theorem:
sage: K.<x>=FunctionField(GF(4)); _.<Y> = K[] sage: L.<y>=K.extension(Y^3 + x^3*Y + x) sage: d = 3*L.places()[0] sage: l = len(d.basis_function_space()) sage: i = len(d.basis_differential_space()) sage: l == d.degree() + 1 - L.genus() + i True
>>> from sage.all import * >>> K = FunctionField(GF(Integer(4)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1) >>> L = K.extension(Y**Integer(3) + x**Integer(3)*Y + x, names=('y',)); (y,) = L._first_ngens(1) >>> d = Integer(3)*L.places()[Integer(0)] >>> l = len(d.basis_function_space()) >>> i = len(d.basis_differential_space()) >>> l == d.degree() + Integer(1) - L.genus() + i True
- basis_function_space()[source]¶
Return a basis of the Riemann-Roch space of the divisor.
EXAMPLES:
sage: K.<x> = FunctionField(GF(5)); _.<Y> = K[] sage: F.<y> = K.extension(Y^2 - x^3 - 1) sage: O = F.maximal_order() sage: I = O.ideal(x - 2) sage: D = I.divisor() sage: D.basis_function_space() [x/(x + 3), 1/(x + 3)]
>>> from sage.all import * >>> K = FunctionField(GF(Integer(5)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1) >>> F = K.extension(Y**Integer(2) - x**Integer(3) - Integer(1), names=('y',)); (y,) = F._first_ngens(1) >>> O = F.maximal_order() >>> I = O.ideal(x - Integer(2)) >>> D = I.divisor() >>> D.basis_function_space() [x/(x + 3), 1/(x + 3)]
- degree()[source]¶
Return the degree of the divisor.
EXAMPLES:
sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[] sage: L.<y> = K.extension(Y^3 + x^3*Y + x) sage: p1,p2 = L.places()[:2] sage: D = 2*p1 - 3*p2 sage: D.degree() -1
>>> from sage.all import * >>> K = FunctionField(GF(Integer(4)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1) >>> L = K.extension(Y**Integer(3) + x**Integer(3)*Y + x, names=('y',)); (y,) = L._first_ngens(1) >>> p1,p2 = L.places()[:Integer(2)] >>> D = Integer(2)*p1 - Integer(3)*p2 >>> D.degree() -1
- denominator()[source]¶
Return the denominator part of the divisor.
The denominator of a divisor is the negative of the negative part of the divisor.
EXAMPLES:
sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[] sage: L.<y> = K.extension(Y^3 + x^3*Y + x) sage: p1,p2 = L.places()[:2] sage: D = 2*p1 - 3*p2 sage: D.denominator() 3*Place (1/x, 1/x^3*y^2 + 1/x^2*y + 1)
>>> from sage.all import * >>> K = FunctionField(GF(Integer(4)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1) >>> L = K.extension(Y**Integer(3) + x**Integer(3)*Y + x, names=('y',)); (y,) = L._first_ngens(1) >>> p1,p2 = L.places()[:Integer(2)] >>> D = Integer(2)*p1 - Integer(3)*p2 >>> D.denominator() 3*Place (1/x, 1/x^3*y^2 + 1/x^2*y + 1)
- dict()[source]¶
Return the dictionary representing the divisor.
EXAMPLES:
sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[] sage: L.<y> = K.extension(Y^3 + x^3*Y + x) sage: f = x/(y + 1) sage: D = f.divisor() sage: D.dict() {Place (1/x, 1/x^3*y^2 + 1/x): -1, Place (1/x, 1/x^3*y^2 + 1/x^2*y + 1): 1, Place (x, y): 3, Place (x^3 + x + 1, y + 1): -1}
>>> from sage.all import * >>> K = FunctionField(GF(Integer(4)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1) >>> L = K.extension(Y**Integer(3) + x**Integer(3)*Y + x, names=('y',)); (y,) = L._first_ngens(1) >>> f = x/(y + Integer(1)) >>> D = f.divisor() >>> D.dict() {Place (1/x, 1/x^3*y^2 + 1/x): -1, Place (1/x, 1/x^3*y^2 + 1/x^2*y + 1): 1, Place (x, y): 3, Place (x^3 + x + 1, y + 1): -1}
- differential_space()[source]¶
Return the vector space of the differential space \(\Omega(D)\) of the divisor \(D\).
OUTPUT:
a vector space isomorphic to \(\Omega(D)\)
an isomorphism from the vector space to the differential space
the inverse of the isomorphism
EXAMPLES:
sage: K.<x> = FunctionField(GF(5)); _.<Y> = K[] sage: F.<y> = K.extension(Y^2 - x^3 - 1) sage: O = F.maximal_order() sage: I = O.ideal(x - 2) sage: P1 = I.divisor().support()[0] sage: Pinf = F.places_infinite()[0] sage: D = -3*Pinf + P1 sage: V, from_V, to_V = D.differential_space() sage: all(to_V(from_V(e)) == e for e in V) True
>>> from sage.all import * >>> K = FunctionField(GF(Integer(5)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1) >>> F = K.extension(Y**Integer(2) - x**Integer(3) - Integer(1), names=('y',)); (y,) = F._first_ngens(1) >>> O = F.maximal_order() >>> I = O.ideal(x - Integer(2)) >>> P1 = I.divisor().support()[Integer(0)] >>> Pinf = F.places_infinite()[Integer(0)] >>> D = -Integer(3)*Pinf + P1 >>> V, from_V, to_V = D.differential_space() >>> all(to_V(from_V(e)) == e for e in V) True
- dimension()[source]¶
Return the dimension of the Riemann-Roch space of the divisor.
