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nieudane proby coordinates dla cech_drw

This commit is contained in:
jgarnek 2023-03-08 13:54:07 +00:00
parent b79484be4f
commit 5f82d7db59
10 changed files with 14452 additions and 26766 deletions
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41001
sage/.run.term-0.term
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File diff suppressed because one or more lines are too long

54
sage/as_covers/as_cover_class.sage
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@ -1,5 +1,5 @@
class as_cover:
def __init__(self, C, list_of_fcts, prec = 10):
def __init__(self, C, list_of_fcts, branch_points = [], prec = 10):
self.quotient = C
self.functions = list_of_fcts
self.height = len(list_of_fcts)
@ -22,24 +22,25 @@ class as_cover:
r = f.degree()
delta = GCD(m, r)
self.nb_of_pts_at_infty = delta
self.branch_points = list(range(delta)) + branch_points
Rxy.<x, y> = PolynomialRing(F, 2)
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
all_x_series = []
all_y_series = []
all_z_series = []
all_dx_series = []
all_jumps = []
all_x_series = {}
all_y_series = {}
all_z_series = {}
all_dx_series = {}
all_jumps = {}
for i in range(delta):
x_series = superelliptic_function(C, x).expansion_at_infty(place = i, prec=prec)
y_series = superelliptic_function(C, y).expansion_at_infty(place = i, prec=prec)
for pt in self.branch_points:
x_series = superelliptic_function(C, x).expansion(pt=pt, prec=prec)
y_series = superelliptic_function(C, y).expansion(pt=pt, prec=prec)
z_series = []
jumps = []
n = len(list_of_fcts)
list_of_power_series = [g.expansion_at_infty(place = i, prec=prec) for g in list_of_fcts]
for i in range(n):
power_series = list_of_power_series[i]
list_of_power_series = [g.expansion(pt=pt, prec=prec) for g in list_of_fcts]
for j in range(n):
power_series = list_of_power_series[j]
jump, correction, t_old, z = artin_schreier_transform(power_series, prec = prec)
x_series = x_series(t = t_old)
y_series = y_series(t = t_old)
@ -48,11 +49,11 @@ class as_cover:
jumps += [jump]
list_of_power_series = [g(t = t_old) for g in list_of_power_series]
all_jumps += [jumps]
all_x_series += [x_series]
all_y_series += [y_series]
all_z_series += [z_series]
all_dx_series += [x_series.derivative()]
all_jumps[pt] = jumps
all_x_series[pt] = x_series
all_y_series[pt] = y_series
all_z_series[pt] = z_series
all_dx_series[pt] = x_series.derivative()
self.jumps = all_jumps
self.x_series = all_x_series
self.y_series = all_y_series
@ -70,8 +71,9 @@ class as_cover:
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.z = [as_function(self, z[j]) for j in range(n)]
self.dx = as_form(self, 1)
self.one = as_function(self, 1)
def __repr__(self):
@ -89,9 +91,9 @@ class as_cover:
jumps = self.jumps
gY = self.quotient.genus()
n = self.height
delta = self.nb_of_pts_at_infty
branch_pts = self.branch_points
p = self.characteristic
return p^n*gY + (p^n - 1)*(delta - 1) + sum(p^(n-j-1)*(jumps[i][j]-1)*(p-1)/2 for j in range(n) for i in range(delta))
return p^n*gY + (p^n - 1)*(len(branch_pts) - 1) + sum(p^(n-j-1)*(jumps[pt][j]-1)*(p-1)/2 for j in range(n) for pt in branch_pts)
def exponent_of_different(self, place = 0):
jumps = self.jumps
@ -130,17 +132,17 @@ class as_cover:
for j in range(0, m):
for k in product(*pr):
eta = as_form(self, x^i * prod(z[i1]^(k[i1]) for i1 in range(n))/y^j)
eta_exp = eta.expansion_at_infty()
eta_exp = eta.expansion(pt=self.branch_points[0])
S += [(eta, eta_exp)]
forms = holomorphic_combinations(S)
for i in range(1, delta):
forms = [(omega, omega.expansion_at_infty(place = i)) for omega in forms]
for pt in self.branch_points[1:]:
forms = [(omega, omega.expansion(pt=pt)) for omega in forms]
forms = holomorphic_combinations(forms)
if len(forms) < self.genus():
print("I haven't found all forms.")
print("I haven't found all forms, only ", len(forms), " of ", self.genus())
return holomorphic_differentials_basis(self, threshold = threshold + 1)
if len(forms) > self.genus():
print("Increase precision.")
@ -236,8 +238,8 @@ class as_cover:
forms = holomorphic_combinations_forms(S, pole_order)
for i in range(1, delta):
forms = [(omega, omega.expansion_at_infty(place = i)) for omega in forms]
for pt in self.branch_points[1:]:
forms = [(omega, omega.expansion(pt=pt)) for omega in forms]
forms = holomorphic_combinations_forms(forms, pole_order)
return forms

