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praca nad as_cech coordinate; przed poprawa drugiej czesci

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jgarnek 2022-12-23 12:52:17 +00:00
parent 6494187dfe
commit d77addeaf8
5 changed files with 11731 additions and 54 deletions
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11662
sage/.run.term-0.term
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File diff suppressed because one or more lines are too long

38
sage/as_covers/as_cech_class.sage
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@ -12,6 +12,9 @@ class as_cech:
RxyzQ = FractionField(Rxyz)
self.omega0 = omega
self.f = f
self.omega8 = self.omega0 - self.f.diffn()
if self.omega0.form not in Rxyz or self.omega8.valuation() < 0:
raise ValueError('cech cocycle not regular')
def __repr__(self):
return "( " + str(self.omega0)+", " + str(self.f) + " )"
@ -41,6 +44,7 @@ class as_cech:
def coordinates(self, threshold=10, basis = 0):
'''Find coordinates of self in the de Rham cohomology basis. Threshold is an argument passed to AS.de_rham_basis().'''
AS = self.curve
RxyzQ, Rxyz, x, y, z = AS.fct_field
if basis == 0:
basis = [AS.holomorphic_differentials_basis(), AS.cohomology_of_structure_sheaf_basis(), AS.de_rham_basis(threshold=threshold)]
holo_diffs = basis[0]
@ -48,25 +52,37 @@ class as_cech:
dR = basis[2]
F = AS.base_ring
f_products = []
for i, f in enumerate(coh_basis):
f_products += [[]]
for omega in holo_diffs:
f_products[i] += [sum((f*omega).residue(place = _) for _ in range(AS.nb_of_pts_at_infty))]
for f in coh_basis:
f_products += [[omega.serre_duality_pairing(f) for omega in holo_diffs]]
print(f_products)
product_of_fct_and_omegas = []
fct = self.f
for omega in holo_diffs:
product_of_fct_and_omegas += [sum((fct*omega).residue(place = _) for _ in range(AS.nb_of_pts_at_infty))]
product_of_fct_and_omegas = [omega.serre_duality_pairing(fct) 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) #coeficients of self in the basis elts coming from cohomology of OX
for i in range(AS.genus()):
self -= coh_coordinates[i]*dR[i+AS.genus()]
#We remove now from f the summands which are obviously regular at infty
print(self, [])
f_num = numerator(self.f.function)
f_den = denominator(self.f.function)
v_f_den = as_function(AS, f_den).valuation()
for a in f_num.monomials():
if as_function(AS, a).valuation() >= v_f_den:
self.f.function -= f_num.monomial_coefficient(a)*a/f_den
f_num = numerator(self.f.function)
f_den = denominator(self.f.function)
quo, rem = f_num.quo_rem(f_den)
if as_function(AS, rem/f_den).valuation() >= 0:
self.f = as_function(AS, quo)
hol_form = self.omega0 - self.f.diffn() #now this should be a holomorphic form
hol_form = as_form(AS, as_reduction(AS, hol_form.form))
print('hol_form', hol_form)
return hol_form.coordinates() + coh_coordinates
print(self, [omega.serre_duality_pairing(self.f) for omega in holo_diffs])
raise ValueError('I arrived at a form (omega, 0), in which omega is not regular on U0. I hoped this wouldn t happen.')
hol_form = self.omega0 + self.f.diffn() #now this should be a holomorphic form
hol_form = hol_form
print(hol_form)
return hol_form.coordinates() + coh_coordinates
def group_action(self, g):
AS = self.curve

