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superelliptic drw podzielone na wiecej plikow

This commit is contained in:
jgarnek 2023-03-09 08:54:01 +00:00
parent b03c679f7d
commit ce0ac0dcec
8 changed files with 442 additions and 421 deletions
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22
sage/.run.term-0.term
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@ -34630,4 +34630,24 @@ omega8_regular 2 (0, 2/x)
V(((-x^18 - x^16 - x^14 - x^10 - x^8 - x^4 - x^2 - 1)/(x^10*y - x^8*y)) dx) + dV([((x^10 + x^8 + x^6 + 2*x^4 + 2*x^2 + 2)/x^6)*y])
[?2004h[?25l[?7lsage: [?7h[?12l[?25h[?25l[?7l[?7h[?12l[?25h[?25l[?7lq[?7h[?12l[?25h[?25l[?7lu[?7h[?12l[?25h[?25l[?7li[?7h[?12l[?25h[?25l[?7lt[?7h[?12l[?25h[?25l[?7l([?7h[?12l[?25h[?25l[?7l()[?7h[?12l[?25h[?25l[?7lsage: quit()
[?7h[?12l[?25h[?2004l]0;~/Research/2021 De Rham/DeRhamComputation/sage~/Research/2021 De Rham/DeRhamComputation/sage$ cd ..
]0;~/Research/2021 De Rham/DeRhamComputation~/Research/2021 De Rham/DeRhamComputation$ gicd ..sagegit pushcommit -m "Poprawiony i sprawdzony dy_w oraz autom." add -u
]0;~/Research/2021 De Rham/DeRhamComputation~/Research/2021 De Rham/DeRhamComputation$ gicd ..sagegit pushcommit -m "Poprawiony i sprawdzony dy_w oraz autom." add -udra -u
]0;~/Research/2021 De Rham/DeRhamComputation~/Research/2021 De Rham/DeRhamComputation$ git git commit -m "przeniesione pliki"
[master 7bcfa5d] przeniesione pliki
6 files changed, 208 insertions(+), 516 deletions(-)
delete mode 100644 sage/drafty/second_patch.sage
delete mode 100644 sage/drafty/superelliptic_drw.sage
delete mode 100644 sage/superelliptic/decomposition_into_g0_g8.sage
]0;~/Research/2021 De Rham/DeRhamComputation~/Research/2021 De Rham/DeRhamComputation$ git add sage/drafty/superelliptic_drw/superelliptic_drw.sage
]0;~/Research/2021 De Rham/DeRhamComputation~/Research/2021 De Rham/DeRhamComputation$ git add sage/superelliptic_drw/superelliptic_drw.sageencond_patch.sage sage/superelliptic_drw/decomposition_into_g0_g8.sage
]0;~/Research/2021 De Rham/DeRhamComputation~/Research/2021 De Rham/DeRhamComputation$ git add sage/superelliptic_drw/second_patch.sage sage/superelliptic_drw/decomposition_into_g0_g8.sage uperelliptic_drw.sage commit -m "przeniesione pliki" "2"
[master b03c679] przeniesione pliki 2
3 files changed, 515 insertions(+)
create mode 100644 sage/superelliptic_drw/decomposition_into_g0_g8.sage
create mode 100644 sage/superelliptic_drw/second_patch.sage
create mode 100644 sage/superelliptic_drw/superelliptic_drw.sage
]0;~/Research/2021 De Rham/DeRhamComputation~/Research/2021 De Rham/DeRhamComputation$ git commit -m "przeniesione pliki 2"git add sage/superelliptic_drw/superelliptic_witt.sage sage/superelliptic_drw/superelliptic_drw_cech.sage sage/superelliptic_drw/superelliptic_drw_
superelliptic_drw_cech.sage superelliptic_drw_form.sage superelliptic_drw_tests/
]0;~/Research/2021 De Rham/DeRhamComputation~/Research/2021 De Rham/DeRhamComputation$ git add sage/superelliptic_drw/superelliptic_witt.sage sage/superelliptic_drw/superelliptic_drw_cech.sage sage/superelliptic_drw/superelliptic_drw_
