83 KiB
83 KiB
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):
Rx = f.parent()
x = Rx.gen()
F = Rx.base()
Rx.<x> = PolynomialRing(F)
Rxy.<x, y> = PolynomialRing(F, 2)
Fxy = FractionField(Rxy)
self.polynomial = Rx(f)
self.exponent = m
self.base_ring = F
self.characteristic = F.characteristic()
r = Rx(f).degree()
delta = GCD(r, m)
def __repr__(self):
f = self.polynomial
m = self.exponent
F = self.base_ring
return 'Superelliptic curve with the equation y^' + str(m) + ' = ' + str(f)+' over ' + str(F)
#Auxilliary algorithm that returns the basis of holomorphic differentials
#of the curve and (as a second argument) the list of pairs (i, j)
#such that x^i dx/y^j is holomorphic.
def basis_holomorphic_differentials_degree(self):
f = self.polynomial
m = self.exponent
r = f.degree()
delta = GCD(r, m)
F = self.base_ring
Rx.<x> = PolynomialRing(F)
Rxy.<x, y> = PolynomialRing(F, 2)
Fxy = FractionField(Rxy)
#########basis of holomorphic differentials and de Rham
basis_holo = []
degrees0 = {}
k = 0
for j in range(1, m):
for i in range(1, r):
if (r*j - m*i >= delta):
basis_holo += [superelliptic_form(self, Fxy(x^(i-1)/y^j))]
degrees0[k] = (i-1, j)
k = k+1
return(basis_holo, degrees0)
#Returns the basis of holomorphic differentials using the previous algorithm.
def holomorphic_differentials_basis(self):
basis_holo, degrees0 = self.basis_holomorphic_differentials_degree()
return basis_holo
#Returns the list of pairs (i, j) such that x^i dx/y^j is holomorphic.
def degrees_holomorphic_differentials(self):
basis_holo, degrees0 = self.basis_holomorphic_differentials_degree()
return degrees0
def basis_de_rham_degrees(self):
f = self.polynomial
m = self.exponent
r = f.degree()
delta = GCD(r, m)
F = self.base_ring
Rx.<x> = PolynomialRing(F)
Rxy.<x, y> = PolynomialRing(F, 2)
Fxy = FractionField(Rxy)
basis_holo = self.holomorphic_differentials_basis()
basis = []
#First g_X elements of basis are holomorphic differentials.
for k in range(0, len(basis_holo)):
basis += [superelliptic_cech(self, basis_holo[k], superelliptic_function(self, 0))]
## Next elements do not come from holomorphic differentials.
t = len(basis)
degrees0 = {}
degrees1 = {}
for j in range(1, m):
for i in range(1, r):
if (r*(m-j) - m*i >= delta):
s = Rx(m-j)*Rx(x)*Rx(f.derivative()) - Rx(m)*Rx(i)*f
psi = Rx(cut(s, i))
basis += [superelliptic_cech(self, superelliptic_form(self, Fxy(psi/y^j)), superelliptic_function(self, Fxy(m*y^(m-j)/x^i)))]
degrees0[t] = (psi.degree(), j)
degrees1[t] = (-i, m-j)
t += 1
return basis, degrees0, degrees1
def de_rham_basis(self):
basis, degrees0, degrees1 = self.basis_de_rham_degrees()
return basis
def degrees_de_rham0(self):
basis, degrees0, degrees1 = self.basis_de_rham_degrees()
return degrees0
def degrees_de_rham1(self):
basis, degrees0, degrees1 = self.basis_de_rham_degrees()
return degrees1
def is_smooth(self):
f = self.polynomial
if f.discriminant() == 0:
return 0
return 1
def genus(self):
r = self.polynomial.degree()
m = self.exponent
delta = GCD(r, m)
return 1/2*((r-1)*(m-1) - delta + 1)
def verschiebung_matrix(self):
basis = self.de_rham_basis()
g = self.genus()
p = self.characteristic
F = self.base_ring
M = matrix(F, 2*g, 2*g)
for i in range(0, len(basis)):
w = basis[i]
v = w.verschiebung().coordinates()
M[i, :] = v
return M
def frobenius_matrix(self):
basis = self.de_rham_basis()
g = self.genus()
p = self.characteristic
F = self.base_ring
M = matrix(F, 2*g, 2*g)
for i in range(0, len(basis)):
w = basis[i]
v = w.frobenius().coordinates()
M[i, :] = v
return M
def cartier_matrix(self):
basis = self.holomorphic_differentials_basis()
g = self.genus()
p = self.characteristic
F = self.base_ring
M = matrix(F, g, g)
for i in range(0, len(basis)):
w = basis[i]
v = w.cartier().coordinates()
M[i, :] = v
return M
# def p_rank(self):
# return self.cartier_matrix().rank()
def a_number(self):
g = C.genus()
return g - self.cartier_matrix().rank()
def final_type(self, test = 0):
Fr = self.frobenius_matrix()
V = self.verschiebung_matrix()
p = self.characteristic
return flag(Fr, V, p, test)
#Auxilliary. Given a superelliptic curve C : y^m = f(x) and a polynomial g(x, y)
#it replaces repeteadly all y^m's in g(x, y) by f(x). As a result
#you obtain \sum_{i = 0}^{m-1} y^i g_i(x).
