511 lines
20 KiB
Python
511 lines
20 KiB
Python
class heisenberg_cover:
|
|
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)
|
|
F = C.base_ring
|
|
self.base_ring = F
|
|
p = C.characteristic
|
|
self.characteristic = p
|
|
self.prec = prec
|
|
#group acting
|
|
n = self.height
|
|
from itertools import product
|
|
pr = [list(GF(p)) for _ in range(n)]
|
|
self.group = [(a, b, c) for a in range(p) for b in range(p) for c in range(p)]
|
|
#########
|
|
f = C.polynomial
|
|
m = C.exponent
|
|
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 = {}
|
|
|
|
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(pt=pt, prec=prec) for g in list_of_fcts]
|
|
for j in range(n):
|
|
if j == 2:
|
|
list_of_power_series[j] += (z_series[0] - z_series[1])*list_of_power_series[1]
|
|
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)
|
|
z_series = [zi(t = t_old) for zi in z_series]
|
|
z_series += [z]
|
|
jumps += [jump]
|
|
list_of_power_series = [g(t = t_old) for g in list_of_power_series]
|
|
|
|
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
|
|
self.z_series = all_z_series
|
|
self.dx_series = all_dx_series
|
|
##############
|
|
#Function field
|
|
variable_names = 'x, y'
|
|
for i in range(n):
|
|
variable_names += ', z' + str(i)
|
|
Rxyz = PolynomialRing(F, n+2, variable_names)
|
|
x, y = Rxyz.gens()[:2]
|
|
z = Rxyz.gens()[2:]
|
|
RxyzQ = FractionField(Rxyz)
|
|
self.fct_field = (RxyzQ, Rxyz, x, y, z)
|
|
self.x = heisenberg_function(self, x)
|
|
self.y = heisenberg_function(self, y)
|
|
self.z = [heisenberg_function(self, z[j]) for j in range(n)]
|
|
self.dx = heisenberg_form(self, 1)
|
|
self.one = heisenberg_function(self, 1)
|
|
|
|
|
|
def __repr__(self):
|
|
n = self.height
|
|
p = self.characteristic
|
|
if n==1:
|
|
return "(Z/p)-cover of " + str(self.quotient)+" with the equation:\n z^" + str(p) + " - z = " + str(self.functions[0])
|
|
|
|
result = "E(p^3)-cover of " + str(self.quotient)+" with the equations:\n"
|
|
for i in range(n):
|
|
if i !=2:
|
|
result += 'z' + str(i) + "^" + str(p) + " - z" + str(i) + " = " + str(self.functions[i]) + "\n"
|
|
else:
|
|
result += 'z2'+ "^" + str(p) + ' - z2 = ' + str(self.functions[2]) + ' + ' + '(z0 - z1) * (' + str(self.functions[1]) + ")"
|
|
return result
|
|
|
|
def genus(self):
|
|
jumps = self.jumps
|
|
gY = self.quotient.genus()
|
|
n = self.height
|
|
branch_pts = self.branch_points
|
|
p = self.characteristic
|
|
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
|
|
n = self.height
|
|
delta = self.nb_of_pts_at_infty
|
|
p = self.characteristic
|
|
return sum(p^(n-j-1)*(jumps[place][j]+1)*(p-1) for j in range(n))
|
|
|
|
def exponent_of_different_prim(self, place = 0):
|
|
jumps = self.jumps
|
|
n = self.height
|
|
delta = self.nb_of_pts_at_infty
|
|
p = self.characteristic
|
|
return sum(p^(n-j-1)*(jumps[place][j])*(p-1) for j in range(n))
|
|
|
|
def holomorphic_differentials_basis(self, threshold = 8):
|
|
from itertools import product
|
|
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
|
|
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
|
|
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 = []
|
|
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_form(self, x^i * prod(z[i1]^(k[i1]) for i1 in range(n))/y^j)
|
|
eta_exp = eta.expansion(pt=self.branch_points[0])
|
|
S += [(eta, eta_exp)]
|
|
|
|
forms = heisenberg_holomorphic_combinations(S)
|
|
print("next iteration")
|
|
for pt in self.branch_points:
|
|
for g in [(0, i, 0) for i in range(p)]:
|
|
print("next iteration")
|
|
if pt != self.branch_points[0] or g != (0, 0, 0):
|
|
forms = [(omega, omega.group_action(g).expansion(pt=pt)) for omega in forms]
|
|
forms = heisenberg_holomorphic_combinations(forms)
|
|
|
|
return forms
|
|
if len(forms) < self.genus():
|
|
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():
|
|
raise ValueError("Increase precision.")
