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naprawione regular form; superelliptyczne maja C.x_series itd

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jgarnek 2023-03-30 15:49:22 +00:00
parent eda1cca0c2
commit 64fe2ee228
10 changed files with 6605 additions and 33234 deletions
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39711
sage/.run.term-0.term
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File diff suppressed because one or more lines are too long

12
sage/drafty/draft.sage
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@ -1,14 +1,14 @@
p = 5
m = 2
F = GF(p)
Rx.<x> = PolynomialRing(F)
Rxx.<x> = PolynomialRing(F)
#f = (x^3 - x)^3 + x^3 - x
f = x^3 + x
f1 = f(x = x^5 - x)
C = superelliptic(f, m)
C1 = superelliptic(f1, m)
C1 = superelliptic(f1, m, prec = 500)
B = C.crystalline_cohomology_basis(prec = 100, info = 1)
B1 = C1.crystalline_cohomology_basis(prec = 500, info = 1)
B1 = C1.crystalline_cohomology_basis(prec = 100, info = 1)
def crystalline_matrix(C, prec = 50):
B = C.crystalline_cohomology_basis(prec = prec)
@ -20,6 +20,12 @@ def crystalline_matrix(C, prec = 50):
M[i, :] = vector(autom(b).coordinates(basis = B))
return M
for b in B:
print(b.regular_form())
for b in B1:
print(b.regular_form())
#M = crystalline_matrix(C, prec = 150)
#print(M)
#print(M^3)

27
sage/superelliptic/superelliptic_class.sage
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@ -1,7 +1,7 @@
class superelliptic:
"""Class of a superelliptic curve. Given a polynomial f(x) with coefficient field F, it constructs
the curve y^m = f(x)"""
def __init__(self, f, m):
def __init__(self, f, m, prec = 100):
Rx = f.parent()
x = Rx.gen()
F = Rx.base()
@ -20,6 +20,31 @@ class superelliptic:
self.y = superelliptic_function(self, Rxy(y))
self.dx = superelliptic_form(self, Rxy(1))
self.one = superelliptic_function(self, Rxy(1))
# We compute now expansions at infinity of x and y.
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
RptW.<W> = PolynomialRing(Rt)
RptWQ = FractionField(RptW)
Rxy.<x, y> = PolynomialRing(F)
RxyQ = FractionField(Rxy)
delta, a, b = xgcd(m, r)
a = -a
M = m/delta
R = r/delta
while a<0:
a += R
b += M
g = (x^r*f(x = 1/x))
gW = RptWQ(g(x = t^M * W^b)) - W^(delta)
self.x_series = []
self.y_series = []
self.dx_series = []
for place in range(delta):
ww = naive_hensel(gW, F, start = root_of_unity(F, delta)^place, prec = prec)
xx = Rt(1/(t^M*ww^b))
yy = 1/(t^R*ww^a)
self.x_series += [xx]
self.y_series += [yy]
self.dx_series += [xx.derivative()]
def __repr__(self):
f = self.polynomial

8
sage/superelliptic/superelliptic_form_class.sage
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@ -137,10 +137,10 @@ class superelliptic_form:
g = self.form
C = self.curve
g = superelliptic_function(C, g)
g = g.expansion_at_infty(place = place, prec=prec)
x_series = C.x.expansion_at_infty(place = place, prec=prec)
dx_series = x_series.derivative()
return g*dx_series
F = C.base_ring
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
g = Rt(g.expansion_at_infty(place = place, prec=prec))
return g*C.dx_series[place]
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.'''