def reduction(C, g):
p = C.characteristic
F = C.base_ring
Rxy.<x, y> = PolynomialRing(F, 2)
Fxy = FractionField(Rxy)
f = C.polynomial
r = f.degree()
m = C.exponent
g = Fxy(g)
g1 = g.numerator()
g2 = g.denominator()
Rx.<x> = PolynomialRing(F)
Fx = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)
(A, B, C) = xgcd(FxRy(g2), FxRy(y^m - f))
g = FxRy(g1*B/A)
while(g.degree(Rxy(y)) >= m):
d = g.degree(Rxy(y))
G = coff(g, d)
i = floor(d/m)
g = g - G*y^d + f^i * y^(d%m) *G
return(FxRy(g))
#Auxilliary. Given a superelliptic curve C : y^m = f(x) and a polynomial g(x, y)
#it replaces repeteadly all y^m's in g(x, y) by f(x). As a result
#you obtain \sum_{i = 0}^{m-1} g_i(x)/y^i. This is needed for reduction of
#superelliptic forms.
def reduction_form(C, g):
F = C.base_ring
Rxy.<x, y> = PolynomialRing(F, 2)
Fxy = FractionField(Rxy)
f = C.polynomial
r = f.degree()
m = C.exponent
g = reduction(C, g)
g1 = Rxy(0)
Rx.<x> = PolynomialRing(F)
Fx = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)
g = FxRy(g)
for j in range(0, m):
if j==0:
G = coff(g, 0)
g1 += FxRy(G)
else:
G = coff(g, j)
g1 += Fxy(y^(j-m)*f*G)
return(g1)
#Class of rational functions on a superelliptic curve C. g = g(x, y) is a polynomial
#defining the function.
class superelliptic_function:
def __init__(self, C, g):
F = C.base_ring
Rxy.<x, y> = PolynomialRing(F, 2)
Fxy = FractionField(Rxy)
f = C.polynomial
r = f.degree()
m = C.exponent
self.curve = C
g = reduction(C, g)
self.function = g
def __repr__(self):
return str(self.function)
def jth_component(self, j):
g = self.function
C = self.curve
F = C.base_ring
Rx.<x> = PolynomialRing(F)
Fx.<x> = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)
g = FxRy(g)
return coff(g, j)
def __add__(self, other):
C = self.curve
g1 = self.function
g2 = other.function
g = reduction(C, g1 + g2)
return superelliptic_function(C, g)
def __sub__(self, other):
C = self.curve
g1 = self.function
g2 = other.function
g = reduction(C, g1 - g2)
return superelliptic_function(C, g)
def __mul__(self, other):
C = self.curve
g1 = self.function
g2 = other.function
g = reduction(C, g1 * g2)
return superelliptic_function(C, g)
def __truediv__(self, other):
C = self.curve
g1 = self.function
g2 = other.function
g = reduction(C, g1 / g2)
return superelliptic_function(C, g)
def __pow__(self, exp):
C = self.curve
g = self.function
return superelliptic_function(C, g^(exp))
def diffn(self):
C = self.curve
f = C.polynomial
m = C.exponent
F = C.base_ring
g = self.function
Rxy.<x, y> = PolynomialRing(F, 2)
Fxy = FractionField(Rxy)
g = Fxy(g)
A = g.derivative(x)
B = g.derivative(y)*f.derivative(x)/(m*y^(m-1))
return superelliptic_form(C, A+B)
def expansion_at_infty(self, i = 0, prec=10):
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)
b = -b
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)^i, prec = prec)
xx = Rt(1/(t^M*ww^b))
yy = 1/(t^R*ww^a)
return Rt(fct(x = Rt(xx), y = Rt(yy)))
def naive_hensel(fct, F, start = 1, prec=10):
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
RtQ = FractionField(Rt)
RptW.<W> = PolynomialRing(RtQ)
RptWQ = FractionField(RptW)
fct = RptWQ(fct)
fct = RptW(numerator(fct))
#return(fct)
#while fct not in RptW:
# print(fct)
# fct *= W
alpha = (fct.derivative())(W = start)
w0 = Rt(start)
i = 1
while(i < prec):
w0 = w0 - fct(W = w0)/alpha + O(t^(prec))
i += 1
return w0
class superelliptic_form:
def __init__(self, C, g):
F = C.base_ring
Rxy.<x, y> = PolynomialRing(F, 2)
Fxy = FractionField(Rxy)
g = Fxy(reduction_form(C, g))
self.form = g
self.curve = C
def __add__(self, other):
C = self.curve
g1 = self.form
g2 = other.form
g = reduction(C, g1 + g2)
return superelliptic_form(C, g)
def __sub__(self, other):
C = self.curve
g1 = self.form
g2 = other.form
g = reduction(C, g1 - g2)
return superelliptic_form(C, g)
def __repr__(self):
g = self.form
if len(str(g)) == 1:
return str(g) + ' dx'
return '('+str(g) + ') dx'
def __rmul__(self, constant):
C = self.curve
omega = self.form
return superelliptic_form(C, constant*omega)
def cartier(self):
C = self.curve
m = C.exponent
p = C.characteristic
f = C.polynomial
F = C.base_ring
Rx.<x> = PolynomialRing(F)
Fx = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)
Fxy = FractionField(FxRy)
result = superelliptic_form(C, FxRy(0))
mult_order = Integers(m)(p).multiplicative_order()
M = Integer((p^(mult_order)-1)/m)
for j in range(1, m):
fct_j = self.jth_component(j)
h = Rx(fct_j*f^(M*j))
j1 = (p^(mult_order-1)*j)%m
B = floor(p^(mult_order-1)*j/m)
result += superelliptic_form(C, polynomial_part(p, h)/(f^B*y^(j1)))
return result
def coordinates(self):
C = self.curve
F = C.base_ring
m = C.exponent
Rx.<x> = PolynomialRing(F)