superelliptic_drw_cech.sage superelliptic_drw_form.sage superelliptic_drw_tests/
]0;~/Research/2021 De Rham/DeRhamComputation~/Research/2021 De Rham/DeRhamComputation$ git add sage/superelliptic_drw/superelliptic_witt.sage sage/superelliptic_drw/superelliptic_drw_cech.sage sage/superelliptic_drw/superelliptic_drw_form.sage sage/superelliptic_drw/de_rham_witt_lift.sage sage/superelliptic_drw/automorphism.sage
]0;~/Research/2021 De Rham/DeRhamComputation~/Research/2021 De Rham/DeRhamComputation$ g

6
sage/init.sage
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@ -15,7 +15,11 @@ load('as_covers/combination_components.sage')
load('as_covers/group_action_matrices.sage')
load('superelliptic_drw/second_patch.sage')
load('superelliptic_drw/decomposition_into_g0_g8.sage')
load('superelliptic_drw/superelliptic_drw.sage')
load('superelliptic_drw/superelliptic_witt.sage')
load('superelliptic_drw/superelliptic_drw_form.sage')
load('superelliptic_drw/superelliptic_drw_cech.sage')
load('superelliptic_drw/de_rham_witt_lift.sage')
load('superelliptic_drw/automorphism.sage')
load('auxilliaries/reverse.sage')
load('auxilliaries/hensel.sage')
load('auxilliaries/linear_combination_polynomials.sage')

23
sage/superelliptic_drw/automorphism.sage Normal file
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@ -0,0 +1,23 @@
def autom(self):
'''Given a fct/form/cocycle self (Witt or usual), return phi(self), where phi(x) = x+1. Destined
for curves y^2 = f(x^p - x).'''
C = self.curve
F = C.base_ring
Rxy.<x, y> = PolynomialRing(F, 2)
Fxy = FractionField(Rxy)
if isinstance(self, superelliptic_function):
result = superelliptic_function(C, Fxy(self.function).subs({x:x+1, y:y}))
return result
if isinstance(self, superelliptic_form):
result = superelliptic_form(C, Fxy(self.form).subs({x:x+1, y:y}))
return result
if isinstance(self, superelliptic_witt):
result = superelliptic_witt(autom(self.t), autom(self.f))
return result
if isinstance(self, superelliptic_drw_form):
result = superelliptic_drw_form(0*C.x, autom(self.omega), autom(self.h2))
result += autom(self.h1).teichmuller()*(C.x + C.one).teichmuller().diffn()
return result
if isinstance(self, superelliptic_drw_cech):
result = superelliptic_drw_cech(autom(self.omega0), autom(self.f))
return result

36
sage/superelliptic_drw/de_rham_witt_lift.sage Normal file
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@ -0,0 +1,36 @@
def de_rham_witt_lift(cech_class, prec = 50):
C = cech_class.curve
g = C.genus()
omega0 = cech_class.omega0
omega8 = cech_class.omega8
fct = cech_class.f
omega0_regular = regular_form(omega0) #Present omega0 in the form P dx + Q dy
print('omega0_regular', omega0_regular)
omega0_lift = omega0_regular[0].teichmuller()*(C.x.teichmuller().diffn()) + omega0_regular[1].teichmuller()*(C.y.teichmuller().diffn())
#Now the obvious lift of omega0 = P dx + Q dy to de Rham-Witt is [P] d[x] + [Q] d[y]
print('omega8', omega8, 'second_patch(omega8)', second_patch(omega8))
omega8_regular = regular_form(second_patch(omega8)) # The same for omega8.