28
sage/superelliptic/superelliptic_function_class.sage
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@ -108,31 +108,11 @@ class superelliptic_function:
def expansion_at_infty(self, place = 0, prec=20):
C = self.curve
f = C.polynomial
m = C.exponent
F = C.base_ring
Rx.<x> = PolynomialRing(F)
f = Rx(f)
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
RptW.<W> = PolynomialRing(Rt)
RptWQ = FractionField(RptW)
Rxy.<x, y> = PolynomialRing(F)
RxyQ = FractionField(Rxy)
fct = self.function
fct = RxyQ(fct)
r = f.degree()
delta, a, b = xgcd(m, r)
a = -a
M = m/delta
R = r/delta
while a<0:
a += R
b += M
g = (x^r*f(x = 1/x))
gW = RptWQ(g(x = t^M * W^b)) - W^(delta)
ww = naive_hensel(gW, F, start = root_of_unity(F, delta)^place, prec = prec)
xx = Rt(1/(t^M*ww^b))
yy = 1/(t^R*ww^a)
F = C.base_ring
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
xx = C.x_series[place]
yy = C.y_series[place]
return Rt(fct(x = Rt(xx), y = Rt(yy)))
def expansion(self, pt, prec = 50):

2
sage/superelliptic_drw/de_rham_witt_lift.sage
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@ -34,10 +34,10 @@ def de_rham_witt_lift(cech_class, prec = 50):
def crystalline_cohomology_basis(self, prec = 50, info = 0):
result = []
prec1 = prec
for i, a in enumerate(self.de_rham_basis()):
if info:
print("Computing " + str(i) +". basis element")
prec1 = prec
while True:
try:
result += [de_rham_witt_lift(a, prec = prec1)]

4
sage/superelliptic_drw/regular_form.sage
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@ -74,12 +74,14 @@ class superelliptic_regular_drw_form:
def regular_drw_form(omega):
C = omega.curve
p = C.characteristic
omega_aux = omega.r()
omega_aux = omega_aux.regular_form()
aux = omega - omega_aux.dx.teichmuller()*C.x.teichmuller().diffn() - omega_aux.dy.teichmuller()*C.y.teichmuller().diffn()
aux.omega, fct = decomposition_omega0_hpdh(aux.omega)
aux.h2 += fct^p
aux.h2 = decomposition_g0_p2th_power(aux.h2)[0]
aux.h2, A = decomposition_g0_pth_power(aux.h2)
aux.omega += (A.diffn()).inv_cartier()
result = superelliptic_regular_drw_form(omega_aux.dx, omega_aux.dy, aux.omega.regular_form(), aux.h2)
return result

14
sage/superelliptic_drw/superelliptic_drw_auxilliaries.sage
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@ -1,4 +1,8 @@
def decomposition_g0_pth_power(fct):
C = fct.curve
Fxy, Rxy, xy, y = C.fct_field
if fct.function in Rxy:
return (fct, 0*C.x)
'''Decompose fct as g0 + A^p, if possible. Output: (g0, A).'''
omega = fct.diffn().regular_form()
g0 = omega.int()
@ -7,6 +11,8 @@ def decomposition_g0_pth_power(fct):
def decomposition_g0_p2th_power(fct):
'''Decompose fct as g0 + A^(p^2), if possible. Output: (g0, A).'''
C = fct.curve
p = C.characteristic
g0, A = decomposition_g0_pth_power(fct)
A0, A1 = decomposition_g0_pth_power(A)
return (g0 + A0^p, A1)
@ -14,6 +20,9 @@ def decomposition_g0_p2th_power(fct):
def decomposition_omega0_hpdh(omega):
'''Decompose omega = (regular on U0) + h^(p-1) dh, so that Cartier(omega) = (regular on U0) + dh.
Result: (regular on U0, h)'''
C = omega.curve
if omega.is_regular_on_U0():
return (omega, 0*C.x)
omega1 = omega.cartier().cartier()
omega1 = omega1.inv_cartier().inv_cartier()
fct = (omega.cartier() - omega1.cartier()).int()
@ -27,6 +36,9 @@ def decomposition_omega8_hpdh(omega, prec = 50):
Fxy, Rxy, x, y = C.fct_field
F = C.base_ring
p = C.characteristic
C = omega.curve
if omega.is_regular_on_Uinfty():
return (omega, 0*C.x)
Rt.<t> = LaurentSeriesRing(F)
omega_analytic = Rt(laurent_analytic_part(omega.expansion_at_infty(prec = prec)))
Cv = C.uniformizer()
@ -44,6 +56,8 @@ def decomposition_g8_pth_power(fct, prec = 50):
F = C.base_ring
Rt.<t> = LaurentSeriesRing(F)
Fxy, Rxy, x, y = C.fct_field
if fct.expansion_at_infty().valuation() >= 0:
return (fct, 0*C.x)
A = laurent_analytic_part(fct.expansion_at_infty(prec = prec))
Cv = C.uniformizer()
v = Cv.function