Fx = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)
g = C.genus()
degrees_holo = C.degrees_holomorphic_differentials()
degrees_holo_inv = {b:a for a, b in degrees_holo.items()}
basis = C.holomorphic_differentials_basis()
for j in range(1, m):
omega_j = Fx(self.jth_component(j))
if omega_j != Fx(0):
d = degree_of_rational_fctn(omega_j, F)
index = degrees_holo_inv[(d, j)]
a = coeff_of_rational_fctn(omega_j, F)
a1 = coeff_of_rational_fctn(basis[index].jth_component(j), F)
elt = self - (a/a1)*basis[index]
return elt.coordinates() + a/a1*vector([F(i == index) for i in range(0, g)])
return vector(g*[0])
def jth_component(self, j):
g = self.form
C = self.curve
F = C.base_ring
Rx.<x> = PolynomialRing(F)
Fx = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)
Fxy = FractionField(FxRy)
Ryinv.<y_inv> = PolynomialRing(Fx)
g = Fxy(g)
g = g(y = 1/y_inv)
g = Ryinv(g)
return coff(g, j)
def is_regular_on_U0(self):
C = self.curve
F = C.base_ring
m = C.exponent
Rx.<x> = PolynomialRing(F)
for j in range(1, m):
if self.jth_component(j) not in Rx:
return 0
return 1
def is_regular_on_Uinfty(self):
C = self.curve
F = C.base_ring
m = C.exponent
f = C.polynomial
r = f.degree()
delta = GCD(m, r)
M = m/delta
R = r/delta
for j in range(1, m):
A = self.jth_component(j)
d = degree_of_rational_fctn(A, F)
if(-d*M + j*R -(M+1)<0):
return 0
return 1
def expansion_at_infty(self, i = 0, prec=10):
g = self.form
C = self.curve
g = superelliptic_function(C, g)
g = g.expansion_at_infty(i = i, prec=prec)
x_series = superelliptic_function(C, x).expansion_at_infty(i = i, prec=prec)
dx_series = x_series.derivative()
return g*dx_series
class superelliptic_cech:
def __init__(self, C, omega, fct):
self.omega0 = omega
self.omega8 = omega - fct.diffn()
self.f = fct
self.curve = C
def __add__(self, other):
C = self.curve
return superelliptic_cech(C, self.omega0 + other.omega0, self.f + other.f)
def __sub__(self, other):
C = self.curve
return superelliptic_cech(C, self.omega0 - other.omega0, self.f - other.f)
def __rmul__(self, constant):
C = self.curve
w1 = self.omega0.form
f1 = self.f.function
w2 = superelliptic_form(C, constant*w1)
f2 = superelliptic_function(C, constant*f1)
return superelliptic_cech(C, w2, f2)
def __repr__(self):
return "(" + str(self.omega0) + ", " + str(self.f) + ", " + str(self.omega8) + ")"
def verschiebung(self):
C = self.curve
omega = self.omega0
F = C.base_ring
Rx.<x> = PolynomialRing(F)
return superelliptic_cech(C, omega.cartier(), superelliptic_function(C, Rx(0)))
def frobenius(self):
C = self.curve
fct = self.f.function
p = C.characteristic
Rx.<x> = PolynomialRing(F)
return superelliptic_cech(C, superelliptic_form(C, Rx(0)), superelliptic_function(C, fct^p))
def coordinates(self):
C = self.curve
F = C.base_ring
m = C.exponent
Rx.<x> = PolynomialRing(F)
Fx = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)
g = C.genus()
degrees_holo = C.degrees_holomorphic_differentials()
degrees_holo_inv = {b:a for a, b in degrees_holo.items()}
degrees0 = C.degrees_de_rham0()
degrees0_inv = {b:a for a, b in degrees0.items()}
degrees1 = C.degrees_de_rham1()
degrees1_inv = {b:a for a, b in degrees1.items()}
basis = C.de_rham_basis()
omega = self.omega0
fct = self.f
if fct.function == Rx(0) and omega.form != Rx(0):
for j in range(1, m):
omega_j = Fx(omega.jth_component(j))
if omega_j != Fx(0):
d = degree_of_rational_fctn(omega_j, F)
index = degrees_holo_inv[(d, j)]
a = coeff_of_rational_fctn(omega_j, F)
a1 = coeff_of_rational_fctn(basis[index].omega0.jth_component(j), F)
elt = self - (a/a1)*basis[index]
return elt.coordinates() + a/a1*vector([F(i == index) for i in range(0, 2*g)])
for j in range(1, m):
fct_j = Fx(fct.jth_component(j))
if (fct_j != Rx(0)):
d = degree_of_rational_fctn(fct_j, p)
if (d, j) in degrees1.values():
index = degrees1_inv[(d, j)]
a = coeff_of_rational_fctn(fct_j, F)
elt = self - (a/m)*basis[index]
return elt.coordinates() + a/m*vector([F(i == index) for i in range(0, 2*g)])
if d<0:
a = coeff_of_rational_fctn(fct_j, F)
h = superelliptic_function(C, FxRy(a*y^j*x^d))
elt = superelliptic_cech(C, self.omega0, self.f - h)
return elt.coordinates()
if (fct_j != Rx(0)):
G = superelliptic_function(C, y^j*x^d)
a = coeff_of_rational_fctn(fct_j, F)
elt =self - a*superelliptic_cech(C, diffn(G), G)
return elt.coordinates()
return vector(2*g*[0])
def is_cocycle(self):
w0 = self.omega0
w8 = self.omega8
fct = self.f
if not w0.is_regular_on_U0() and not w8.is_regular_on_Uinfty():
return('w0 & w8')
if not w0.is_regular_on_U0():
return('w0')
if not w8.is_regular_on_Uinfty():
return('w8')
if w0.is_regular_on_U0() and w8.is_regular_on_Uinfty():
return 1
return 0
#Auxilliary. If f = f1/f2 is a rational function, return deg f_1 - deg f_2.