print('omega8_regular 1', omega8_regular)
omega8_regular = (second_patch(omega8_regular[0]), second_patch(omega8_regular[1]))
print('omega8_regular 2', omega8_regular)
u = (C.x)^(-1)
v = (C.y)/(C.x)^(g+1)
omega8_lift = omega8_regular[0].teichmuller()*(u.teichmuller().diffn()) + omega8_regular[1].teichmuller()*(v.teichmuller().diffn())
aux = omega0_lift - omega8_lift - fct.teichmuller().diffn() # now aux is of the form (V(smth) + dV(smth), V(smth))
return aux
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())
return result.reduce()
def crystalline_cohomology_basis(self, prec = 50):
result = []
for a in self.de_rham_basis():
result += [de_rham_witt_lift(a, prec = prec)]
return result
superelliptic.crystalline_cohomology_basis = crystalline_cohomology_basis

419
sage/superelliptic_drw/superelliptic_drw.sage
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@ -1,419 +0,0 @@
class superelliptic_witt:
def __init__(self, t, f):
''' Define Witt function on C of the form [t] + V(f). '''
self.curve = t.curve
C = t.curve
p = C.characteristic
self.t = t #superelliptic_function
self.f = f #superelliptic_function
def __repr__(self):
f = self.f
t = self.t
if f.function == 0:
return "[" + str(t) + "]"
if t.function == 0:
return "V(" + str(f) + ")"
return "[" + str(t) + "] + V(" + str(f) + ")"
def __neg__(self):
f = self.f
t = self.t
return superelliptic_witt(-t, -f)
def __add__(self, other):
C = self.curve
second_coor = 0*C.x
X = self.t
Y = other.t
for i in range(1, p):
second_coor -= binomial_prim(p, i)*X^i*Y^(p-i)
return superelliptic_witt(self.t + other.t, self.f + other.f + second_coor)
def __sub__(self, other):
return self + (-other)
def __rmul__(self, other):
p = self.curve.characteristic
if other in ZZ:
if other == 0:
return superelliptic_witt(0*C.x, 0*C.x)
if other > 0:
return self + (other-1)*self
if other < 0:
return (-other)*(-self)
if other in QQ:
other_integer = Integers(p^2)(other)
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
t2 = other.t
f2 = other.f
return superelliptic_witt(t1*t2, t1^p*f2 + t2^p*f1)
if isinstance(other, superelliptic_drw_form):
h1 = other.h1
h2 = other.h2
omega = other.omega
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
return superelliptic_drw_form(t*h1, aux_form, t^p*h2)
def __eq__(self, other):
return self.t == other.t and self.f == other.f
def diffn(self, dy_w = 0):
if dy_w == 0:
dy_w = self.curve.dy_w()
C = self.curve
t = self.t
f = self.f
fC = C.polynomial
F = C.base_ring
Rxy.<x, y> = PolynomialRing(F, 2)
if t.function == 0:
return superelliptic_drw_form(0*C.x, 0*C.dx, f)
t_polynomial = t.function
num = t_polynomial.numerator()
den = t_polynomial.denominator()
num_t_fct = superelliptic_function(C, num)
den_t_fct = superelliptic_function(C, den)
inv_den_t_fct = superelliptic_function(C, 1/den)
if den != 1:
# d([N/D] + V(f)) = [1/D]*d([N]) - [N]*[D^(-2)]*d([D]) + dV(f)
return ((den_t_fct)^(-1)).teichmuller()*num_t_fct.teichmuller().diffn() - ((den_t_fct)^(-2)).teichmuller()*num_t_fct.teichmuller()*den_t_fct.teichmuller().diffn() + superelliptic_drw_form(0*C.x, 0*C.dx, f)
t_polynomial = Rxy(t_polynomial)
M = t_polynomial.monomials()[0]
a = t_polynomial.monomial_coefficient(M)
#[P] = [aM] + Q, where Q = ([P] - [aM])
aM_fct = superelliptic_function(C, a*M)
