216 lines
7.1 KiB
Python
216 lines
7.1 KiB
Python
class heisenberg_function:
|
|
def __init__(self, C, g):
|
|
self.curve = C
|
|
F = C.base_ring
|
|
n = C.height
|
|
variable_names = 'x, y'
|
|
for i in range(n):
|
|
variable_names += ', z' + str(i)
|
|
Rxyz = PolynomialRing(F, n+2, variable_names)
|
|
x, y = Rxyz.gens()[:2]
|
|
z = Rxyz.gens()[2:]
|
|
RxyzQ = FractionField(Rxyz)
|
|
self.function = RxyzQ(g)
|
|
#self.function = heisenberg_reduction(AS, RxyzQ(g))
|
|
|
|
def __repr__(self):
|
|
return str(self.function)
|
|
|
|
def __eq__(self, other):
|
|
AS = self.curve
|
|
RxyzQ, Rxyz, x, y, z = AS.fct_field
|
|
aux = self - other
|
|
aux = RxyzQ(aux.function)
|
|
aux = aux.numerator()
|
|
aux = heisenberg_function(AS, aux)
|
|
aux = aux.expansion_at_infty()
|
|
if aux.valuation() >= 1:
|
|
return True
|
|
return False
|
|
|
|
def __add__(self, other):
|
|
C = self.curve
|
|
g1 = self.function
|
|
g2 = other.function
|
|
return heisenberg_function(C, g1 + g2)
|
|
|
|
def __sub__(self, other):
|
|
C = self.curve
|
|
g1 = self.function
|
|
g2 = other.function
|
|
return heisenberg_function(C, g1 - g2)
|
|
|
|
def __rmul__(self, constant):
|
|
C = self.curve
|
|
g = self.function
|
|
return heisenberg_function(C, constant*g)
|
|
|
|
def __neg__(self):
|
|
C = self.curve
|
|
g = self.function
|
|
return heisenberg_function(C, -g)
|
|
|
|
def __mul__(self, other):
|
|
if isinstance(other, heisenberg_function):
|
|
C = self.curve
|
|
g1 = self.function
|
|
g2 = other.function
|
|
return heisenberg_function(C, g1*g2)
|
|
if isinstance(other, heisenberg_form):
|
|
C = self.curve
|
|
g1 = self.function
|
|
g2 = other.form
|
|
return heisenberg_form(C, g1*g2)
|
|
|
|
def __truediv__(self, other):
|
|
C = self.curve
|
|
g1 = self.function
|
|
g2 = other.function
|
|
return heisenberg_function(C, g1/g2)
|
|
|
|
def __pow__(self, exponent):
|
|
C = self.curve
|
|
g1 = self.function
|
|
return heisenberg_function(C, g1^(exponent))
|
|
|
|
def expansion_at_infty(self, place = 0):
|
|
C = self.curve
|
|
delta = C.nb_of_pts_at_infty
|
|
F = C.base_ring
|
|
x_series = C.x_series[place]
|
|
y_series = C.y_series[place]
|
|
z_series = C.z_series[place]
|
|
n = C.height
|
|
variable_names = 'x, y'
|
|
for j in range(n):
|
|
variable_names += ', z' + str(j)
|
|
Rxyz = PolynomialRing(F, n+2, variable_names)
|
|
x, y = Rxyz.gens()[:2]
|
|
z = Rxyz.gens()[2:]
|
|
RxyzQ = FractionField(Rxyz)
|
|
prec = C.prec
|
|
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
|
|
g = self.function
|
|
g = RxyzQ(g)
|
|
sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}
|
|
return g.substitute(sub_list)
|
|
|
|
def expansion(self, pt = 0):
|
|
C = self.curve
|
|
delta = C.nb_of_pts_at_infty
|
|
F = C.base_ring
|
|
x_series = C.x_series[pt]
|
|
y_series = C.y_series[pt]
|
|
z_series = C.z_series[pt]
|
|
n = C.height
|
|
variable_names = 'x, y'
|
|
for j in range(n):
|
|
variable_names += ', z' + str(j)
|
|
Rxyz = PolynomialRing(F, n+2, variable_names)
|
|
x, y = Rxyz.gens()[:2]
|
|
z = Rxyz.gens()[2:]
|
|
RxyzQ = FractionField(Rxyz)
|
|
prec = C.prec
|
|
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
|
|
g = self.function
|
|
g = RxyzQ(g)
|
|
sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}
