DeRhamComputation/heisenberg_covers/heisenberg_form_class.sage

293 lines
11 KiB
Python

class heisenberg_form:
def __init__(self, C, g):
self.curve = C
n = C.height
F = C.base_ring
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.form = RxyzQ(g)
def __repr__(self):
return "(" + str(self.form)+") * dx"
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]
dx_series = C.dx_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.form
sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}
return g.substitute(sub_list)*dx_series
def expansion(self, pt = 0):
'''Same code as expansion_at_infty.'''
C = self.curve
F = C.base_ring
x_series = C.x_series[pt]
y_series = C.y_series[pt]
z_series = C.z_series[pt]
dx_series = C.dx_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.form
sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}
return g.substitute(sub_list)*dx_series
def __add__(self, other):
C = self.curve
g1 = self.form
g2 = other.form
return heisenberg_form(C, g1 + g2)
def __sub__(self, other):
C = self.curve
g1 = self.form
g2 = other.form
return heisenberg_form(C, g1 - g2)
def __neg__(self):
C = self.curve
g = self.form
return heisenberg_form(C, -g)
def __rmul__(self, constant):
C = self.curve
omega = self.form
return heisenberg_form(C, constant*omega)
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.form
return heisenberg_form(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.form
return heisenberg_form(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.form
return heisenberg_form(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 coordinates(self, basis = 0):
"""Find coordinates of the given holomorphic form self in terms of the basis forms in a list holo."""
C = self.curve
if basis == 0:
basis = C.holomorphic_differentials_basis()
RxyzQ, Rxyz, x, y, z = C.fct_field
# We need to have only polynomials to use monomial_coefficients in linear_representation_polynomials,
# and sometimes basis elements have denominators. Thus we multiply by them.
fct = heisenberg_function(C, self.form)
fct = heisenberg_reduction(C, fct)
self = heisenberg_form(C, fct)
denom = LCM([denominator(omega.form) for omega in basis])
basis = [denom*omega for omega in basis]
self_with_no_denominator = denom*self
return linear_representation_polynomials(Rxyz(self_with_no_denominator.form), [Rxyz(omega.form) for omega in basis])
def trace(self):
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_form(C, 0)
G = C.group
for a in G:
result += self.group_action(a)
result = result.form
Rxy.<x, y> = PolynomialRing(F, 2)
Qxy = FractionField(Rxy)
result = heisenberg_reduction(AS, result)
return superelliptic_form(C_super, Qxy(result))
def residue(self, place=0):
return self.expansion_at_infty(place = place).residue()
def valuation(self, place=0):
return self.expansion_at_infty(place = place).valuation()
def serre_duality_pairing(self, fct):
AS = self.curve
return sum((fct*self).residue(place = _) for _ in range(AS.nb_of_pts_at_infty))
def cartier(self):
C = self.curve
F = C.base_ring
n = C.height
ff = C.functions
p = F.characteristic()
C_super = C.quotient
(RxyzQ, Rxyz, x, y, z) = C.fct_field
fct = self.form
Rxy.<x, y> = PolynomialRing(F, 2)
RxyQ = FractionField(Rxy)
x, y = Rxyz.gens()[0], Rxyz.gens()[1]
z = Rxyz.gens()[2:]
num = Rxyz(fct.numerator())
den = Rxyz(fct.denominator())
result = RxyzQ(0)
#return (num, den, z, fct)
if den in Rxy:
sub_list = {x : x, y : y} | {z[j] : (z[j]^p - RxyzQ(ff[j].function)) for j in range(n)}
num = RxyzQ(num.substitute(sub_list))
den1 = Rxyz(num.denominator())
num = Rxyz(num*den1^p)
for monomial in Rxyz(num).monomials():
degrees = [monomial.degree(z[i]) for i in range(n)]
product_of_z = prod(z[i]^(degrees[i]) for i in range(n))
monomial_divided_by_z = monomial/product_of_z
product_of_z_no_p = prod(z[i]^(degrees[i]/p) for i in range(n))
aux_form = superelliptic_form(C_super, RxyQ(monomial_divided_by_z/den))
aux_form = aux_form.cartier()
result += product_of_z_no_p * Rxyz(num).monomial_coefficient(monomial) * aux_form.form/den1
return heisenberg_form(C, result)
raise ValueError("Please present first your form as sum z^i omega_i, where omega_i are forms on quotient curve.")