Q = self - aM_fct.teichmuller()
exp_x = M.exponents()[0][0]
exp_y = M.exponents()[0][1]
return Q.diffn() + exp_x*superelliptic_drw_form(aM_fct/C.x, 0*C.dx, 0*C.x) + exp_y*(aM_fct/C.y).teichmuller()*dy_w
def binomial_prim(p, i):
return binomial(p, i)/p
def reduce_rational_fct(fct, p):
Rxy.<x, y> = PolynomialRing(QQ)
Fxy = FractionField(Rxy)
fct = Fxy(fct)
num = Rxy(fct.numerator())
den = Rxy(fct.denominator())
num1 = Rxy(0)
for m in num.monomials():
a = num.monomial_coefficient(m)
num1 += (a%p^2)*m
den1 = Rxy(0)
for m in den.monomials():
a = den.monomial_coefficient(m)
den1 += (a%p^2)*m
return num1/den1
def teichmuller(fct):
C = fct.curve
return superelliptic_witt(fct, 0*C.x)
superelliptic_function.teichmuller = teichmuller
#dy = [f(x)]'/2*y dx
#[f1 + M] = [f1] + [M] + V(cos)
#d[f1 + M] = d[f1] + d[M] + dV(f1*M)
#M = b x^a
#d[M] = a*[b x^(a-1)]
def auxilliary_derivative(P):
'''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
F = C.base_ring
Rx.<x> = PolynomialRing(F)
P0 = Rx(P0)
P1 = Rx(P1)
if P0 == 0:
return superelliptic_drw_form(0*C.x, 0*C.dx, P.f)
M = P0.monomials()[0]
a = P0.monomial_coefficient(M)
#[P] = [aM] + Q, where Q = ([P] - [aM])
aM_fct = superelliptic_function(C, a*M)
Q = P - aM_fct.teichmuller()
exp = M.exponents()[0]
return auxilliary_derivative(Q) + exp*superelliptic_drw_form(aM_fct/C.x, 0*C.dx, 0*C.x)
class superelliptic_drw_form:
def __init__(self, h1, omega, h2):
'''Form [h1] d[x] + V(omega) + dV([h])'''
self.curve = h1.curve
self.h1 = h1
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:
H = (self.h2 - other.h2).pth_root()
except:
return False
eq2 = ((self.omega - other.omega).cartier() - H.diffn()) == 0*self.curve.dx
if eq1 and eq2:
return True
return False
def __repr__(self):
C = self.curve
h1 = self.h1
omega = self.omega
h2 = self.h2
result = ""
if h1.function != 0:
result += "[" + str(h1) + "] d[x]"
if h1.function !=0 and omega.form != 0:
result += " + "
if omega.form != 0:
result += "V(" + str(omega) + ")"
if h2.function !=0 and omega.form != 0:
result += " + "
if h2.function != 0:
result += "dV([" + str(h2) +"])"
if result == "":
result += "0"
return result
def __rmul__(self, other):
h1 = self.h1
h2 = self.h2
omega = self.omega
p = self.curve.characteristic
if other in ZZ:
if other == 0:
return superelliptic_drw_form(0*C.x, 0*C.dx, 0*C.x)
if other > 0:
return self + (other-1)*self
if other < 0:
return (-other)*(-self)
if other in QQ:
other_integer = Integers(p^2)(other)
return other_integer*self
t = other.t
f = other.f
aux_form = t^p*omega - h2*t^(p-1)*t.diffn() + f*h1^p*(C.x)^(p-1)*C.dx
return superelliptic_drw_form(t*h1, aux_form, t^p*h2)
def __neg__(self):
C = self.curve
h1 = self.h1
h2 = self.h2
omega = self.omega
return superelliptic_drw_form(-h1, -omega, -h2)
def __sub__(self, other):
return self + (-other)
def __add__(self, other):
C = self.curve
h1 = self.h1
h2 = self.h2
omega = self.omega
H1 = other.h1
H2 = other.h2
OMEGA = other.omega
aux = (teichmuller(h1) + teichmuller(H1))*superelliptic_drw_form(C.one, 0*C.dx, 0*C.x)
h1_aux = aux.h1
h2_aux = aux.h2
omega_aux = aux.omega
return superelliptic_drw_form(h1_aux, omega + OMEGA + omega_aux, h2 + H2 + h2_aux)
def frobenius(self):
C = self.curve
h1 = self.h1
h2 = self.h2
p = C.characteristic