|
|
return forms
|
|
|
|
def cartier_matrix(self, prec=50):
|
|
g = self.genus()
|
|
F = self.base_ring
|
|
M = matrix(F, g, g)
|
|
for i, omega in enumerate(self.holomorphic_differentials_basis()):
|
|
M[:, i] = vector(omega.cartier().coordinates())
|
|
return M
|
|
|
|
def at_most_poles(self, pole_order, threshold = 8):
|
|
""" 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_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_order)
|
|
|
|
for i in range(delta):
|
|
for g in [(0, i, 0) for i in range(p)]:
|
|
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
|
|
|
|
def magical_element(self, threshold = 8):
|
|
list_of_elts = self.at_most_poles(self.exponent_of_different_prim(), threshold)
|
|
result = []
|
|
for a in list_of_elts:
|
|
if a.trace().function != 0:
|
|
result += [a]
|
|
return result
|
|
|
|
def pseudo_magical_element(self, threshold = 8):
|
|
list_of_elts = self.at_most_poles(self.exponent_of_different(), threshold)
|
|
result = []
|
|
for a in list_of_elts:
|
|
if a.trace().function != 0:
|
|
result += [a]
|
|
return result
|
|
|
|
def at_most_poles_forms(self, pole_order, threshold = 8):
|
|
"""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."""
|
|
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
|
|
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_form(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_forms(S, pole_order)
|
|
|
|
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
|
|
|
|
def uniformizer(self, place = 0):
|
|
'''Return uniformizer of curve self at place-th place at infinity.'''
|
|
p = self.characteristic
|
|
n = self.height
|
|
F = self.base_ring
|
|
RxyzQ, Rxyz, x, y, z = self.fct_field
|
|
fx = heisenberg_function(self, x)
|
|
z = [heisenberg_function(self, zi) for zi in z]
|
|
# We create a list of functions. We add there all variables...
|
|
list_of_fcts = [fx]+z
|
|
vfx = fx.valuation(place)
|
|
vz = [zi.valuation(place) for zi in z]
|
|
|
|
# Then we subtract powers of variables with the same valuation (so that 1/t^(kp) cancels) and add to this list.
|
|
for j1 in range(n):
|
|
for j2 in range(n):
|
|
if j1>j2:
|
|
a = gcd(vz[j1] , vz[j2])
|
|
vz1 = vz[j1]/a
|
|
vz2 = vz[j2]/a
|
|
for b in F:
|
|
if (z[j1]^(vz2) - b*z[j2]^(vz1)).valuation(place) > (z[j2]^(vz1)).valuation(place):
|
|
list_of_fcts += [z[j1]^(vz2) - b*z[j2]^(vz1)]
|
|
for j1 in range(n):
|
|
a = gcd(vz[j1], vfx)
|
|
vzj = vz[j1] /a
|
|
vfx = vfx/a
|
|
for b in F:
|
|
if (fx^(vzj) - b*z[j1]^(vfx)).valuation(place) > (z[j1]^(vfx)).valuation(place):
|
|
list_of_fcts += [fx^(vzj) - b*z[j1]^(vfx)]
|
|
#Finally, we check if on the list there are two elements with the same valuation.
|
|
for f1 in list_of_fcts:
|
|
for f2 in list_of_fcts:
|
|
d, a, b = xgcd(f1.valuation(place), f2.valuation(place))
|
|
if d == 1:
|
|
return f1^a*f2^b
|
|
raise ValueError("My method of generating fcts with relatively prime valuation failed.")
|
|
|
|
|
|
def ith_ramification_gp(self, i, place = 0):
|
|
'''Find ith ramification group at place at infty of nb place.'''
|
|
G = self.group
|
|
t = self.uniformizer(place)
|
|
Gi = [G[0]]
|
|
for g in G:
|
|
if g != G[0]:
|
|
tg = t.group_action(g)
|
|
v = (tg - t).valuation(place)
|
|
if v >= i+1:
|
|
Gi += [g]
|
|
return Gi
|
|
|
|
def ramification_jumps(self, place = 0):
|
|
'''Return list of lower ramification jumps at at place at infty of nb place.'''
