as transform for unramified points fixed

This commit is contained in:
jgarnek 2024-02-10 12:41:04 +00:00
parent 9585d6995b
commit 780c1f03f2
7 changed files with 131 additions and 138 deletions

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@ -176,49 +176,6 @@ class as_form:
return True return True
return False return False
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
z^p - z = power_series, where z = 1/t_new^(jump) and express z in terms of the new uniformizer."""
correction = 0
F = power_series.parent().base()
p = F.characteristic()
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
RtQ = FractionField(Rt)
power_series = RtQ(power_series)
if power_series.valuation() == +Infinity:
raise ValueError("Precision is too low.")
if power_series.valuation() >= 0:
# THIS IS WRONG - THERE ARE SEVERAL PLACES OVER THIS PLACE, AND IT DEPENDS
aux = t^p - t
z = new_reverse(aux, prec = prec)
z = z(t = power_series)
return(0, 0, t, z)
while(power_series.valuation() % p == 0 and power_series.valuation() < 0):
M = -power_series.valuation()/p
coeff = power_series.list()[0] #wspolczynnik a_(-p) w f_AS
correction += coeff.nth_root(p)*t^(-M)
power_series = power_series - (coeff*t^(-p*M) - coeff.nth_root(p)*t^(-M))
jump = max(-(power_series.valuation()), 0)
try:
if jump != 0:
T = nth_root2((power_series)^(-1), jump, prec=prec) #T is defined by power_series = 1/T^m
except:
print("no ", str(jump), "-th root; divide by", power_series.list()[0])
return (jump, power_series.list()[0])
if jump != 0:
T_rev = new_reverse(T, prec = prec)
t_old = T_rev(t^p/nth_root2(1 - t^((p-1)*jump), jump, prec=prec))
z = 1/t^(jump) + Rt(correction)(t = t_old)
return(jump, correction, t_old, z)
if jump == 0:
aux = t^p - t
z = new_reverse(aux, prec = prec)
z = z(t = power_series)
return(0, correction, t, z)
def are_forms_linearly_dependent(set_of_forms): def are_forms_linearly_dependent(set_of_forms):
from sage.rings.polynomial.toy_variety import is_linearly_dependent from sage.rings.polynomial.toy_variety import is_linearly_dependent
C = set_of_forms[0].curve C = set_of_forms[0].curve

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@ -0,0 +1,45 @@
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
z^p - z = power_series, where z = 1/t_new^(jump) and express z in terms of the new uniformizer."""
print('as transform for ', power_series)
correction = 0
F = power_series.parent().base()
p = F.characteristic()
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
RtQ = FractionField(Rt)
power_series = RtQ(power_series)
if power_series.valuation() == +Infinity:
raise ValueError("Precision is too low.")
if power_series.valuation() >= 0:
# THIS IS WRONG - THERE ARE SEVERAL PLACES OVER THIS PLACE, AND IT DEPENDS
aux = t^p - t
z = new_reverse(aux, prec = prec)
z = z(t = power_series)
print("a")
return(0, 0, t, z)
while(power_series.valuation() % p == 0 and power_series.valuation() < 0):
M = -power_series.valuation()/p
coeff = power_series.list()[0] #wspolczynnik a_(-p) w f_AS
correction += coeff.nth_root(p)*t^(-M)
power_series = power_series - (coeff*t^(-p*M) - coeff.nth_root(p)*t^(-M))
jump = max(-(power_series.valuation()), 0)
try:
if jump != 0:
T = nth_root2((power_series)^(-1), jump, prec=prec) #T is defined by power_series = 1/T^m
except:
print("no ", str(jump), "-th root; divide by", power_series.list()[0])
return (jump, power_series.list()[0])
if jump != 0:
T_rev = new_reverse(T, prec = prec)
t_old = T_rev(t^p/nth_root2(1 - t^((p-1)*jump), jump, prec=prec))
z = 1/t^(jump) + Rt(correction)(t = t_old)
return(jump, correction, t_old, z)
if jump == 0:
aux = t^p - t
z = new_reverse(aux, prec = prec)
z = z(t = power_series)
z = z + correction
print('corr', correction, 'ps', power_series, 'z', z)
return(0, correction, t, z)

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@ -1,12 +1,13 @@
p = 3 p = 3
F = GF(3^2, 'a') F = GF(3)
#F.<a> = GF(3^2)
Rx.<x> = PolynomialRing(F) Rx.<x> = PolynomialRing(F)
P1 = superelliptic(x^2 + 1, 2) P1 = superelliptic(x^2 + 1, 2)
fct1 = (P1.x)^2 fct1 = (P1.x)^2
fct2 = fct1 + (P1.x)/(P1.y - P1.x) fct2 = (fct1 + P1.one/(P1.y - P1.x))
fct3 = (P1.x)^4 fct3 = 0*P1.x
C = heisenberg_cover(P1, [1/2*fct1, fct2, fct3], prec=500) C = heisenberg_cover(P1, [fct1, fct2, fct3], prec=200)
print(C) print(C, '\n', C.genus(), '\n', C.jumps)
B = C.holomorphic_differentials_basis() #B = C.holomorphic_differentials_basis()
#print("Computed basis") #print("Computed basis")
#a1, b1, c1 = heisenberg_group_action_matrices_holo(C, basis = B) #a1, b1, c1 = heisenberg_group_action_matrices_holo(C, basis = B)