return h1^p*C.x^(p-1)*C.dx + h2.diffn()
def mult_by_p(omega):
C = omega.curve
fct = omega.form
Fxy, Rxy, x, y = C.fct_field
omega = superelliptic_form(C, fct^p * x^(p-1))
result = superelliptic_drw_form(0*C.x, omega, 0*C.x)
return result
def verschiebung(elt):
C = elt.curve
if isinstance(elt, superelliptic_function):
return superelliptic_witt(0*C.x, elt)
if isinstance(elt, superelliptic_form):
return superelliptic_drw_form(0*C.x, elt, 0*C.x)
superelliptic_form.verschiebung = verschiebung
superelliptic_function.verschiebung = verschiebung
class superelliptic_drw_cech:
def __init__(self, omega0, f):
self.curve = omega0.curve
self.omega0 = omega0
self.omega8 = omega0 - f.diffn()
self.f = f
def reduce(self):
C = self.curve
fct = self.f
f_first_comp = fct.t
f_second_comp = fct.f
decomp_first_comp = decomposition_g0_g8(f_first_comp)
decomp_second_comp = decomposition_g0_g8(f_second_comp)
new = self
new.omega0 -= decomposition_g0_g8(f_first_comp)[0].teichmuller().diffn()
new.omega0 -= decomposition_g0_g8(f_second_comp)[0].verschiebung().diffn()
new.f = decomposition_g0_g8(f_first_comp)[2].teichmuller() + decomposition_g0_g8(f_second_comp)[2].verschiebung()
new.omega8 = new.omega0 - new.f.diffn()
return new
def __repr__(self):
return("(" + str(self.omega0) + ", "+ str(self.f) + ", " + str(self.omega8) + ")")
def __add__(self, other):
C = self.curve
omega0 = self.omega0
f = self.f
omega0_1 = other.omega0
f_1 = other.f
return superelliptic_drw_cech(omega0 + omega0_1, f + f_1)
def __sub__(self, other):
C = self.curve
omega0 = self.omega0
f = self.f
omega0_1 = other.omega0
f_1 = other.f
return superelliptic_drw_cech(omega0 - omega0_1, f - f_1)
def __neg__(self):
C = self.curve
omega0 = self.omega0
f = self.f
return superelliptic_drw_cech(-omega0, -f)
def __rmul__(self, other):
omega0 = self.omega0
f = self.f
return superelliptic_drw_cech(other*omega0, other*f)
def r(self):
omega0 = self.omega0
f = self.f
C = self.curve
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 = C.crystalline_cohomology_basis()
aux = self
for i, a in enumerate(basis):
aux -= coord_lifted[i]*a
print('aux before reduce', aux)
#aux = aux.reduce() # Now aux = p*cech class.
# 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 is_regular(self):
print(self.omega0.r().is_regular_on_U0(), self.omega8.r().is_regular_on_Uinfty(), self.omega0.frobenius().is_regular_on_U0(), self.omega8.frobenius().is_regular_on_Uinfty())
eq1 = self.omega0.r().is_regular_on_U0() and self.omega8.r().is_regular_on_Uinfty()
eq2 = self.omega0.frobenius().is_regular_on_U0() and self.omega8.frobenius().is_regular_on_Uinfty()
return eq1 and eq2
def de_rham_witt_lift(cech_class, prec = 50):
C = cech_class.curve
g = C.genus()
omega0 = cech_class.omega0
omega8 = cech_class.omega8
fct = cech_class.f
omega0_regular = regular_form(omega0) #Present omega0 in the form P dx + Q dy
print('omega0_regular', omega0_regular)
omega0_lift = omega0_regular[0].teichmuller()*(C.x.teichmuller().diffn()) + omega0_regular[1].teichmuller()*(C.y.teichmuller().diffn())
#Now the obvious lift of omega0 = P dx + Q dy to de Rham-Witt is [P] d[x] + [Q] d[y]
print('omega8', omega8, 'second_patch(omega8)', second_patch(omega8))
omega8_regular = regular_form(second_patch(omega8)) # The same for omega8.