29
sage/as_covers/as_form_class.sage
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@ -37,6 +37,28 @@ class as_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
@ -152,6 +174,13 @@ def artin_schreier_transform(power_series, prec = 10):
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

22
sage/as_covers/as_function_class.sage
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@ -77,6 +77,28 @@ class as_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, ZN_tuple):
C = self.curve

18
sage/drafty/draft.sage
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@ -2,13 +2,15 @@ p = 3
m = 2
F = GF(p)
Rx.<x> = PolynomialRing(F)
f = x^3 - x + 1
f = x^3 - x
C = superelliptic(f, m)
C1 = patch(C)
#C1 = patch(C)
#print(C1.crystalline_cohomology_basis())
g1 = C1.polynomial
g_AS = g1(x^p - x)
C2 = superelliptic(g_AS, 2)
print(convert_super_into_AS(C2))
converted = (C2.x)^4 - (C2.x)^2
print(convert_super_fct_into_AS(converted))
#g1 = C1.polynomial
#g_AS = g1(x^p - x)
#C2 = superelliptic(g_AS, 2)
#print(convert_super_into_AS(C2))
#converted = (C2.x)^4 - (C2.x)^2
#print(convert_super_fct_into_AS(converted))
b = C.crystalline_cohomology_basis()
print(autom(b[0]).coordinates(basis = b))