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@ -194,9 +194,11 @@ class heisenberg_cover:
forms = holomorphic_combinations_fcts(S, pole_order) forms = holomorphic_combinations_fcts(S, pole_order)
for i in range(1, delta): for i in range(delta):
forms = [(omega, omega.expansion_at_infty(place = i)) for omega in forms] for g in [(0, i, 0) for i in range(p)]:
forms = holomorphic_combinations_fcts(forms, pole_order) if i!=0 or g != (0, 0, 0):
forms = [(omega, omega.expansion_at_infty(place = i)) for omega in forms]
forms = holomorphic_combinations_fcts(forms, pole_order)
return forms return forms
@ -217,7 +219,7 @@ class heisenberg_cover:
return result return result
def at_most_poles_forms(self, pole_order, threshold = 8): def at_most_poles_forms(self, pole_order, threshold = 8):
"""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. """Find forms with pole order in all the points at infty equal 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.""" If you suspect that you haven't found all the functions, you may increase it."""
from itertools import product from itertools import product
x_series = self.x_series x_series = self.x_series
@ -405,6 +407,75 @@ class heisenberg_cover:
result += [heisenberg_cech(self, omega, f)] result += [heisenberg_cech(self, omega, f)]
return result return result
def stabilizer(self, place = 0):
result = []
for g in self.group:
flag = 1
for i in range(self.height):
if self.z[i].valuation(place = place) > 0:
fct = self.z[i]
elif self.z[i].valuation(place = place) < 0:
fct = self.one/self.z[i]
if fct.group_action(g).valuation(place = place) <= 0:
flag = 0
if flag:
result += [g]
return result
def stabilizer_coset_reps(self, place = 0):
result = [(0, 0, 0)]
p = self.characteristic
H = self.stabilizer(place = place)
for g in self.group:
flag = 1
for v in result:
if heisenberg_mult(g, heisenberg_inv(v, p), p) in H:
flag = 0
if flag:
result += [g]
return result
##############
def at_most_poles2(self, pole_orders, threshold = 8):
""" Find fcts with pole order in infty's at most pole_order from the given dictionary. The keys of the dictionary are pairs (place_at_infty, group element). The items are the poles orders at those places.
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_series
y_series = self.y_series
z_series = self.z_series
delta = self.nb_of_pts_at_infty
p = self.characteristic
n = self.height
prec = self.prec
C = self.quotient
F = self.base_ring
m = C.exponent
r = C.polynomial.degree()
RxyzQ, Rxyz, x, y, z = self.fct_field
F = C.base_ring
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
#Tworzymy zbiór S form z^i x^j y^k dx/y o waluacji >= waluacja z^(p-1)*dx/y
S = []
RQxyz = FractionField(Rxyz)
pr = [list(GF(p)) for _ in range(n)]
for i in range(0, threshold*r):
for j in range(0, m):
for k in product(*pr):
eta = heisenberg_function(self, x^i * prod(z[i1]^(k[i1]) for i1 in range(n))*y^j)
eta_exp = eta.expansion_at_infty()
S += [(eta, eta_exp)]
forms = holomorphic_combinations_fcts(S, pole_orders[(0, (0, 0, 0))])
for i in range(delta):
for g in self.stabilizer_coset_reps(place = i):
if i!=0 or g != (0, 0, 0):
forms = [(omega, omega.expansion_at_infty(place = i)) for omega in forms]
forms = holomorphic_combinations_fcts(forms, pole_orders[(i, g)])
return forms
##############
def heisenberg_holomorphic_combinations(S): def heisenberg_holomorphic_combinations(S):
"""Given a list S of pairs (form, corresponding Laurent series at some pt), find their combinations holomorphic at that pt.""" """Given a list S of pairs (form, corresponding Laurent series at some pt), find their combinations holomorphic at that pt."""
C_AS = S[0][0].curve C_AS = S[0][0].curve