def degree_of_rational_fctn(f, F):
Rx.<x> = PolynomialRing(F)
Fx = FractionField(Rx)
f = Fx(f)
f1 = f.numerator()
f2 = f.denominator()
d1 = f1.degree()
d2 = f2.degree()
return(d1 - d2)
#Auxilliary. If f = f1/f2 is a rational function, return (leading coeff of f1)/(leading coeff of f2).
def coeff_of_rational_fctn(f, F):
Rx.<x> = PolynomialRing(F)
Fx = FractionField(Rx)
f = Fx(f)
if f == Rx(0):
return 0
f1 = f.numerator()
f2 = f.denominator()
d1 = f1.degree()
d2 = f2.degree()
a1 = f1.coefficients(sparse = false)[d1]
a2 = f2.coefficients(sparse = false)[d2]
return(a1/a2)
#Auxilliary. Given polynomial f(x) and integer d, return
#coefficient of x^d in f (and 0 is deg(f)<d).
def coff(f, d):
lista = f.coefficients(sparse = false)
if len(lista) <= d:
return 0
return lista[d]
#Auxilliary. Given polynomial f(x) = \sum_i a_i x^i and integer i, return
#only \sum_{j >= i+1} a_j x^{j - i -1}
def cut(f, i):
R = f.parent()
coeff = f.coefficients(sparse = false)
return sum(R(x^(j-i-1)) * coeff[j] for j in range(i+1, f.degree() + 1))
def polynomial_part(p, h):
F = GF(p)
Rx.<x> = PolynomialRing(F)
h = Rx(h)
result = Rx(0)
for i in range(0, h.degree()+1):
if (i%p) == p-1:
power = Integer((i-(p-1))/p)
result += Integer(h[i]) * x^(power)
return result
#Find delta-th root of unity in field F
def root_of_unity(F, delta):
Rx.<x> = PolynomialRing(F)
cyclotomic = x^(delta) - 1
for root, a in cyclotomic.roots():
powers = [root^d for d in delta.divisors() if d!= delta]
if 1 not in powers:
return root
def preimage(U, V, M): #preimage of subspace U under M
basis_preimage = M.right_kernel().basis()
imageU = U.intersection(M.transpose().image())
basis = imageU.basis()
for v in basis:
w = M.solve_right(v)
basis_preimage = basis_preimage + [w]
return V.subspace(basis_preimage)
def image(U, V, M):
basis = U.basis()
basis_image = []
for v in basis:
basis_image += [M*v]
return V.subspace(basis_image)
def flag(F, V, p, test = 0):
dim = F.dimensions()[0]
space = VectorSpace(GF(p), dim)
flag_subspaces = (dim+1)*[0]
flag_used = (dim+1)*[0]
final_type = (dim+1)*['?']
flag_subspaces[dim] = space
flag_used[dim] = 1
while 1 in flag_used:
index = flag_used.index(1)
flag_used[index] = 0
U = flag_subspaces[index]
U_im = image(U, space, V)
d_im = U_im.dimension()
final_type[index] = d_im
U_pre = preimage(U, space, F)
d_pre = U_pre.dimension()
if flag_subspaces[d_im] == 0:
flag_subspaces[d_im] = U_im
flag_used[d_im] = 1
if flag_subspaces[d_pre] == 0:
flag_subspaces[d_pre] = U_pre
flag_used[d_pre] = 1
if test == 1:
print('(', final_type, ')')
for i in range(0, dim+1):
if final_type[i] == '?' and final_type[dim - i] != '?':
i1 = dim - i
final_type[i] = final_type[i1] - i1 + dim/2
final_type[0] = 0
for i in range(1, dim+1):
if final_type[i] == '?':
prev = final_type[i-1]
if prev != '?' and prev in final_type[i+1:]:
final_type[i] = prev
for i in range(1, dim+1):
if final_type[i] == '?':
final_type[i] = min(final_type[i-1] + 1, dim/2)
if is_final(final_type, dim/2):
return final_type[1:dim/2 + 1]
print('error:', final_type[1:dim/2 + 1])
def is_final(final_type, dim):
n = len(final_type)
if final_type[0] != 0:
return 0
if final_type[n-1] != dim:
return 0
for i in range(1, n):
if final_type[i] != final_type[i - 1] and final_type[i] != final_type[i - 1] + 1:
return 0
return 1class as_cover:
def __init__(self, C, list_of_fcts, 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
f = C.polynomial
m = C.exponent
r = f.degree()
delta = GCD(m, r)
self.nb_of_pts_at_infty = delta
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 i in range(delta):
x_series = superelliptic_function(C, x).expansion_at_infty(i = i, prec=prec)
y_series = superelliptic_function(C, y).expansion_at_infty(i = i, prec=prec)
z_series = []
jumps = []
n = len(list_of_fcts)
list_of_power_series = [g.expansion_at_infty(i = i, prec=prec) for g in list_of_fcts]
for i in range(n):
power_series = list_of_power_series[i]
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 += [jumps]
all_x_series += [x_series]
all_y_series += [y_series]
all_z_series += [z_series]
all_dx_series += [x_series.derivative()]
self.jumps = all_jumps
self.x = all_x_series
self.y = all_y_series
self.z = all_z_series
self.dx = all_dx_series
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 = "(Z/p)^"+str(self.height)+ "-cover of " + str(self.quotient)+" with the equations:\n"
for i in range(n):
result += 'z' + str(i) + "^" + str(p) + " - z" + str(i) + " = " + str(self.functions[i]) + "\n"
return result
def genus(self):
jumps = self.jumps
gY = self.quotient.genus()
n = self.height
delta = self.nb_of_pts_at_infty
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))
def exponent_of_different(self, i = 0):
jumps = self.jumps
n = self.height
delta = self.nb_of_pts_at_infty
p = self.characteristic
return sum(p^(n-j-1)*(jumps[i][j]+1)*(p-1) for j in range(n))
def exponent_of_different_prim(self, i = 0):
jumps = self.jumps
n = self.height
delta = self.nb_of_pts_at_infty
p = self.characteristic
return sum(p^(n-j-1)*(jumps[i][j])*(p-1) for j in range(n))
def holomorphic_differentials_basis(self, threshold = 8):
from itertools import product
x_series = self.x
y_series = self.y
z_series = self.z
dx_series = self.dx
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()
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)
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 = as_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(S)
for i in range(1, delta):
forms = [(omega, omega.expansion_at_infty(i = i)) for omega in forms]
forms = holomorphic_combinations(forms)
if len(forms) < self.genus():
print("I haven't found all forms.")