|
|
G = self.group
|
|
ramification_jps = []
|
|
i = 0
|
|
while len(G) > 1:
|
|
Gi = self.ith_ramification_gp(i+1, place)
|
|
if len(Gi) < len(G):
|
|
ramification_jps += [i]
|
|
G = Gi
|
|
i+=1
|
|
return ramification_jps
|
|
|
|
def a_number(self):
|
|
g = self.genus()
|
|
return g - self.cartier_matrix().rank()
|
|
|
|
def cohomology_of_structure_sheaf_basis(self, holo_basis = 0, threshold = 8):
|
|
if holo_basis == 0:
|
|
holo_basis = self.holomorphic_differentials_basis(threshold = threshold)
|
|
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
|
|
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
|
|
result_fcts = []
|
|
V = VectorSpace(F,self.genus())
|
|
S = V.subspace([])
|
|
RQxyz = FractionField(Rxyz)
|
|
pr = [list(GF(p)) for _ in range(n)]
|
|
i = 0
|
|
while len(result_fcts) < self.genus():
|
|
for j in range(0, m):
|
|
for k in product(*pr):
|
|
f = heisenberg_function(self, prod(z[i1]^(k[i1]) for i1 in range(n))/x^i*y^j)
|
|
f_products = [omega.serre_duality_pairing(f) for omega in holo_basis]
|
|
if vector(f_products) not in S:
|
|
S = S+V.subspace([V(f_products)])
|
|
result_fcts += [f]
|
|
i += 1
|
|
return result_fcts
|
|
|
|
def lift_to_de_rham(self, fct, threshold = 30):
|
|
'''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)'''
|
|
from itertools import product
|
|
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
|
|
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
|
|
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 = [(fct.diffn(), fct.diffn().expansion_at_infty())]
|
|
pr = [list(GF(p)) for _ in range(n)]
|
|
holo = self.holomorphic_differentials_basis(threshold = threshold)
|
|
for i in range(0, threshold*r):
|
|
for j in range(0, m):
|
|
for k in product(*pr):
|
|
eta = heisenberg_form(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 = heisenberg_holomorphic_combinations(S)
|
|
if len(forms) <= self.genus():
|
|
raise ValueError("Increase threshold!")
|
|
for omega in forms:
|
|
for a in F:
|
|
if (a*omega + fct.diffn()).is_regular_on_U0():
|
|
return a*omega + fct.diffn()
|
|
raise ValueError("Unknown.")
|
|
|
|
def de_rham_basis(self, holo_basis = 0, cohomology_basis = 0, threshold = 30):
|
|
if holo_basis == 0:
|
|
holo_basis = self.holomorphic_differentials_basis(threshold = threshold)
|
|
if cohomology_basis == 0:
|
|
cohomology_basis = self.cohomology_of_structure_sheaf_basis(holo_basis = holo_basis, threshold = threshold)
|
|
result = []
|
|
for omega in holo_basis:
|
|
result += [heisenberg_cech(self, omega, heisenberg_function(self, 0))]
|
|
for f in cohomology_basis:
|
|
omega = self.lift_to_de_rham(f, threshold = threshold)
|
|
result += [heisenberg_cech(self, omega, f)]
|
|
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):
|
|
"""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
|
|
p = C_AS.characteristic
|
|
F = C_AS.base_ring
|
|
prec = C_AS.prec
|
|
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
|
|
RtQ = FractionField(Rt)
|
|
minimal_valuation = min([g[1].valuation() for g in S])
|
|
if minimal_valuation >= 0:
|
|
return [s[0] for s in S]
|
|
list_of_lists = [] #to będzie lista złożona z list współczynników część nieholomorficznych rozwinięcia form z S
|
|
for eta, eta_exp in S:
|
|
a = -minimal_valuation + eta_exp.valuation()
|
|
list_coeffs = a*[0] + eta_exp.list() + (-minimal_valuation)*[0]
|
|
list_coeffs = list_coeffs[:-minimal_valuation]
|
|
list_of_lists += [list_coeffs]
|
|
M = matrix(F, list_of_lists)
|
|
V = M.kernel() #chcemy wyzerować części nieholomorficzne, biorąc kombinacje form z S
|
|
|
|
|
|
# Sprawdzamy, jakim formom odpowiadają elementy V.
|
|
forms = []
|
|
for vec in V.basis():
|
|
forma_holo = heisenberg_form(C_AS, 0)
|
|
forma_holo_power_series = Rt(0)
|
|
for vec_wspolrzedna, elt_S in zip(vec, S):
|
|
eta = elt_S[0]
|
|
#eta_exp = elt_S[1]
|
|
forma_holo += vec_wspolrzedna*eta
|
|
#forma_holo_power_series += vec_wspolrzedna*eta_exp
|
|
forms += [forma_holo]
|
|
return forms |