print('omega8_regular 1', omega8_regular)
omega8_regular = (second_patch(omega8_regular[0]), second_patch(omega8_regular[1]))
print('omega8_regular 2', omega8_regular)
u = (C.x)^(-1)
v = (C.y)/(C.x)^(g+1)
omega8_lift = omega8_regular[0].teichmuller()*(u.teichmuller().diffn()) + omega8_regular[1].teichmuller()*(v.teichmuller().diffn())
aux = omega0_lift - omega8_lift - fct.teichmuller().diffn() # now aux is of the form (V(smth) + dV(smth), V(smth))
return aux
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())
return result.reduce()
def crystalline_cohomology_basis(self, prec = 50):
result = []
for a in self.de_rham_basis():
result += [de_rham_witt_lift(a, prec = prec)]
return result
superelliptic.crystalline_cohomology_basis = crystalline_cohomology_basis
def autom(self):
C = self.curve
F = C.base_ring
Rxy.<x, y> = PolynomialRing(F, 2)
Fxy = FractionField(Rxy)
if isinstance(self, superelliptic_function):
result = superelliptic_function(C, Fxy(self.function).subs({x:x+1, y:y}))
return result
if isinstance(self, superelliptic_form):
result = superelliptic_form(C, Fxy(self.form).subs({x:x+1, y:y}))
return result
if isinstance(self, superelliptic_witt):
result = superelliptic_witt(autom(self.t), autom(self.f))
return result
if isinstance(self, superelliptic_drw_form):
result = superelliptic_drw_form(0*C.x, autom(self.omega), autom(self.h2))
result += autom(self.h1).teichmuller()*(C.x + C.one).teichmuller().diffn()
return result
if isinstance(self, superelliptic_drw_cech):
result = superelliptic_drw_cech(autom(self.omega0), autom(self.f))
return result
def dy_w(C):
'''Return d[y].'''
fC = C.polynomial
fC = superelliptic_function(C, fC)
fC = fC.teichmuller()
dy_w = 1/2* ((C.y)^(-1)).teichmuller()*auxilliary_derivative(fC)
return dy_w
superelliptic.dy_w = dy_w

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class superelliptic_drw_cech:
def __init__(self, omega0, f):
self.curve = omega0.curve
self.omega0 = omega0
self.omega8 = omega0 - f.diffn()
self.f = f
def reduce(self):
C = self.curve
fct = self.f
f_first_comp = fct.t
f_second_comp = fct.f
decomp_first_comp = decomposition_g0_g8(f_first_comp)
decomp_second_comp = decomposition_g0_g8(f_second_comp)
new = self
new.omega0 -= decomposition_g0_g8(f_first_comp)[0].teichmuller().diffn()
new.omega0 -= decomposition_g0_g8(f_second_comp)[0].verschiebung().diffn()
new.f = decomposition_g0_g8(f_first_comp)[2].teichmuller() + decomposition_g0_g8(f_second_comp)[2].verschiebung()
new.omega8 = new.omega0 - new.f.diffn()
return new
def __repr__(self):
return("(" + str(self.omega0) + ", "+ str(self.f) + ", " + str(self.omega8) + ")")
def __add__(self, other):
C = self.curve
omega0 = self.omega0
f = self.f
omega0_1 = other.omega0
f_1 = other.f
return superelliptic_drw_cech(omega0 + omega0_1, f + f_1)
def __sub__(self, other):
C = self.curve
omega0 = self.omega0
f = self.f
omega0_1 = other.omega0
f_1 = other.f
return superelliptic_drw_cech(omega0 - omega0_1, f - f_1)
def __neg__(self):
C = self.curve
omega0 = self.omega0
f = self.f
return superelliptic_drw_cech(-omega0, -f)
def __rmul__(self, other):
omega0 = self.omega0
f = self.f
return superelliptic_drw_cech(other*omega0, other*f)
def r(self):
omega0 = self.omega0
f = self.f
C = self.curve
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 = C.crystalline_cohomology_basis()
aux = self
for i, a in enumerate(basis):
aux -= coord_lifted[i]*a
print('aux before reduce', aux)
#aux = aux.reduce() # Now aux = p*cech class.