EXAMPLES:
sage: K.<x> = FunctionField(GF(5)); _.<Y> = K[] sage: F.<y> = K.extension(Y^2 - x^3 - 1) sage: O = F.maximal_order() sage: I = O.ideal(x - 2) sage: P1 = I.divisor().support()[0] sage: Pinf = F.places_infinite()[0] sage: D = 3*Pinf + 2*P1 sage: D.dimension() 5
>>> from sage.all import * >>> K = FunctionField(GF(Integer(5)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1) >>> F = K.extension(Y**Integer(2) - x**Integer(3) - Integer(1), names=('y',)); (y,) = F._first_ngens(1) >>> O = F.maximal_order() >>> I = O.ideal(x - Integer(2)) >>> P1 = I.divisor().support()[Integer(0)] >>> Pinf = F.places_infinite()[Integer(0)] >>> D = Integer(3)*Pinf + Integer(2)*P1 >>> D.dimension() 5
- function_space()[source]¶
Return the vector space of the Riemann-Roch space of the divisor.
OUTPUT:
a vector space, an isomorphism from the vector space to the Riemann-Roch space, and its inverse.
EXAMPLES:
sage: K.<x> = FunctionField(GF(5)); _.<Y> = K[] sage: F.<y> = K.extension(Y^2-x^3-1) sage: O = F.maximal_order() sage: I = O.ideal(x - 2) sage: D = I.divisor() sage: V, from_V, to_V = D.function_space() sage: all(to_V(from_V(e)) == e for e in V) True
>>> from sage.all import * >>> K = FunctionField(GF(Integer(5)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1) >>> F = K.extension(Y**Integer(2)-x**Integer(3)-Integer(1), names=('y',)); (y,) = F._first_ngens(1) >>> O = F.maximal_order() >>> I = O.ideal(x - Integer(2)) >>> D = I.divisor() >>> V, from_V, to_V = D.function_space() >>> all(to_V(from_V(e)) == e for e in V) True
- is_effective()[source]¶
Return
True
if this divisor has nonnegative multiplicity at all places.EXAMPLES:
sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[] sage: L.<y> = K.extension(Y^3 + x^3*Y + x) sage: p1, p2 = L.places()[:2] sage: D = 2*p1 + 3*p2 sage: D.is_effective() True sage: E = D - 4*p2 sage: E.is_effective() False
>>> from sage.all import * >>> K = FunctionField(GF(Integer(4)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1) >>> L = K.extension(Y**Integer(3) + x**Integer(3)*Y + x, names=('y',)); (y,) = L._first_ngens(1) >>> p1, p2 = L.places()[:Integer(2)] >>> D = Integer(2)*p1 + Integer(3)*p2 >>> D.is_effective() True >>> E = D - Integer(4)*p2 >>> E.is_effective() False
- list()[source]¶
Return the list of place and multiplicity pairs of the divisor.
EXAMPLES:
sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[] sage: L.<y> = K.extension(Y^3 + x^3*Y + x) sage: f = x/(y + 1) sage: D = f.divisor() sage: D.list() [(Place (1/x, 1/x^3*y^2 + 1/x), -1), (Place (1/x, 1/x^3*y^2 + 1/x^2*y + 1), 1), (Place (x, y), 3), (Place (x^3 + x + 1, y + 1), -1)]
>>> from sage.all import * >>> K = FunctionField(GF(Integer(4)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1) >>> L = K.extension(Y**Integer(3) + x**Integer(3)*Y + x, names=('y',)); (y,) = L._first_ngens(1) >>> f = x/(y + Integer(1)) >>> D = f.divisor() >>> D.list() [(Place (1/x, 1/x^3*y^2 + 1/x), -1), (Place (1/x, 1/x^3*y^2 + 1/x^2*y + 1), 1), (Place (x, y), 3), (Place (x^3 + x + 1, y + 1), -1)]
- multiplicity(place)[source]¶
Return the multiplicity of the divisor at the place.
INPUT:
place
– place of a function field
EXAMPLES:
sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[] sage: L.<y> = K.extension(Y^3 + x^3*Y + x) sage: p1,p2 = L.places()[:2] sage: D = 2*p1 - 3*p2 sage: D.multiplicity(p1) 2 sage: D.multiplicity(p2) -3
>>> from sage.all import * >>> K = FunctionField(GF(Integer(4)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1) >>> L = K.extension(Y**Integer(3) + x**Integer(3)*Y + x, names=('y',)); (y,) = L._first_ngens(1) >>> p1,p2 = L.places()[:Integer(2)] >>> D = Integer(2)*p1 - Integer(3)*p2 >>> D.multiplicity(p1) 2 >>> D.multiplicity(p2) -3
- numerator()[source]¶
Return the numerator part of the divisor.
