Differentials of function fields

Sage provides arithmetic with differentials of function fields.

EXAMPLES:

The module of differentials on a function field forms an one-dimensional vector space over the function field:

sage: # needs sage.rings.finite_rings sage.rings.function_field
sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x + x^3*Y)
sage: f = x + y
sage: g = 1 / y
sage: df = f.differential()
sage: dg = g.differential()
sage: dfdg = f.derivative() / g.derivative()
sage: df == dfdg * dg
True
sage: df
(x*y^2 + 1/x*y + 1) d(x)
sage: df.parent()
Space of differentials of Function field in y defined by y^3 + x^3*y + x

We can compute a canonical divisor:

sage: # needs sage.rings.finite_rings sage.rings.function_field
sage: k = df.divisor()
sage: k.degree()
4
sage: k.degree() == 2 * L.genus() - 2
True

Exact differentials vanish and logarithmic differentials are stable under the Cartier operation:

sage: # needs sage.rings.finite_rings sage.rings.function_field
sage: df.cartier()
0
sage: w = 1/f * df
sage: w.cartier() == w
True

AUTHORS:

• Kwankyu Lee (2017-04-30): initial version

class sage.rings.function_field.differential.DifferentialsSpace(field, category=None)

Bases: UniqueRepresentation, Parent

Space of differentials of a function field.

INPUT:

• field – function field

EXAMPLES:

sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[]                                 # needs sage.rings.finite_rings
sage: L.<y> = K.extension(Y^3 + x^3*Y + x)                                      # needs sage.rings.finite_rings sage.rings.function_field
sage: L.space_of_differentials()                                                # needs sage.rings.finite_rings sage.rings.function_field
Space of differentials of Function field in y defined by y^3 + x^3*y + x

The space of differentials is a one-dimensional module over the function field. So a base differential is chosen to represent differentials. Usually the generator of the base rational function field is a separating element and used to generate the base differential. Otherwise a separating element is automatically found and used to generate the base differential relative to which other differentials are denoted:

sage: # needs sage.rings.function_field
sage: K.<x> = FunctionField(GF(5))
sage: R.<y> = K[]
sage: L.<y> = K.extension(y^5 - 1/x)
sage: L(x).differential()
0
sage: y.differential()
d(y)
sage: (y^2).differential()
(2*y) d(y)

Element

alias of FunctionFieldDifferential

basis()

Return a basis.

EXAMPLES:

sage: # needs sage.rings.finite_rings sage.rings.function_field
sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x^3*Y + x)
sage: S = L.space_of_differentials()
sage: S.basis()
Family (d(x),)

function_field()

Return the function field to which the space of differentials is attached.

EXAMPLES:

sage: # needs sage.rings.finite_rings sage.rings.function_field
sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x^3*Y + x)
sage: S = L.space_of_differentials()
sage: S.function_field()
Function field in y defined by y^3 + x^3*y + x

class sage.rings.function_field.differential.DifferentialsSpaceInclusion

Bases: Morphism

Inclusion morphisms for extensions of function fields.

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)                                    # needs sage.rings.function_field
sage: OK = K.space_of_differentials()
sage: OL = L.space_of_differentials()                                           # needs sage.rings.function_field
sage: OL.coerce_map_from(OK)                                                    # needs sage.rings.function_field
Inclusion morphism:
  From: Space of differentials of Rational function field in x over Rational Field
  To:   Space of differentials of Function field in y defined by y^2 - x*y + 4*x^3

is_injective()

Return True, since the inclusion morphism is injective.

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)                                # needs sage.rings.function_field
sage: OK = K.space_of_differentials()
sage: OL = L.space_of_differentials()                                       # needs sage.rings.function_field
sage: OL.coerce_map_from(OK).is_injective()                                 # needs sage.rings.function_field
True

is_surjective()

Return True if the inclusion morphism is surjective.

EXAMPLES:

sage: # needs sage.rings.function_field
sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)
sage: OK = K.space_of_differentials()
sage: OL = L.space_of_differentials()
sage: OL.coerce_map_from(OK).is_surjective()
False
sage: S.<z> = L[]
sage: M.<z> = L.extension(z - 1)
sage: OM = M.space_of_differentials()
sage: OM.coerce_map_from(OL).is_surjective()
True

class sage.rings.function_field.differential.DifferentialsSpace_global(field, category=None)

Bases: DifferentialsSpace

Space of differentials of a global function field.

INPUT:

• field – function field

EXAMPLES:

sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[]                                 # needs sage.rings.finite_rings
sage: L.<y> = K.extension(Y^3 + x^3*Y + x)                                      # needs sage.rings.finite_rings sage.rings.function_field
sage: L.space_of_differentials()                                                # needs sage.rings.finite_rings sage.rings.function_field
Space of differentials of Function field in y defined by y^3 + x^3*y + x

Element

alias of FunctionFieldDifferential_global

class sage.rings.function_field.differential.FunctionFieldDifferential(parent, f, t=None)

Bases: ModuleElement

Base class for differentials on function fields.

INPUT:

• f – element of the function field

• t – element of the function field; if \(t\) is not specified, the generator of the base differential is assumed

EXAMPLES:

sage: F.<x> = FunctionField(QQ)
sage: f = x/(x^2 + x + 1)
sage: f.differential()
((-x^2 + 1)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1)) d(x)

sage: K.<x> = FunctionField(QQ); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x + x^3*Y)                                      # needs sage.rings.function_field
sage: L(x).differential()                                                       # needs sage.rings.function_field
d(x)
sage: y.differential()                                                          # needs sage.rings.function_field
((21/4*x/(x^7 + 27/4))*y^2 + ((3/2*x^7 + 9/4)/(x^8 + 27/4*x))*y + 7/2*x^4/(x^7 + 27/4)) d(x)

divisor()

Return the divisor of the differential.