55
sage/as_covers/as_cover_class.sage
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@ -54,10 +54,10 @@ class as_cover:
all_z_series += [z_series]
all_dx_series += [x_series.derivative()]
self.jumps = all_jumps
self.x = all_x_series
self.y = all_y_series
self.z = all_z_series
self.dx = all_dx_series
self.x_series = all_x_series
self.y_series = all_y_series
self.z_series = all_z_series
self.dx_series = all_dx_series
##############
#Function field
variable_names = 'x, y'
@ -68,6 +68,11 @@ class as_cover:
z = Rxyz.gens()[2:]
RxyzQ = FractionField(Rxyz)
self.fct_field = (RxyzQ, Rxyz, x, y, z)
self.x = as_function(self, x)
self.y = as_function(self, y)
self.z = [as_function(self, z[i]) for i in range(n)]
self.dx = as_form(self, 1)
def __repr__(self):
n = self.height
@ -104,10 +109,10 @@ class as_cover:
def holomorphic_differentials_basis(self, threshold = 8):
from itertools import product
x_series = self.x
y_series = self.y
z_series = self.z
dx_series = self.dx
x_series = self.x_series
y_series = self.y_series
z_series = self.z_series
dx_series = self.dx_series
delta = self.nb_of_pts_at_infty
p = self.characteristic
n = self.height
@ -145,9 +150,9 @@ class as_cover:
""" Find fcts with pole order in infty's at most pole_order. Threshold gives a bound on powers of x in the function.
If you suspect that you haven't found all the functions, you may increase it."""
from itertools import product
x_series = self.x
y_series = self.y
z_series = self.z
x_series = self.x_series
y_series = self.y_series
z_series = self.z_series
delta = self.nb_of_pts_at_infty
p = self.characteristic
n = self.height
@ -197,9 +202,9 @@ class as_cover:
"""Find forms with pole order in all the points at infty equat at most to pole_order. Threshold gives a bound on powers of x in the form.
If you suspect that you haven't found all the functions, you may increase it."""
from itertools import product
x_series = self.x
y_series = self.y
z_series = self.z
x_series = self.x_series
y_series = self.y_series
z_series = self.z_series
delta = self.nb_of_pts_at_infty
p = self.characteristic
n = self.height
@ -297,9 +302,9 @@ class as_cover:
def cohomology_of_structure_sheaf_basis(self, threshold = 8):
holo_diffs = self.holomorphic_differentials_basis(threshold = threshold)
from itertools import product
x_series = self.x
y_series = self.y
z_series = self.z
x_series = self.x_series
y_series = self.y_series
z_series = self.z_series
delta = self.nb_of_pts_at_infty
p = self.characteristic
n = self.height
@ -321,9 +326,7 @@ class as_cover:
for j in range(0, m):
for k in product(*pr):
f = as_function(self, prod(z[i1]^(k[i1]) for i1 in range(n))/x^i*y^j)
f_products = []
for omega in holo_diffs:
f_products += [sum((f*omega).residue(place = _) for _ in range(self.nb_of_pts_at_infty))]
f_products = [omega.serre_duality_pairing(f) for omega in holo_diffs]
if vector(f_products) not in S:
S = S+V.subspace([V(f_products)])
result_fcts += [f]
@ -334,10 +337,10 @@ class as_cover:
'''Given function fct, find form eta regular on affine part such that eta - d(fct) is regular in infty. (Works for one place at infty now)'''
print(fct)
from itertools import product
x_series = self.x
y_series = self.y
z_series = self.z
dx_series = self.dx
x_series = self.x_series
y_series = self.y_series
z_series = self.z_series
dx_series = self.dx_series
delta = self.nb_of_pts_at_infty
p = self.characteristic
n = self.height
@ -363,8 +366,8 @@ class as_cover:
raise ValueError("Increase threshold!")
for omega in forms:
for a in F:
if (a*omega - fct.diffn()).form in Rxyz:
return a*omega
if (a*omega + fct.diffn()).form in Rxyz:
return a*omega + fct.diffn()
raise ValueError("Unknown.")
def de_rham_basis(self, threshold = 30):

22
sage/as_covers/as_form_class.sage
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@ -19,10 +19,10 @@ class as_form:
C = self.curve
delta = C.nb_of_pts_at_infty
F = C.base_ring
x_series = C.x[i]
y_series = C.y[i]
z_series = C.z[i]
dx_series = C.dx[i]
x_series = C.x_series[i]
y_series = C.y_series[i]
z_series = C.z_series[i]
dx_series = C.dx_series[i]
n = C.height
variable_names = 'x, y'
for j in range(n):
@ -57,14 +57,7 @@ class as_form:
def group_action(self, ZN_tuple):
C = self.curve
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)
RxyzQ, Rxyz, x, y, z = C.fct_field
sub_list = {x : x, y : y} | {z[j] : z[j]+ZN_tuple[j] for j in range(n)}
g = self.form
return as_form(C, g.substitute(sub_list))
@ -80,6 +73,7 @@ class as_form:
denom = LCM([denominator(omega.form) for omega in basis])
basis = [denom*omega for omega in basis]
self_with_no_denominator = denom*self
print(self_with_no_denominator, basis)
return linear_representation_polynomials(Rxyz(self_with_no_denominator.form), [Rxyz(omega.form) for omega in basis])
def trace(self):
@ -110,6 +104,10 @@ class as_form:
def valuation(self, place=0):
return self.expansion_at_infty(i = 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 artin_schreier_transform(power_series, prec = 10):
"""Given a power_series, find correction such that power_series - (correction)^p +correction has valuation
-jump non divisible by p. Also, express t (the variable) in terms of the uniformizer at infty on the curve

6
sage/as_covers/as_function_class.sage
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@ -60,9 +60,9 @@ class as_function:
C = self.curve
delta = C.nb_of_pts_at_infty
F = C.base_ring
x_series = C.x[i]
y_series = C.y[i]
z_series = C.z[i]
x_series = C.x_series[i]
y_series = C.y_series[i]
z_series = C.z_series[i]
n = C.height
variable_names = 'x, y'
for j in range(n):