def is_regular_on_U0(self):
AS = self.curve
C = AS.quotient
m = C.exponent
RxyzQ, Rxyz, x, y, z = AS.fct_field
if y^(m-1)*self.form in Rxyz:
return True
return False
def are_forms_linearly_dependent(set_of_forms):
from sage.rings.polynomial.toy_variety import is_linearly_dependent
C = set_of_forms[0].curve
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)
denominators = prod(denominator(omega.form) for omega in set_of_forms)
return is_linearly_dependent([Rxyz(denominators*omega.form) for omega in set_of_forms])
def holomorphic_combinations_fcts(S, pole_order):
'''given a set S of (form, corresponding Laurent series at some pt), find their combinations holomorphic at that pt'''
C_AS = S[0][0].curve
p = C_AS.characteristic
F = C_AS.base_ring
prec = C_AS.prec
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
RtQ = FractionField(Rt)
minimal_valuation = min([Rt(g[1]).valuation() for g in S])
if minimal_valuation >= -pole_order:
return [s[0] for s in S]
list_of_lists = [] #to będzie lista złożona z list współczynników część nieholomorficznych rozwinięcia form z S
for eta, eta_exp in S:
a = -minimal_valuation + Rt(eta_exp).valuation()
list_coeffs = a*[0] + Rt(eta_exp).list() + (-minimal_valuation)*[0]
list_coeffs = list_coeffs[:-minimal_valuation - pole_order]
list_of_lists += [list_coeffs]
M = matrix(F, list_of_lists)
V = M.kernel() #chcemy wyzerować części nieholomorficzne, biorąc kombinacje form z S
# Sprawdzamy, jakim formom odpowiadają elementy V.
forms = []
for vec in V.basis():
forma_holo = heisenberg_function(C_AS, 0)
forma_holo_power_series = Rt(0)
for vec_wspolrzedna, elt_S in zip(vec, S):
eta = elt_S[0]
#eta_exp = elt_S[1]
forma_holo += vec_wspolrzedna*eta
#forma_holo_power_series += vec_wspolrzedna*eta_exp
forms += [forma_holo]
return forms
#given a set S of (form, corresponding Laurent series at some pt), find their combinations holomorphic at that pt
def holomorphic_combinations_forms(S, pole_order):
C_AS = S[0][0].curve
p = C_AS.characteristic
F = C_AS.base_ring
prec = C_AS.prec
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
RtQ = FractionField(Rt)
minimal_valuation = min([Rt(g[1]).valuation() for g in S])
if minimal_valuation >= -pole_order:
return [s[0] for s in S]
list_of_lists = [] #to będzie lista złożona z list współczynników część nieholomorficznych rozwinięcia form z S
for eta, eta_exp in S:
a = -minimal_valuation + Rt(eta_exp).valuation()
list_coeffs = a*[0] + Rt(eta_exp).list() + (-minimal_valuation)*[0]
list_coeffs = list_coeffs[:-minimal_valuation - pole_order]
list_of_lists += [list_coeffs]
M = matrix(F, list_of_lists)
V = M.kernel() #chcemy wyzerować części nieholomorficzne, biorąc kombinacje form z S
# Sprawdzamy, jakim formom odpowiadają elementy V.
forms = []
for vec in V.basis():
forma_holo = heisenberg_form(C_AS, 0)
forma_holo_power_series = Rt(0)
for vec_wspolrzedna, elt_S in zip(vec, S):
eta = elt_S[0]
#eta_exp = elt_S[1]
forma_holo += vec_wspolrzedna*eta
#forma_holo_power_series += vec_wspolrzedna*eta_exp
forms += [forma_holo]
return forms
#print only forms that are log at the branch pts, but not holomorphic
def only_log_forms(C_AS):
list1 = AS.at_most_poles_forms(0)
list2 = AS.at_most_poles_forms(1)
result = []
for a in list2:
if not(are_forms_linearly_dependent(list1 + result + [a])):
result += [a]
return result