47
sage/drafty/superelliptic_drw.sage
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@ -47,10 +47,11 @@ class superelliptic_witt:
return other_integer*self
def __mul__(self, other):
C = self.curve
p = C.characteristic
if isinstance(other, superelliptic_witt):
t1 = self.t
f1 = self.f
p = self.curve.characteristic
t2 = other.t
f2 = other.f
return superelliptic_witt(t1*t2, t1^p*f2 + t2^p*f1)
@ -58,7 +59,6 @@ class superelliptic_witt:
h1 = other.h1
h2 = other.h2
omega = other.omega
p = other.curve.characteristic
t = self.t
f = self.f
aux_form = t^p*omega - h2*t^(p-1)*t.diffn() + f*h1^p*(C.x)^(p-1)*C.dx
@ -130,7 +130,7 @@ superelliptic_function.teichmuller = teichmuller
#d[M] = a*[b x^(a-1)]
def auxilliary_derivative(P):
'''Return "derivative" of P, where P depends only on x. '''
'''Return "derivative" of P, where P depends only on x. In other words d[P(x)].'''
P0 = P.t.function
P1 = P.f.function
C = P.curve
@ -156,6 +156,11 @@ class superelliptic_drw_form:
self.omega = omega
self.h2 = h2
def r(self):
C = self.curve
h1 = self.h1
return superelliptic_form(C, h1.function)
def __eq__(self, other):
eq1 = (self.h1 == self.h1)
try:
@ -317,16 +322,29 @@ class superelliptic_drw_cech:
return superelliptic_cech(C, omega0.h1*C.dx, f.t)
def coordinates(self, basis = 0):
C = self.curve
g = C.genus()
coord_mod_p = self.r().coordinates()
print(coord_mod_p)
coord_lifted = [lift(a) for a in coord_mod_p]
if basis == 0:
basis = self.curve().crystalline_cohomology_basis()
basis = C.crystalline_cohomology_basis()
aux = self
for i, a in enumerate(basis):
aux -= coord_lifted[i]*a
aux = aux.reduce()
return aux
print('aux before reduce', aux)
#aux = aux.reduce() # Now aux = p*cech class.
h02 = aux.omega0.h2
h82 = aux.omega8.h2
aux -= superelliptic_drw_cech(h02.verschiebung().diffn(), (h02 - h82).verschiebung())
# Now aux should be of the form (V(smth), V(smth), V(smth))
print('aux V(smth)', aux)
aux_divided_by_p = superelliptic_cech(C, aux.omega0.omega.cartier(), aux.f.f.pth_root())
print('aux.omega0.omega.cartier()', aux.omega0.omega.cartier())
coord_aux_divided_by_p = aux_divided_by_p.coordinates()
coord_aux_divided_by_p = [ZZ(a) for a in coord_aux_divided_by_p]
coordinates = [ (coord_lifted[i] + p*coord_aux_divided_by_p[i])%p^2 for i in range(2*g)]
return coordinates
def de_rham_witt_lift(cech_class, prec = 50):
@ -335,19 +353,24 @@ def de_rham_witt_lift(cech_class, prec = 50):
omega0 = cech_class.omega0
omega8 = cech_class.omega8
fct = cech_class.f
omega0_regular = regular_form(omega0)
omega0_regular = regular_form(omega0) #Present omega0 in the form P dx + Q dy
print(omega0_regular)
omega0_lift = omega0_regular[0].teichmuller()*(C.x.teichmuller().diffn()) + omega0_regular[1].teichmuller()*(C.y.teichmuller().diffn())
omega8_regular = regular_form(second_patch(omega8))
#Now the obvious lift of omega0 = P dx + Q dy to de Rham-Witt is [P] d[x] + [Q] d[y]
omega8_regular = regular_form(second_patch(omega8)) # The same for omega8.
omega8_regular = (second_patch(omega8_regular[0]), second_patch(omega8_regular[1]))
u = (C.x)^(-1)
v = (C.y)/(C.x)^(g+1)
omega8_lift = omega0_regular[0].teichmuller()*(u.teichmuller().diffn()) + omega0_regular[1].teichmuller()*(v.teichmuller().diffn())
aux = omega0_lift - omega8_lift - fct.teichmuller().diffn()
decom_aux_h2 = decomposition_g0_g8(aux.h2, prec=prec)
omega8_lift = omega8_regular[0].teichmuller()*(u.teichmuller().diffn()) + omega8_regular[1].teichmuller()*(v.teichmuller().diffn())
print('omega8_lift.frobenius().expansion_at_infty()', omega8_lift.frobenius().expansion_at_infty())
aux = omega0_lift - omega8_lift - fct.teichmuller().diffn() # now aux is of the form (V(smth) + dV(smth), V(smth))
if aux.h1.function != 0:
raise ValueError('Something went wrong - aux is not of the form (V(smth) + dV(smth), V(smth)).')
decom_aux_h2 = decomposition_g0_g8(aux.h2, prec=prec) #decompose dV(smth) in aux as smth regular on U0 - smth regular on U8.
aux_h2 = decom_aux_h2[0]
aux_f = decom_aux_h2[2]
aux_omega0 = decomposition_omega0_omega8(aux.omega, prec=prec)[0]
result = superelliptic_drw_cech(omega0_lift + aux_h2.verschiebung().diffn() + aux_omega0.verschiebung(), fct.teichmuller() + aux_f.verschiebung())
result = superelliptic_drw_cech(omega0_lift - aux_h2.verschiebung().diffn() - aux_omega0.verschiebung(), fct.teichmuller() + aux_f.verschiebung())
return result.reduce()
def crystalline_cohomology_basis(self, prec = 50):

3
sage/init.sage
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@ -25,6 +25,7 @@ load('drafty/regular_on_U0.sage')
load('drafty/lift_to_de_rham.sage')
#load('drafty/superelliptic_cohomology_class.sage')
load('drafty/superelliptic_drw.sage')
load('drafty/draft.sage')
#load('drafty/draft_klein_covers.sage')
load('drafty/2gpcovers.sage')
#load('drafty/2gpcovers.sage')
load('drafty/pole_numbers.sage')