return holomorphic_differentials_basis(self, threshold = threshold + 1)
if len(forms) > self.genus():
print("Increase precision.")
return forms
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
y_series = self.y
z_series = self.z
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()
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)
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 = as_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(1, delta):
forms = [(omega, omega.expansion_at_infty(i = 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 equat 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
y_series = self.y
z_series = self.z
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()
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)
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 = as_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 i in range(1, delta):
forms = [(omega, omega.expansion_at_infty(i = i)) for omega in forms]
forms = holomorphic_combinations_forms(forms, pole_order)
return forms
def 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 = as_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
class as_function:
def __init__(self, C, g):
self.curve = C
F = C.base_ring
n = C.height
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.function = RxyzQ(g)
#self.function = as_reduction(AS, RxyzQ(g))
def __repr__(self):
return str(self.function)
def __add__(self, other):
C = self.curve
g1 = self.function
g2 = other.function
return as_function(C, g1 + g2)
def __sub__(self, other):
C = self.curve
g1 = self.function
g2 = other.function
return as_function(C, g1 - g2)
def __rmul__(self, constant):
C = self.curve
g = self.function
return as_function(C, constant*g)
def __mul__(self, other):
C = self.curve
g1 = self.function
g2 = other.function
return as_function(C, g1*g2)
def expansion_at_infty(self, i = 0):
C = self.curve
delta = C.nb_of_pts_at_infty
F = C.base_ring
x_series = C.x[i]
y_series = C.y[i]
z_series = C.z[i]
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
n = C.height
F = C.base_ring
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)
sub_list = {x : x, y : y} | {z[j] : z[j]+ZN_tuple[j] for j in range(n)}
g = self.function
return as_function(C, g.substitute(sub_list))
def trace(self):
C = self.curve
C_super = C.quotient
n = C.height
F = C.base_ring
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)
g = self.function
g = as_reduction(C, g)
result = RxyzQ(0)
g_num = Rxyz(numerator(g))
g_den = Rxyz(denominator(g))
z = prod(z[i] for i in range(n))^(p-1)
for a in g_num.monomials():
if (z.divides(a)):
result += g_num.monomial_coefficient(a)*a/z
result /= g_den
result = as_reduction(C, result)
Rxy.<x, y> = PolynomialRing(F, 2)
Qxy = FractionField(Rxy)
return superelliptic_function(C_super, Qxy(result))
def trace2(self):
C = self.curve
C_super = C.quotient
n = C.height
F = C.base_ring
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)
result = as_function(C, 0)
for i in range(0, p):
for j in range(0, p):
result += self.group_action([i, j])
result = result.function
Rxy.<x, y> = PolynomialRing(F, 2)
Qxy = FractionField(Rxy)
return superelliptic_function(C_super, Qxy(result))
class as_form:
def __init__(self, C, g):
self.curve = C
n = C.height
F = C.base_ring
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.form = RxyzQ(g)
def __repr__(self):
return "(" + str(self.form)+") * dx"
def expansion_at_infty(self, i = 0):
C = self.curve
delta = C.nb_of_pts_at_infty
F = C.base_ring
x_series = C.x[i]
y_series = C.y[i]
z_series = C.z[i]
dx_series = C.dx[i]
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
g2 = other.form
return as_form(C, g1 + g2)
def __sub__(self, other):
C = self.curve
g1 = self.form
g2 = other.form
return as_form(C, g1 - g2)
def __rmul__(self, constant):
C = self.curve
omega = self.form
return as_form(C, constant*omega)
def group_action(self, ZN_tuple):
C = self.curve
n = C.height
F = C.base_ring
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)
sub_list = {x : x, y : y} | {z[j] : z[j]+ZN_tuple[j] for j in range(n)}
g = self.form
return as_form(C, g.substitute(sub_list))
def coordinates(self, holo):
"""Find coordinates of the given form self in terms of the basis forms in a list holo."""