# 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 is_regular(self):
print(self.omega0.r().is_regular_on_U0(), self.omega8.r().is_regular_on_Uinfty(), self.omega0.frobenius().is_regular_on_U0(), self.omega8.frobenius().is_regular_on_Uinfty())
eq1 = self.omega0.r().is_regular_on_U0() and self.omega8.r().is_regular_on_Uinfty()
eq2 = self.omega0.frobenius().is_regular_on_U0() and self.omega8.frobenius().is_regular_on_Uinfty()
return eq1 and eq2

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class superelliptic_drw_form:
def __init__(self, h1, omega, h2):
'''Form [h1] d[x] + V(omega) + dV([h])'''
self.curve = h1.curve
self.h1 = h1
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:
H = (self.h2 - other.h2).pth_root()
except:
return False
eq2 = ((self.omega - other.omega).cartier() - H.diffn()) == 0*self.curve.dx
if eq1 and eq2:
return True
return False
def __repr__(self):
C = self.curve
h1 = self.h1
omega = self.omega
h2 = self.h2
result = ""
if h1.function != 0:
result += "[" + str(h1) + "] d[x]"
if h1.function !=0 and omega.form != 0:
result += " + "
if omega.form != 0:
result += "V(" + str(omega) + ")"
if h2.function !=0 and omega.form != 0:
result += " + "
if h2.function != 0:
result += "dV([" + str(h2) +"])"
if result == "":
result += "0"
return result
def __rmul__(self, other):
h1 = self.h1
h2 = self.h2
omega = self.omega
p = self.curve.characteristic
if other in ZZ:
if other == 0:
return superelliptic_drw_form(0*C.x, 0*C.dx, 0*C.x)
if other > 0:
return self + (other-1)*self
if other < 0:
return (-other)*(-self)
if other in QQ:
other_integer = Integers(p^2)(other)
return other_integer*self
t = other.t
f = other.f
aux_form = t^p*omega - h2*t^(p-1)*t.diffn() + f*h1^p*(C.x)^(p-1)*C.dx
return superelliptic_drw_form(t*h1, aux_form, t^p*h2)
def __neg__(self):
C = self.curve
h1 = self.h1
h2 = self.h2
omega = self.omega
return superelliptic_drw_form(-h1, -omega, -h2)
def __sub__(self, other):
return self + (-other)
def __add__(self, other):
C = self.curve
h1 = self.h1
h2 = self.h2
omega = self.omega
H1 = other.h1
H2 = other.h2
OMEGA = other.omega
aux = (teichmuller(h1) + teichmuller(H1))*superelliptic_drw_form(C.one, 0*C.dx, 0*C.x)
h1_aux = aux.h1
h2_aux = aux.h2
omega_aux = aux.omega
return superelliptic_drw_form(h1_aux, omega + OMEGA + omega_aux, h2 + H2 + h2_aux)
def frobenius(self):
C = self.curve
h1 = self.h1
h2 = self.h2
p = C.characteristic
return h1^p*C.x^(p-1)*C.dx + h2.diffn()

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class superelliptic_witt:
def __init__(self, t, f):
''' Define Witt function on C of the form [t] + V(f). '''
self.curve = t.curve
C = t.curve
p = C.characteristic
self.t = t #superelliptic_function
self.f = f #superelliptic_function
def __repr__(self):
f = self.f
t = self.t
if f.function == 0:
return "[" + str(t) + "]"
if t.function == 0:
return "V(" + str(f) + ")"
return "[" + str(t) + "] + V(" + str(f) + ")"
def __neg__(self):
f = self.f
t = self.t
return superelliptic_witt(-t, -f)
def __add__(self, other):
C = self.curve
second_coor = 0*C.x
X = self.t
Y = other.t
for i in range(1, p):
second_coor -= binomial_prim(p, i)*X^i*Y^(p-i)
return superelliptic_witt(self.t + other.t, self.f + other.f + second_coor)
def __sub__(self, other):
return self + (-other)
def __rmul__(self, other):
p = self.curve.characteristic
if other in ZZ:
if other == 0:
return superelliptic_witt(0*C.x, 0*C.x)
if other > 0:
return self + (other-1)*self
if other < 0:
return (-other)*(-self)