7
sage/superelliptic/decomposition_into_g0_g8.sage
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@ -1,5 +1,6 @@
def decomposition_g0_g8(fct, prec = 50):
'''Writes fct as a difference g0 - g8, with g0 regular on the affine patch and g8 at the points in infinity.'''
'''Writes fct as a difference g0 - g8 + f, with g0 regular on the affine patch and g8 at the points in infinity
and f is combination of basis of H^1(X, OX). Output is (g0, g8, f).'''
C = fct.curve
g = C.genus()
coord = fct.coordinates()
@ -7,8 +8,6 @@ def decomposition_g0_g8(fct, prec = 50):
for i, a in enumerate(C.cohomology_of_structure_sheaf_basis()):
nontrivial_part += coord[i]*a
fct -= nontrivial_part
if fct.coordinates(prec=prec) != g*[0]:
raise ValueError("The given function cannot be written as g0 - g8.")
Fxy, Rxy, x, y = C.fct_field
fct = Fxy(fct.function)
@ -20,7 +19,7 @@ def decomposition_g0_g8(fct, prec = 50):
for monomial in num.monomials():
aux = superelliptic_function(C, monomial)
if aux.expansion_at_infty().valuation() >= aux_den.expansion_at_infty().valuation():
g8 += num.monomial_coefficient(monomial)*aux/aux_den
g8 -= num.monomial_coefficient(monomial)*aux/aux_den
else:
g0 += num.monomial_coefficient(monomial)*aux/aux_den
return (g0, g8, nontrivial_part)

13
sage/superelliptic/superelliptic_form_class.sage
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@ -100,6 +100,8 @@ class superelliptic_form:
FxRy.<y> = PolynomialRing(Fx)
g = reduction(C, y^m*g)
g = FxRy(g)
if j == 0:
return g.monomial_coefficient(y^(0))/C.polynomial
return g.monomial_coefficient(y^(m-j))
def is_regular_on_U0(self):
@ -107,7 +109,7 @@ class superelliptic_form:
F = C.base_ring
m = C.exponent
Rx.<x> = PolynomialRing(F)
for j in range(1, m):
for j in range(0, m):
if self.jth_component(j) not in Rx:
return 0
return 1
@ -138,6 +140,15 @@ class superelliptic_form:
dx_series = x_series.derivative()
return g*dx_series
def expansion(self, pt, prec = 50):
'''Expansion in the completed ring of the point pt. If pt is an integer, it means the corresponding place at infinity.'''
if pt in ZZ:
return self.expansion_at_infty(place=pt, prec=prec)
C = self.curve
dx_series = C.x.expansion(pt = pt, prec=prec).derivative()
aux_fct = superelliptic_function(C, self.form)
return aux_fct.expansion(pt=pt, prec=prec)*dx_series
def residue(self, place = 0, prec=30):
return self.expansion_at_infty(place = place, prec=prec)[-1]

24
sage/superelliptic/superelliptic_function_class.sage
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@ -135,6 +135,30 @@ class superelliptic_function:
xx = Rt(1/(t^M*ww^b))
yy = 1/(t^R*ww^a)
return Rt(fct(x = Rt(xx), y = Rt(yy)))
def expansion(self, pt, prec = 50):
'''Expansion in the completed ring of the point pt. If pt is an integer, it means the corresponding place at infinity.'''
if pt in ZZ:
return self.expansion_at_infty(place=pt, prec=prec)
x0, y0 = pt[0], pt[1]
C = self.curve
f = C.polynomial
F = C.base_ring
m = C.exponent
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
Rxy.<x, y> = PolynomialRing(F, 2)
Fxy = FractionField(Rxy)
if y0 !=0 and f.derivative()(x0) != 0:
y_series = f(x = t + x0).nth_root(m)
return Rt(self.function(x = t + x0, y = y_series))
if f.derivative()(x0) == 0: # then x - x0 is a uniformizer
y_series = Rt(f(x = t+x0).nth_root(m))
return Rt(self.function(x = t + x0, y = y_series))
if y0 == 0: #then y is a uniformizer
f1 = f(x = x+x0) - y0
x_series = new_reverse(f1(x = t), prec = prec)
x_series = x_series(t = t^m - y0) + x0
return self.function(x = x_series, y = t)
def pth_root(self):
'''Compute p-th root of given function. This uses the following fact: if h = H^p, then C(h*dx/x) = H*dx/x.'''