27
sage/superelliptic_drw/superelliptic_drw_cech.sage
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@ -56,13 +56,19 @@ class superelliptic_drw_cech:
C = self.curve
return superelliptic_cech(C, omega0.h1*C.dx, f.t)
def div_by_p(self):
def div_by_p(self, info = 0):
'''Given a regular cocycle of the form (V(omega) + dV(h), [f] + V(t), ...), where [f] = 0 in H^1(X, OX),
find de Rham cocycle (xi0, f, xi8) such that (V(omega) + dV(h), [f] + V(t), ...) = p*(xi0, f, xi8).'''
#
if info: print("Computing " + str(self) + " divided by p.")
#
C = self.curve
aux = self
Fxy, Rxy, x, y = C.fct_field
aux_f_t_0 = decomposition_g0_g8(aux.f.t, prec=50)[0]
#
if info: print("Computed decomposition_g0_g8 of self.f.t.")
#
aux.f.t = 0*C.x
aux.omega0 -= aux_f_t_0.teichmuller().diffn()
aux.omega8 = aux.omega0 - aux.f.diffn()
@ -72,9 +78,17 @@ class superelliptic_drw_cech:
omega = aux.omega0.omega
aux.omega0.omega, fct = decomposition_omega0_hpdh(aux.omega0.omega)
aux.omega0.h2 += fct^p
#
if info: print("Computed decomposition_omega0_hpdh of self.omega0.omega.")
#
# Now we have to ensure that aux.omega0.h2.function in Rxy...
# In other words, we decompose h2 = (regular on U0) + A^(p^2).
aux.omega0.h2 = decomposition_g0_p2th_power(aux.omega0.h2)[0]
# In other words, we decompose h2 = (regular on U0) + A^p.
# Then we replace h2 by (regular on U0) and omega by omega + inverse Cartier(dA)
aux.omega0.h2, A = decomposition_g0_pth_power(aux.omega0.h2)
aux.omega0.omega += (A.diffn()).inv_cartier()
#
if info: print("Computed decomposition_g0_p2th_power(aux.omega0.h2).")
#
# Now we can reduce: (... + dV(h2), V(f), ...) --> (..., V(f - h2), ...)
aux.f -= aux.omega0.h2.verschiebung()
aux.omega0.h2 = 0*C.x
@ -87,18 +101,21 @@ class superelliptic_drw_cech:
aux_divided_by_p = superelliptic_cech(C, aux.omega0.omega.cartier(), aux.f.f.pth_root())
return aux_divided_by_p
def coordinates(self, basis = 0, prec = 50):
def coordinates(self, basis = 0, prec = 50, info = 0):
if info: print("Computing coordinates of " + str(self))
C = self.curve
g = C.genus()
coord_mod_p = self.r().coordinates()
if info: print("Computed coordinates 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
aux_divided_by_p = aux.div_by_p()
aux_divided_by_p = aux.div_by_p(info = info)
coord_aux_divided_by_p = aux_divided_by_p.coordinates()
if info: print("Computed coordinates mod p of (self - lift)/p.")
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

6
sage/tests.sage
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@ -1,6 +1,6 @@
load('init.sage')
#print("Expansion at infty test:")
#load('superelliptic/tests/expansion_at_infty.sage')
#load('init.sage')
print("Expansion at infty test:")
load('superelliptic/tests/expansion_at_infty.sage')
#print("superelliptic form coordinates test:")
#load('superelliptic/tests/form_coordinates_test.sage')
#print("p-th root test:")