The numerator of a divisor is the positive part of the divisor.
EXAMPLES:
sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[] sage: L.<y> = K.extension(Y^3 + x^3*Y + x) sage: p1,p2 = L.places()[:2] sage: D = 2*p1 - 3*p2 sage: D.numerator() 2*Place (1/x, 1/x^3*y^2 + 1/x)
>>> from sage.all import * >>> K = FunctionField(GF(Integer(4)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1) >>> L = K.extension(Y**Integer(3) + x**Integer(3)*Y + x, names=('y',)); (y,) = L._first_ngens(1) >>> p1,p2 = L.places()[:Integer(2)] >>> D = Integer(2)*p1 - Integer(3)*p2 >>> D.numerator() 2*Place (1/x, 1/x^3*y^2 + 1/x)
- support()[source]¶
Return the support of the divisor.
EXAMPLES:
sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[] sage: L.<y> = K.extension(Y^3 + x^3*Y + x) sage: f = x/(y + 1) sage: D = f.divisor() sage: D.support() [Place (1/x, 1/x^3*y^2 + 1/x), Place (1/x, 1/x^3*y^2 + 1/x^2*y + 1), Place (x, y), Place (x^3 + x + 1, y + 1)]
>>> from sage.all import * >>> K = FunctionField(GF(Integer(4)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1) >>> L = K.extension(Y**Integer(3) + x**Integer(3)*Y + x, names=('y',)); (y,) = L._first_ngens(1) >>> f = x/(y + Integer(1)) >>> D = f.divisor() >>> D.support() [Place (1/x, 1/x^3*y^2 + 1/x), Place (1/x, 1/x^3*y^2 + 1/x^2*y + 1), Place (x, y), Place (x^3 + x + 1, y + 1)]
- valuation(place)[source]¶
Return the multiplicity of the divisor at the place.
INPUT:
place
– place of a function field
EXAMPLES:
sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[] sage: L.<y> = K.extension(Y^3 + x^3*Y + x) sage: p1,p2 = L.places()[:2] sage: D = 2*p1 - 3*p2 sage: D.multiplicity(p1) 2 sage: D.multiplicity(p2) -3
>>> from sage.all import * >>> K = FunctionField(GF(Integer(4)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1) >>> L = K.extension(Y**Integer(3) + x**Integer(3)*Y + x, names=('y',)); (y,) = L._first_ngens(1) >>> p1,p2 = L.places()[:Integer(2)] >>> D = Integer(2)*p1 - Integer(3)*p2 >>> D.multiplicity(p1) 2 >>> D.multiplicity(p2) -3
- sage.rings.function_field.divisor.divisor(field, data)[source]¶
Construct a divisor from the data.
INPUT:
field
– function fielddata
– dictionary of place and multiplicity pairs
EXAMPLES:
sage: K.<x> = FunctionField(GF(2)); R.<t> = K[] sage: F.<y> = K.extension(t^3 - x^2*(x^2 + x + 1)^2) sage: from sage.rings.function_field.divisor import divisor sage: p, q, r = F.places() sage: divisor(F, {p: 1, q: 2, r: 3}) Place (1/x, 1/x^2*y + 1) + 2*Place (x, (1/(x^3 + x^2 + x))*y^2) + 3*Place (x + 1, y + 1)
>>> from sage.all import * >>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); R = K['t']; (t,) = R._first_ngens(1) >>> F = K.extension(t**Integer(3) - x**Integer(2)*(x**Integer(2) + x + Integer(1))**Integer(2), names=('y',)); (y,) = F._first_ngens(1) >>> from sage.rings.function_field.divisor import divisor >>> p, q, r = F.places() >>> divisor(F, {p: Integer(1), q: Integer(2), r: Integer(3)}) Place (1/x, 1/x^2*y + 1) + 2*Place (x, (1/(x^3 + x^2 + x))*y^2) + 3*Place (x + 1, y + 1)
- sage.rings.function_field.divisor.prime_divisor(field, place, m=1)[source]¶
Construct a prime divisor from the place.
INPUT:
field
– function fieldplace
– place of the function fieldm
– (default: 1) a positive integer; multiplicity at the place
EXAMPLES:
sage: K.<x> = FunctionField(GF(2)); R.<t> = K[] sage: F.<y> = K.extension(t^3 - x^2*(x^2 + x + 1)^2) sage: p = F.places()[0] sage: from sage.rings.function_field.divisor import prime_divisor sage: d = prime_divisor(F, p) sage: 3 * d == prime_divisor(F, p, 3) True
>>> from sage.all import * >>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); R = K['t']; (t,) = R._first_ngens(1) >>> F = K.extension(t**Integer(3) - x**Integer(2)*(x**Integer(2) + x + Integer(1))**Integer(2), names=('y',)); (y,) = F._first_ngens(1) >>> p = F.places()[Integer(0)] >>> from sage.rings.function_field.divisor import prime_divisor >>> d = prime_divisor(F, p) >>> Integer(3) * d == prime_divisor(F, p, Integer(3)) True