EXAMPLES:

sage: K.<x> = FunctionField(GF(5)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x + x^3*Y)                                  # needs sage.rings.function_field
sage: w = (1/y) * y.differential()                                          # needs sage.rings.function_field
sage: w.divisor()                                                           # needs sage.rings.function_field
- 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 + 2, y + 3)
 + Place (x^6 + 3*x^5 + 4*x^4 + 2*x^3 + x^2 + 3*x + 4, y + x^5)

sage: F.<x> = FunctionField(QQ)
sage: w = (1/x).differential()
sage: w.divisor()                                                           # needs sage.libs.pari
-2*Place (x)

monomial_coefficients(copy=True)

Return a dictionary whose keys are indices of basis elements in the support of self and whose values are the corresponding coefficients.

EXAMPLES:

sage: K.<x> = FunctionField(GF(5)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x + x^3*Y)                                  # needs sage.rings.function_field
sage: d = y.differential()                                                  # needs sage.rings.function_field
sage: d                                                                     # needs sage.rings.function_field
((4*x/(x^7 + 3))*y^2 + ((4*x^7 + 1)/(x^8 + 3*x))*y + x^4/(x^7 + 3)) d(x)
sage: d.monomial_coefficients()                                             # needs sage.rings.function_field
{0: (4*x/(x^7 + 3))*y^2 + ((4*x^7 + 1)/(x^8 + 3*x))*y + x^4/(x^7 + 3)}

residue(place)

Return the residue of the differential at the place.

INPUT:

• place – a place of the function field

OUTPUT:

• an element of the residue field of the place

EXAMPLES:

We verify the residue theorem in a rational function field:

sage: # needs sage.rings.finite_rings
sage: F.<x> = FunctionField(GF(4))
sage: f = 0
sage: while f == 0:
....:     f = F.random_element()
sage: w = 1/f * f.differential()
sage: d = f.divisor()
sage: s = d.support()
sage: sum([w.residue(p).trace() for p in s])                                # needs sage.rings.function_field
0

and in an extension field:

sage: # needs sage.rings.function_field
sage: K.<x> = FunctionField(GF(7)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x + x^3*Y)
sage: f = 0
sage: while f == 0:
....:     f = L.random_element()
sage: w = 1/f * f.differential()
sage: d = f.divisor()
sage: s = d.support()
sage: sum([w.residue(p).trace() for p in s])
0

and also in a function field of characteristic zero:

sage: # needs sage.rings.function_field
sage: R.<x> = FunctionField(QQ)
sage: L.<Y> = R[]
sage: F.<y> = R.extension(Y^2 - x^4 - 4*x^3 - 2*x^2 - 1)
sage: a = 6*x^2 + 5*x + 7
sage: b = 2*x^6 + 8*x^5 + 3*x^4 - 4*x^3 - 1
sage: w = y*a/b*x.differential()
sage: d = w.divisor()
sage: sum([QQ(w.residue(p)) for p in d.support()])
0

valuation(place)

Return the valuation of the differential at the place.

INPUT:

• place – a place of the function field

EXAMPLES:

sage: K.<x> = FunctionField(GF(5)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x + x^3*Y)                                  # needs sage.rings.function_field
sage: w = (1/y) * y.differential()                                          # needs sage.rings.function_field
sage: [w.valuation(p) for p in L.places()]                                  # needs sage.rings.function_field
[-1, -1, -1, 0, 1, 0]

class sage.rings.function_field.differential.FunctionFieldDifferential_global(parent, f, t=None)

Bases: FunctionFieldDifferential

Differentials on global function fields.

EXAMPLES:

sage: F.<x> = FunctionField(GF(7))
sage: f = x/(x^2 + x + 1)
sage: f.differential()
((6*x^2 + 1)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1)) d(x)

sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[]                                 # needs sage.rings.finite_rings
sage: L.<y> = K.extension(Y^3 + x + x^3*Y)                                      # needs sage.rings.finite_rings sage.rings.function_field
sage: y.differential()                                                          # needs sage.rings.finite_rings sage.rings.function_field
(x*y^2 + 1/x*y) d(x)

cartier()

Return the image of the differential by the Cartier operator.

The Cartier operator operates on differentials. Let \(x\) be a separating element of the function field. If a differential \(\omega\) is written in prime-power representation \(\omega=(f_0^p+f_1^px+\dots+f_{p-1}^px^{p-1})dx\), then the Cartier operator maps \(\omega\) to \(f_{p-1}dx\). It is known that this definition does not depend on the choice of \(x\).

The Cartier operator has interesting properties. Notably, the set of exact differentials is precisely the kernel of the Cartier operator and logarithmic differentials are stable under the Cartier operation.

EXAMPLES:

sage: # needs sage.rings.finite_rings sage.rings.function_field
sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x + x^3*Y)
sage: f = x/y
sage: w = 1/f*f.differential()
sage: w.cartier() == w
True

sage: # needs sage.rings.finite_rings
sage: F.<x> = FunctionField(GF(4))
sage: f = x/(x^2 + x + 1)
sage: w = 1/f*f.differential()
sage: w.cartier() == w                                                      # needs sage.rings.function_field
True