C = self.curve
n = C.height
gC = C.genus()
F = C.base_ring
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:]
from sage.rings.polynomial.toy_variety import linear_representation
return linear_representation(Rxyz(self.form), holo)
def trace(self):
C = self.curve
C_super = C.quotient
n = C.height
F = C.base_ring
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)
g = self.form
result = RxyzQ(0)
g_num = Rxyz(numerator(g))
g_den = Rxyz(denominator(g))
z = prod(z[i] for i in range(n))^(p-1)
for a in g_num.monomials():
if (z.divides(a)):
result += g_num.monomial_coefficient(a)*a/z
result /= g_den
Rxy.<x, y> = PolynomialRing(F, 2)
return superelliptic_form(C_super, Rxy(result))
def trace2(self):
C = self.curve
C_super = C.quotient
n = C.height
F = C.base_ring
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)
result = as_form(C, 0)
for i in range(0, p):
for j in range(0, p):
result += self.group_action([i, j])
result = result.form
Rxy.<x, y> = PolynomialRing(F, 2)
Qxy = FractionField(Rxy)
return superelliptic_form(C_super, Qxy(result))
# Given power_series, find its reverse (g with g \circ power_series = id) with given precision
def new_reverse(power_series, prec = 10):
F = power_series.parent().base()
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
RtQ = FractionField(Rt)
power_series = RtQ(power_series)
a = power_series.list()[0]
g = 1/a*t
n = 2
while(n <= prec):
aux = power_series(t = g) - t
if aux.valuation() > n:
b = 0
else:
b = aux.list()[0]
g = g - b/a*t^n
n += 1
return g
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()
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
RtQ = FractionField(Rt)
power_series = RtQ(power_series)
if power_series.valuation() == +Infinity:
return(0,0,t,0)
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:
T = ((power_series)^(-1)).nth_root(jump) #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])
T_rev = new_reverse(T, prec = prec)
t_old = T_rev(t^p/(1 - t^((p-1)*jump)).nth_root(jump))
z = 1/t^(jump) + Rt(correction)(t = t_old)
return(jump, correction, t_old, z)
def are_forms_linearly_dependent(set_of_forms):
from sage.rings.polynomial.toy_variety import is_linearly_dependent
C = set_of_forms[0].curve
F = C.base_ring
n = C.height
variable_names = 'x, y'
for i in range(n):
variable_names += ', z' + str(i)
Rxyz = PolynomialRing(F, n+2, variable_names)
denominators = prod(denominator(omega.form) for omega in set_of_forms)
return is_linearly_dependent([Rxyz(denominators*omega.form) for omega in set_of_forms])
def group_action_matrices(C_AS):
F = C_AS.base_ring
n = C_AS.height
holo = C_AS.holomorphic_differentials_basis()
holo_forms = [omega.form for omega in holo]
denom = LCM([denominator(omega) for omega in holo_forms])
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:]
holo_forms = [Rxyz(omega*denom) for omega in holo_forms]
A = [[] for i in range(n)]
for omega in holo:
for i in range(n):
ei = n*[0]
ei[i] = 1
omega1 = omega.group_action(ei)
omega1 = denom * omega1
v1 = omega1.coordinates(holo_forms)
A[i] += [v1]
for i in range(n):
A[i] = matrix(F, A[i])
A[i] = A[i].transpose()
return A
def group_action_matrices_log(C_AS):
F = C_AS.base_ring
n = C_AS.height
holo = C_AS.at_most_poles_forms(1)
holo_forms = [omega.form for omega in holo]
denom = LCM([denominator(omega) for omega in holo_forms])
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:]
holo_forms = [Rxyz(omega*denom) for omega in holo_forms]
A = [[] for i in range(n)]
for omega in holo:
for i in range(n):
ei = n*[0]
ei[i] = 1
omega1 = omega.group_action(ei)
omega1 = denom * omega1
v1 = omega1.coordinates(holo_forms)
A[i] += [v1]
for i in range(n):
A[i] = matrix(F, A[i])
A[i] = A[i].transpose()
return A
#given a set S of (form, corresponding Laurent series at some pt), find their combinations holomorphic at that pt
def holomorphic_combinations_fcts(S, pole_order):
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([Rt(g[1]).valuation() for g in S])
if minimal_valuation >= -pole_order:
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 + Rt(eta_exp).valuation()
list_coeffs = a*[0] + Rt(eta_exp).list() + (-minimal_valuation)*[0]
list_coeffs = list_coeffs[:-minimal_valuation - pole_order]
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 = as_function(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
#given a set S of (form, corresponding Laurent series at some pt), find their combinations holomorphic at that pt
def holomorphic_combinations_forms(S, pole_order):
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([Rt(g[1]).valuation() for g in S])
if minimal_valuation >= -pole_order:
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 + Rt(eta_exp).valuation()
list_coeffs = a*[0] + Rt(eta_exp).list() + (-minimal_valuation)*[0]
list_coeffs = list_coeffs[:-minimal_valuation - pole_order]
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 = as_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
#print only forms that are log at the branch pts, but not holomorphic
def only_log_forms(C_AS):
list1 = AS.at_most_poles_forms(0)
list2 = AS.at_most_poles_forms(1)
result = []
for a in list2:
if not(are_forms_linearly_dependent(list1 + result + [a])):
result += [a]
return result
def magmathis(A, B, text = False):
"""Find decomposition of Z/p^2-module given by matrices A, B into indecomposables using magma.
If text = True, print the command for Magma. Else - return the output of Magma free."""