|
|
return g.substitute(sub_list)
|
|
|
|
def group_action(self, elt):
|
|
AS = self.curve
|
|
RxyzQ, Rxyz, x, y, z = AS.fct_field
|
|
if elt == (0, 0, 0):
|
|
return self
|
|
if elt == (1, 0, 0):
|
|
sub_list = {x : x, y : y} | {z[0] : z[0] + 1, z[1] : z[1], z[2]: z[2] + z[1]}
|
|
g = self.function
|
|
return heisenberg_function(AS, g.substitute(sub_list))
|
|
if elt == (0, 1, 0):
|
|
sub_list = {x : x, y : y} | {z[0] : z[0] + 1, z[1] : z[1] + 1, z[2]: z[2]}
|
|
g = self.function
|
|
return heisenberg_function(AS, g.substitute(sub_list))
|
|
if elt == (0, 0, 1):
|
|
sub_list = {x : x, y : y} | {z[0] : z[0], z[1] : z[1], z[2]: z[2] - 1}
|
|
g = self.function
|
|
return heisenberg_function(AS, g.substitute(sub_list))
|
|
|
|
if elt[0] > 0:
|
|
elt1 = (elt[0] - 1, elt[1], elt[2])
|
|
return self.group_action(elt1).group_action((1, 0, 0))
|
|
|
|
if elt[1] > 0:
|
|
elt1 = (elt[0], elt[1] - 1, elt[2])
|
|
return self.group_action(elt1).group_action((0, 1, 0))
|
|
|
|
if elt[2] > 0:
|
|
elt1 = (elt[0], elt[1], elt[2] - 1)
|
|
return self.group_action(elt1).group_action((0, 0, 1))
|
|
|
|
def trace(self):
|
|
print('trace ')
|
|
C = self.curve
|
|
C_super = C.quotient
|
|
n = C.height
|
|
F = C.base_ring
|
|
variable_names = 'x, y'
|
|
for j in range(n):
|
|
variable_names += ', z' + str(j)
|
|
Rxyz = PolynomialRing(F, n+2, variable_names)
|
|
x, y = Rxyz.gens()[:2]
|
|
z = Rxyz.gens()[2:]
|
|
RxyzQ = FractionField(Rxyz)
|
|
result = heisenberg_function(C, 0)
|
|
G = C.group
|
|
for a in G:
|
|
result += self.group_action(a)
|
|
result = result.function
|
|
Rxy.<x, y> = PolynomialRing(F, 2)
|
|
Qxy = FractionField(Rxy)
|
|
result = heisenberg_reduction(C, result)
|
|
#print(result)
|
|
return superelliptic_function(C_super, Qxy(result))
|
|
|
|
def coordinates(self, prec = 100, basis = 0):
|
|
"Return coordinates in H^1(X, OX)."
|
|
AS = self.curve
|
|
if basis == 0:
|
|
basis = [AS.holomorphic_differentials_basis(), AS.cohomology_of_structure_sheaf_basis()]
|
|
holo_diffs = basis[0]
|
|
coh_basis = basis[1]
|
|
f_products = []
|
|
for f in coh_basis:
|
|
f_products += [[omega.serre_duality_pairing(f) for omega in holo_diffs]]
|
|
product_of_fct_and_omegas = []
|
|
product_of_fct_and_omegas = [omega.serre_duality_pairing(self) for omega in holo_diffs]
|
|
|
|
V = (F^(AS.genus())).span_of_basis([vector(a) for a in f_products])
|
|
coh_coordinates = V.coordinates(product_of_fct_and_omegas)
|
|
return coh_coordinates
|
|
|
|
def diffn(self):
|
|
C = self.curve
|
|
C_super = C.quotient
|
|
n = C.height
|
|
RxyzQ, Rxyz, x, y, z = C.fct_field
|
|
fcts = C.functions
|
|
f = self.function
|
|
y_super = superelliptic_function(C_super, y)
|
|
dy_super = y_super.diffn().form
|
|
dz = []
|
|
for i in range(n):
|
|
dfct = fcts[i].diffn().form
|
|
dz += [-dfct]
|
|
result = f.derivative(x)
|
|
result += f.derivative(y)*dy_super
|
|
for i in range(n):
|
|
result += f.derivative(z[i])*dz[i]
|
|
return heisenberg_form(C, result)
|
|
|
|
def valuation(self, place = 0):
|
|
'''Return valuation at i-th place at infinity.'''
|
|
C = self.curve
|
|
F = C.base_ring
|
|
Rt.<t> = LaurentSeriesRing(F)
|
|
return Rt(self.expansion_at_infty(place = place)).valuation()
|