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@ -201,48 +201,6 @@ class heisenberg_form:
return True return True
return False return False
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
z^p - z = power_series, where z = 1/t_new^(jump) and express z in terms of the new uniformizer."""
correction = 0
F = power_series.parent().base()
p = F.characteristic()
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
RtQ = FractionField(Rt)
power_series = RtQ(power_series)
if power_series.valuation() == +Infinity:
raise ValueError("Precision is too low.")
if power_series.valuation() >= 0:
# THIS IS WRONG - THERE ARE SEVERAL PLACES OVER THIS PLACE, AND IT DEPENDS
aux = t^p - t
z = new_reverse(aux, prec = prec)
z = z(t = power_series)
return(0, 0, t, z)
while(power_series.valuation() % p == 0 and power_series.valuation() < 0):
M = -power_series.valuation()/p
coeff = power_series.list()[0] #wspolczynnik a_(-p) w f_AS
correction += coeff.nth_root(p)*t^(-M)
power_series = power_series - (coeff*t^(-p*M) - coeff.nth_root(p)*t^(-M))
jump = max(-(power_series.valuation()), 0)
try:
if jump != 0:
T = nth_root2((power_series)^(-1), jump, prec=prec) #T is defined by power_series = 1/T^m
except:
print("no ", str(jump), "-th root; divide by", power_series.list()[0])
return (jump, power_series.list()[0])
if jump != 0:
T_rev = new_reverse(T, prec = prec)
t_old = T_rev(t^p/nth_root2(1 - t^((p-1)*jump), jump, prec=prec))
z = 1/t^(jump) + Rt(correction)(t = t_old)
return(jump, correction, t_old, z)
if jump == 0:
aux = t^p - t
z = new_reverse(aux, prec = prec)
z = z(t = power_series)
return(0, correction, t, z)
def are_forms_linearly_dependent(set_of_forms): def are_forms_linearly_dependent(set_of_forms):
from sage.rings.polynomial.toy_variety import is_linearly_dependent from sage.rings.polynomial.toy_variety import is_linearly_dependent
C = set_of_forms[0].curve C = set_of_forms[0].curve

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@ -3,6 +3,7 @@ load('superelliptic/superelliptic_function_class.sage')
load('superelliptic/superelliptic_form_class.sage') load('superelliptic/superelliptic_form_class.sage')
load('superelliptic/superelliptic_cech_class.sage') load('superelliptic/superelliptic_cech_class.sage')
load('superelliptic/frobenius_kernel.sage') load('superelliptic/frobenius_kernel.sage')
load('as_covers/as_transform.sage')
load('as_covers/as_cover_class.sage') load('as_covers/as_cover_class.sage')
load('as_covers/as_function_class.sage') load('as_covers/as_function_class.sage')
load('as_covers/as_form_class.sage') load('as_covers/as_form_class.sage')
@ -43,6 +44,9 @@ load('heisenberg_covers/heisenberg_form_class.sage')
load('heisenberg_covers/heisenberg_polyforms.sage') load('heisenberg_covers/heisenberg_polyforms.sage')
load('heisenberg_covers/heisenberg_reduction.sage') load('heisenberg_covers/heisenberg_reduction.sage')
load('heisenberg_covers/heisenberg_group_action_matrices.sage') load('heisenberg_covers/heisenberg_group_action_matrices.sage')
load('heisenberg_covers/dual_element.sage')
load('heisenberg_covers/ith_magical_component.sage')
load('heisenberg_covers/heisenberg_group.sage')
############## ##############
############## ##############
def init(lista, tests = False, init=True): def init(lista, tests = False, init=True):

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@ -195,49 +195,6 @@ class quaternion_form:
return True return True
return False return False
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
z^p - z = power_series, where z = 1/t_new^(jump) and express z in terms of the new uniformizer."""
correction = 0
F = power_series.parent().base()
p = F.characteristic()
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
RtQ = FractionField(Rt)
power_series = RtQ(power_series)
if power_series.valuation() == +Infinity:
raise ValueError("Precision is too low.")
if power_series.valuation() >= 0:
# THIS IS WRONG - THERE ARE SEVERAL PLACES OVER THIS PLACE, AND IT DEPENDS
aux = t^p - t
z = new_reverse(aux, prec = prec)
z = z(t = power_series)
return(0, 0, t, z)
while(power_series.valuation() % p == 0 and power_series.valuation() < 0):
M = -power_series.valuation()/p
coeff = power_series.list()[0] #wspolczynnik a_(-p) w f_AS
correction += coeff.nth_root(p)*t^(-M)
power_series = power_series - (coeff*t^(-p*M) - coeff.nth_root(p)*t^(-M))
jump = max(-(power_series.valuation()), 0)
try:
if jump != 0:
T = nth_root2((power_series)^(-1), jump, prec=prec) #T is defined by power_series = 1/T^m
except:
print("no ", str(jump), "-th root; divide by", power_series.list()[0])
return (jump, power_series.list()[0])
if jump != 0:
T_rev = new_reverse(T, prec = prec)
t_old = T_rev(t^p/nth_root2(1 - t^((p-1)*jump), jump, prec=prec))
z = 1/t^(jump) + Rt(correction)(t = t_old)
return(jump, correction, t_old, z)
if jump == 0:
aux = t^p - t
z = new_reverse(aux, prec = prec)
z = z(t = power_series)
return(0, correction, t, z)
def are_forms_linearly_dependent(set_of_forms): def are_forms_linearly_dependent(set_of_forms):
from sage.rings.polynomial.toy_variety import is_linearly_dependent from sage.rings.polynomial.toy_variety import is_linearly_dependent
C = set_of_forms[0].curve C = set_of_forms[0].curve