if other in QQ:
other_integer = Integers(p^2)(other)
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
t2 = other.t
f2 = other.f
return superelliptic_witt(t1*t2, t1^p*f2 + t2^p*f1)
if isinstance(other, superelliptic_drw_form):
h1 = other.h1
h2 = other.h2
omega = other.omega
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
return superelliptic_drw_form(t*h1, aux_form, t^p*h2)
def __eq__(self, other):
return self.t == other.t and self.f == other.f
def diffn(self, dy_w = 0):
if dy_w == 0:
dy_w = self.curve.dy_w()
C = self.curve
t = self.t
f = self.f
fC = C.polynomial
F = C.base_ring
Rxy.<x, y> = PolynomialRing(F, 2)
if t.function == 0:
return superelliptic_drw_form(0*C.x, 0*C.dx, f)
t_polynomial = t.function
num = t_polynomial.numerator()
den = t_polynomial.denominator()
num_t_fct = superelliptic_function(C, num)
den_t_fct = superelliptic_function(C, den)
inv_den_t_fct = superelliptic_function(C, 1/den)
if den != 1:
# d([N/D] + V(f)) = [1/D]*d([N]) - [N]*[D^(-2)]*d([D]) + dV(f)
return ((den_t_fct)^(-1)).teichmuller()*num_t_fct.teichmuller().diffn() - ((den_t_fct)^(-2)).teichmuller()*num_t_fct.teichmuller()*den_t_fct.teichmuller().diffn() + superelliptic_drw_form(0*C.x, 0*C.dx, f)
t_polynomial = Rxy(t_polynomial)
M = t_polynomial.monomials()[0]
a = t_polynomial.monomial_coefficient(M)
#[P] = [aM] + Q, where Q = ([P] - [aM])
aM_fct = superelliptic_function(C, a*M)
Q = self - aM_fct.teichmuller()
exp_x = M.exponents()[0][0]
exp_y = M.exponents()[0][1]
return Q.diffn() + exp_x*superelliptic_drw_form(aM_fct/C.x, 0*C.dx, 0*C.x) + exp_y*(aM_fct/C.y).teichmuller()*dy_w
def binomial_prim(p, i):
return binomial(p, i)/p
def reduce_rational_fct(fct, p):
Rxy.<x, y> = PolynomialRing(QQ)
Fxy = FractionField(Rxy)
fct = Fxy(fct)
num = Rxy(fct.numerator())
den = Rxy(fct.denominator())
num1 = Rxy(0)
for m in num.monomials():
a = num.monomial_coefficient(m)
num1 += (a%p^2)*m
den1 = Rxy(0)
for m in den.monomials():
a = den.monomial_coefficient(m)
den1 += (a%p^2)*m
return num1/den1
def teichmuller(fct):
C = fct.curve
return superelliptic_witt(fct, 0*C.x)
superelliptic_function.teichmuller = teichmuller
#dy = [f(x)]'/2*y dx
#[f1 + M] = [f1] + [M] + V(cos)
#d[f1 + M] = d[f1] + d[M] + dV(f1*M)
#M = b x^a
#d[M] = a*[b x^(a-1)]
def auxilliary_derivative(P):
'''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
F = C.base_ring
Rx.<x> = PolynomialRing(F)
P0 = Rx(P0)
P1 = Rx(P1)
if P0 == 0:
return superelliptic_drw_form(0*C.x, 0*C.dx, P.f)
M = P0.monomials()[0]
a = P0.monomial_coefficient(M)
#[P] = [aM] + Q, where Q = ([P] - [aM])
aM_fct = superelliptic_function(C, a*M)
Q = P - aM_fct.teichmuller()
exp = M.exponents()[0]
return auxilliary_derivative(Q) + exp*superelliptic_drw_form(aM_fct/C.x, 0*C.dx, 0*C.x)
def mult_by_p(omega):
C = omega.curve
fct = omega.form
Fxy, Rxy, x, y = C.fct_field
omega = superelliptic_form(C, fct^p * x^(p-1))
result = superelliptic_drw_form(0*C.x, omega, 0*C.x)
return result
def verschiebung(elt):
C = elt.curve
if isinstance(elt, superelliptic_function):
return superelliptic_witt(0*C.x, elt)
if isinstance(elt, superelliptic_form):
return superelliptic_drw_form(0*C.x, elt, 0*C.x)
superelliptic_form.verschiebung = verschiebung
superelliptic_function.verschiebung = verschiebung
def dy_w(C):
'''Return d[y].'''
fC = C.polynomial
fC = superelliptic_function(C, fC)
fC = fC.teichmuller()
dy_w = 1/2* ((C.y)^(-1)).teichmuller()*auxilliary_derivative(fC)
return dy_w
superelliptic.dy_w = dy_w