q = parent(A).base_ring().order()
n = A.dimensions()[0]
A = str(list(A))
B = str(list(B))
A = A.replace("(", "")
A = A.replace(")", "")
B = B.replace("(", "")
B = B.replace(")", "")
result = "A := MatrixAlgebra<GF("+str(q) + "),"+ str(n) + "|"
result += A + "," + B
result += ">;"
result += "M := RModule(RSpace(GF("+str(q)+")," + str(n) + "), A);"
result += "IndecomposableSummands(M);"
if text:
return result
print(magma_free(result))
def as_reduction(AS, fct):
n = AS.height
F = AS.base_ring
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)
ff = AS.functions
ff = [RxyzQ(F.function) for F in ff]
fct = RxyzQ(fct)
fct1 = numerator(fct)
fct2 = denominator(fct)
if fct2 != 1:
return as_reduction(AS, fct1)/as_reduction(AS, fct2)
result = RxyzQ(0)
change = 0
for a in fct1.monomials():
degrees_zi = [a.degree(z[i]) for i in range(n)]
d_div = [a.degree(z[i])//p for i in range(n)]
if d_div != n*[0]:
change = 1
d_rem = [a.degree(z[i])%p for i in range(n)]
monomial = fct1.coefficient(a)*x^(a.degree(x))*y^(a.degree(y))*prod(z[i]^(d_rem[i]) for i in range(n))*prod((z[i] + ff[i])^(d_div[i]) for i in range(n))
result += RxyzQ(fct1.coefficient(a)*x^(a.degree(x))*y^(a.degree(y))*prod(z[i]^(d_rem[i]) for i in range(n))*prod((z[i] + ff[i])^(d_div[i]) for i in range(n)))
if change == 0:
return RxyzQ(result)
else:
print(fct, '\n')
return as_reduction(AS, RxyzQ(result))Testy
p = 5
m = 2
Rx.<x> = PolynomialRing(GF(p))
f = x^3 + x^2 + 1
C_super = superelliptic(f, m)
Rxy.<x, y> = PolynomialRing(GF(p), 2)
fArS1 = superelliptic_function(C_super, y*x)
fArS2 = superelliptic_function(C_super, y*x^2)
fArS3 = superelliptic_function(C_super, y)
AS1 = as_cover(C_super, [fArS1, fArS2, fArS3], prec=500)
print(AS1.genus())
AS2 = as_cover(C_super, [fArS2, fArS3, fArS1], prec=500)
print(AS2.genus())325 325
p = 5
m = 2
Rx.<x> = PolynomialRing(GF(p))
f = x^3 + x^2 + 1
C_super = superelliptic(f, m)
Rxy.<x, y> = PolynomialRing(GF(p), 2)
fArS1 = superelliptic_function(C_super, y*x)
fArS2 = superelliptic_function(C_super, y*x^2)
fArS3 = superelliptic_function(C_super, y)
AS1 = as_cover(C_super, [fArS1, fArS2, fArS3], prec=1000)
print(AS1.exponent_of_different())
omega = as_form(AS1, 1/y)
print(omega.expansion_at_infty())Brudnopis
AS.at_most_poles(14)[1, z1, z1^2, z0, z0*z1, -x*z0^2 + z0*z1^2, z0^2, z0^2*z1, z0^2*z1^2 + x*z0*z1 + x^2, y, -y*z0^2 + y*z1, y*z0, x, -x*z0^2 + x*z1, x*z0]
only_log_forms(AS)[((z0^2*z1^2 + x*z0*z1 + x^2)/y) * dx]
dpp = AS.exponent_of_different_prim()
print(at_most_poles(AS, dpp))I haven't found all forms. [1]
Rxy.<x, y, z0, z1> = PolynomialRing(F, 4)
omega = as_form(AS, z0^2*z1^2/y - (a+1)*z0*x/y)
print(omega - omega.group_action([1, 0]))((z0*z1^2 - z1^2 + (a + 1)*x)/y) * dx
g = as_function(AS, z0^2*z1^2+z0*(a+1)*x)
g.expansion_at_infty(i = 0)a*t^-10 + 2*t^-8 + a*t^-6 + (a + 1)*t^-2 + O(t^288)
p = 7
m = 2
F = GF(p)
Rx.<x> = PolynomialRing(F)
f = x^3 + 1
C_super = superelliptic(f, m)
Rxy.<x, y> = PolynomialRing(F, 2)
f1 = superelliptic_function(C_super, x^2*y)
f2 = superelliptic_function(C_super, x^3)
AS = as_cover(C_super, [f1, f2], prec=1000)
#print(AS.magical_element())
#print(AS.pseudo_magical_element())#A, B = group_action_matrices2_log(AS1)
A, B = group_action_matrices2(AS)
n = A.dimensions()[0]
print(A*B == B*A)
print(A^p == identity_matrix(n))
print(B^p == identity_matrix(n))
magmathis(A, B, p, deg = 1)True True True
print(f1.expansion_at_infty())
print(f2.expansion_at_infty())t^-5 + 2*t + O(t^5) t^-6 + 3 + O(t^4)
AS.magical_element()[z0^2]
AS1.exponent_of_different_prim()68
AS1.jumps[[5, 2], [5, 2]]
magmathis(A, B, 3, deg = 2)[ RModule of dimension 1 over GF(3^2), RModule of dimension 1 over GF(3^2), RModule of dimension 3 over GF(3^2), RModule of dimension 3 over GF(3^2), RModule of dimension 6 over GF(3^2), RModule of dimension 6 over GF(3^2), RModule of dimension 8 over GF(3^2) ]
magmatxtx(A, B, p)WARNING: Some output was deleted.
