Differentials of function fields¶
Sage provides basic arithmetic and advanced computations with differentials on global function fields.
EXAMPLES:
The module of differentials on a function field forms an one-dimensional vector space over the 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: 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: 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:
sage.structure.unique_representation.UniqueRepresentation
,sage.structure.parent.Parent
Space of differentials of a function field.
INPUT:
field
– function field
EXAMPLES:
sage: K.<x>=FunctionField(GF(4)); _.<Y>=K[] sage: L.<y>=K.extension(Y^3+x+x^3*Y) sage: L.space_of_differentials() Space of differentials of Function field in y defined by y^3 + x^3*y + x
-
Element
¶ alias of
FunctionFieldDifferential_global
-
basis
()¶ Return a basis.
EXAMPLES:
sage: K.<x>=FunctionField(GF(4)); _.<Y>=K[] sage: L.<y>=K.extension(Y^3+x+x^3*Y) 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: K.<x>=FunctionField(GF(4)); _.<Y>=K[] sage: L.<y>=K.extension(Y^3+x+x^3*Y) 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.
FunctionFieldDifferential
¶ Bases:
sage.structure.element.ModuleElement
Base class for differentials on function fields.
-
class
sage.rings.function_field.differential.
FunctionFieldDifferential_global
(parent, f, t=None)¶ Bases:
sage.rings.function_field.differential.FunctionFieldDifferential
Differentials on global function fields.
INPUT:
f
– element of the function fieldt
– element of the function field; if \(t\) is not specified, \(t\) is the generator of the base rational function field
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[] sage: L.<y> = K.extension(Y^3 + x + x^3*Y) sage: y.differential() (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 \(\omega=(f_0^p+f_1^px+\dots+f_{p-1}^px^{p-1})dx\) (prime-power representation), 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: 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: F.<x> = FunctionField(GF(4)) sage: f = x/(x^2+x+1) sage: w = 1/f * f.differential() sage: w.cartier() == w True
-
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) sage: w = (1/y) * y.differential() sage: w.divisor() - 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() -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) sage: d = y.differential() sage: d ((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() {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
– 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: 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]) 0
and also in an extension 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