A := MatrixAlgebra<GF(3^1),14|[1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1],[1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]>; M := RModule(RSpace(GF(3^1),14), A); lll := IndecomposableSummands(M); Morphism(lll[2], M);
parent(A).base_ring().order()3
Rx.<x> = PolynomialRing(GF(3))
C = superelliptic(x^3 - x, 2)C.de_rham_basis()[((1/y) dx, 0, (1/y) dx), ((x/y) dx, 2/x*y, ((-1)/(x*y)) dx)]
Rxy.<x, y> = PolynomialRing(GF(p), 2)
omega = superelliptic_form(C, 1/y^3)
ff = superelliptic_function(C, x)
cycle = superelliptic_cech(C, omega, ff)cycle.coordinates()(0, 0)
def dual_elt(AS, zmag):
p = AS.characteristic
n = AS.height
group_elts = [(i, j) for i in range(p) for j in range(p)]
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)
M = matrix(RxyzQ, p^n, p^n)
for i in range(p^n):
for j in range(p^n):
M[i, j] = (zmag.group_action(group_elts[i])*zmag.group_action(group_elts[j])).trace2()
main_det = M.determinant()
zvee = as_function(AS, 0)
for i in range(p^n):
Mprim = matrix(RxyzQ, M)
Mprim[:, i] = vector([(j == 0) for j in range(p^2)])
fi = Mprim.determinant()/main_det
zvee += fi*zmag.group_action(group_elts[i])
return zveep = 5
m = 1
F = GF(p)
Rx.<x> = PolynomialRing(F)
f = x
C_super = superelliptic(f, m)
Rxy.<x, y> = PolynomialRing(F, 2)
f1 = superelliptic_function(C_super, x^2)
f2 = superelliptic_function(C_super, x^3)
AS = as_cover(C_super, [f1, f2], prec=500)
zmag = (AS.magical_element())[0]n = 2
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:]
f3 = as_function(AS, z[0]^p - z[0])zdual = dual_elt(AS, zmag)(zmag*(zdual.group_action([0, 0]))).trace2()1
n = 2
F = GF(5)
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:]def ith_component(omega, zvee, i):
AS = omega.curve
p = AS.characteristic
group_elts = [(j1, j2) for j1 in range(p) for j2 in range(p)]
z_vee_fct = zvee.group_action(group_elts[i]).function
new_form = as_form(AS, z_vee_fct*omega.form)
return new_form.trace2()om = AS.holomorphic_differentials_basis()[4]om(-z0^3*z1^2 - 2*x*z0^2*z1 + z1^4 + 2*x^2*z0) * dx
def combination_components(omega, zmag, w):
AS = omega.curve
p = AS.characteristic
group_elts = [(j1, j2) for j1 in range(p) for j2 in range(p)]
zvee = dual_elt(AS, zmag)
result = as_form(AS, 0)
for i in range(p^2):
omegai = ith_component(omega, zvee, i)
aux_fct1 = w.group_action(group_elts[i]).function
aux_fct2 = omegai.form
result += as_form(AS, aux_fct1*aux_fct2)
return resultcombination_components(om, zmag, zmag)(-z0^3*z1^2 - 2*x*z0^2*z1 + z1^4 + 2*x^2*z0) * dx
om(-z0^3*z1^2 - 2*x*z0^2*z1 + z1^4 + 2*x^2*z0) * dx
zmag.expansion_at_infty()t^-68 + t^-48 + t^-40 + 2*t^-36 + 3*t^-28 + 3*t^-20 + 2*t^-16 + t^-12 + 3*t^-8 + 4*t^-4 + 4*t^4 + t^8 + 3*t^12 + 4*t^20 + t^24 + 3*t^28 + 3*t^32 + 4*t^36 + t^44 + 3*t^48 + t^52 + 2*t^56 + 4*t^60 + 3*t^64 + 2*t^72 + 3*t^76 + 2*t^88 + 2*t^92 + t^96 + O(t^100)
w1 = as_function(AS, z[0]^2*y^3/x - z[0]^2*y^2)
w1.expansion_at_infty()O(t^430)
4*t^-85 + t^-70 + t^-65 + t^-57 + 3*t^-50 + t^-45 + 3*t^-42 + t^-37 + t^-30 + 3*t^-25 + 3*t^-17 + t^-14 + t^-9 + t^-5 + 2*t^-2 + t^10 + t^11 + 3*t^15 + t^19 + 2*t^23 + 3*t^26 + 2*t^30 + t^38 + 2*t^43 + t^47 + 3*t^51 + 4*t^54 + t^55 + t^58 + t^66 + t^70 + t^75 + 4*t^79 + 4*t^83 + 2*t^86 + 2*t^90 + t^94 + 3*t^95 + 2*t^98 + 4*t^103 + 3*t^110 + 4*t^115 + t^122 + 4*t^123 + 3*t^126 + t^130 + 4*t^131 + 3*t^135 + 2*t^138 + 2*t^143 + 3*t^150 + 3*t^151 + 4*t^154 + 4*t^155 + 3*t^158 + 4*t^159 + t^166 + t^170 + t^175 + 3*t^178 + t^186 + 4*t^187 + 2*t^190 + 2*t^191 + 3*t^195 + t^198 + 3*t^206 + t^211 + 3*t^215 + 3*t^218 + 4*t^219 + 4*t^222 + t^223 + 4*t^230 + 3*t^234 + 4*t^235 + t^238 + t^247 + 4*t^250 + 4*t^251 + 3*t^254 + 2*t^258 + 3*t^262 + t^263 + 2*t^266 + t^270 + 2*t^275 + 2*t^279 + 2*t^283 + 4*t^286 + t^290 + 4*t^291 + 3*t^294 + 3*t^295 + t^298 + 4*t^303 + 3*t^310 + 2*t^311 + 2*t^315 + t^318 + 3*t^319 + 4*t^322 + 4*t^323 + 4*t^330 + 2*t^331 + 3*t^335 + t^338 + 4*t^343 + t^346 + 3*t^347 + 2*t^350 + t^351 + 3*t^354 + 3*t^355 + t^358 + t^363 + 4*t^370 + t^374 + 2*t^379 + 4*t^383 + 3*t^387 + 4*t^390 + 4*t^391 + 2*t^398 + t^402 + t^406 + t^410 + 3*t^411 + O(t^415)
om1 = combination_components(om, zmag, w1)print(om1)
print(om1.expansion_at_infty())(0) * dx O(t^474)