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canonical ideal for quaternion covers

This commit is contained in:
jgarnek 2024-01-23 18:53:29 +00:00
parent 1891b447b6
commit a8e8dbbaa3
4 changed files with 63 additions and 71 deletions
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36
as_covers/as_polyforms.sage
View File

@ -91,8 +91,17 @@ class as_symmetric_product_forms:
def __init__(self, forms_and_coeffs):
'''Elements of forms_and_coeffs are of the form [coeff, form1, ..., formn]'''
self.n = len(forms_and_coeffs[0]) - 1
self.tuples = forms_and_coeffs
self.curve = forms_and_coeffs[0][1].curve
forms_and_coeffs1 = []
for atuple in forms_and_coeffs:
if atuple[0] != 0 and atuple[1:] not in [a[1:] for a in forms_and_coeffs1]:
forms_and_coeffs1 += [atuple]
elif atuple[1:] in [a[1:] for a in forms_and_coeffs1]:
i = [a[1:] for a in forms_and_coeffs1].index(atuple[1:])
forms_and_coeffs1[i][0] += atuple[0]
if len(forms_and_coeffs1) == 0:
forms_and_coeffs1 = [[0] + forms_and_coeffs[0][1:]]
self.tuples = forms_and_coeffs1
self.curve = forms_and_coeffs1[0][1].curve
def coordinates(self, basis = 0):
AS = self.curve
@ -172,13 +181,28 @@ class as_symmetric_product_forms:
return result
def group_action(self, elt):
p = self.base_ring.characteristic()
n = self.height
p = self.curve.base_ring.characteristic()
n = self.curve.height
aux_tuples = []
for atuple in self.tuples:
aux_tuple = atuple[0] + [a.group_action(elt) for a in atuple[1:]]
aux_tuple = [atuple[0]] + [a.group_action(elt) for a in atuple[1:]]
aux_tuples += [aux_tuple]
return as_symmetric_product_forms(aux_tuples)
def non_decreasing(L):
return all(x<=y for x, y in zip(L, L[1:]))
return all(x<=y for x, y in zip(L, L[1:]))
def as_matrices_group_action_canonical_ideal(AS, mult, threshold = 8):
K = as_canonical_ideal(AS, mult, threshold = threshold)
n = AS.height
F = AS.base_ring
K_polynomials = [a.polynomial() for a in K]
r = len(K)
matrices = []
for i in range(n):
M = matrix(F, r, r)
K_group_action_polynomials = [a.group_action([j == i for j in range(n)]).polynomial() for a in K]
for i in range(r):
M[i, :] = vector(linear_representation_polynomials(K_group_action_polynomials[i], K_polynomials))
matrices += [M]
return matrices

65
as_covers/group_action_matrices.sage
View File

@ -1,4 +1,4 @@
def group_action_matrices(space, list_of_group_elements, basis):
def group_action_matrices(F, space, list_of_group_elements, basis):
n = len(list_of_group_elements)
d = len(space)
A = [matrix(F, d, d) for i in range(n)]
@ -18,7 +18,8 @@ def group_action_matrices_holo(AS, basis=0, threshold=10):
generators += [ei]
if basis == 0:
basis = AS.holomorphic_differentials_basis(threshold=threshold)
return group_action_matrices(basis, generators, basis = basis)
F = AS.base_ring
return group_action_matrices(F, basis, generators, basis = basis)
def group_action_matrices_dR(AS, threshold=8):
n = AS.height
@ -30,35 +31,9 @@ def group_action_matrices_dR(AS, threshold=8):
holo_basis = AS.holomorphic_differentials_basis(threshold = threshold)
str_basis = AS.cohomology_of_structure_sheaf_basis(holo_basis = holo_basis, threshold = threshold)
dr_basis = AS.de_rham_basis(holo_basis = holo_basis, cohomology_basis = str_basis, threshold=threshold)
F = AS.base_ring
basis = [holo_basis, str_basis, dr_basis]
return group_action_matrices(basis[2], generators, basis = basis)
def group_action_matrices_old(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
return group_action_matrices(F, basis[2], generators, basis = basis)
def group_action_matrices_log(AS):
n = AS.height
@ -67,31 +42,5 @@ def group_action_matrices_log(AS):
ei = n*[0]
ei[i] = 1
generators += [ei]
return group_action_matrices(AS.at_most_poles_forms(1), generators, basis = AS.at_most_poles_forms(1))
def group_action_matrices_log_old(C_AS):
F = C_AS.base_ring
n = C_AS.height
holo = C_AS.at_most_poles_forms(1)
holo_forms = [omega 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
F = AS.base_ring
return group_action_matrices(F, AS.at_most_poles_forms(1), generators, basis = AS.at_most_poles_forms(1))

31
quaternion_covers/quaternion_form_class.sage
View File

@ -81,13 +81,29 @@ class quaternion_form:
omega = self.form
return quaternion_form(C, constant*omega)
def group_action(self, ZN_tuple):
C = self.curve
n = C.height
RxyzQ, Rxyz, x, y, z = C.fct_field
sub_list = {x : x, y : y} | {z[j] : z[j]+ZN_tuple[j] for j in range(n)}
def group_action(self, elt):
Q8 = QuaternionGroup()
Qi, Qj = Q8.gens()
AS = self.curve
RxyzQ, Rxyz, x, y, z = AS.fct_field
if elt == Qi^4:
sub_list = {x : x, y : y} | {z[0] : z[0], z[1] : z[1], z[2]: z[2]}
if elt == Qi:
sub_list = {x : x, y : y} | {z[0] : z[0]+1, z[1] : z[1], z[2]: z[2] + z[0]}
if elt == Qj:
sub_list = {x : x, y : y} | {z[0] : z[0], z[1] : z[1] + 1, z[2]: z[2] + z[1] + z[0]}
if elt == Qi^2:
sub_list = {x : x, y : y} | {z[0] : z[0], z[1] : z[1], z[2]: z[2] + 1}
if elt == Qi * Qj:
sub_list = {x : x, y : y} | {z[0] : z[0] + 1, z[1] : z[1] + 1, z[2]: z[2] + z[1] + 1}
if elt == Qj * Qi:
sub_list = {x : x, y : y} | {z[0] : z[0] + 1, z[1] : z[1] + 1, z[2]: z[2] + z[1]}
if elt == Qi^3:
sub_list = {x : x, y : y} | {z[0] : z[0]+1, z[1] : z[1], z[2]: z[2] + z[0] + 1}
if elt == Qj^3:
sub_list = {x : x, y : y} | {z[0] : z[0], z[1] : z[1] + 1, z[2]: z[2] + z[1] + z[0] + 1}
g = self.form
return quaternion_form(C, g.substitute(sub_list))
return quaternion_form(AS, g.substitute(sub_list))
def coordinates(self, basis = 0):
"""Find coordinates of the given holomorphic form self in terms of the basis forms in a list holo."""
@ -97,6 +113,9 @@ class quaternion_form:
RxyzQ, Rxyz, x, y, z = C.fct_field
# We need to have only polynomials to use monomial_coefficients in linear_representation_polynomials,
# and sometimes basis elements have denominators. Thus we multiply by them.
fct = quaternion_function(C, self.form)
fct = quaternion_reduction(C, fct)
self = quaternion_form(C, fct)
denom = LCM([denominator(omega.form) for omega in basis])
basis = [denom*omega for omega in basis]
self_with_no_denominator = denom*self

2
quaternion_covers/quaternion_reduction.sage
View File

@ -24,7 +24,7 @@ def quaternion_reduction(AS, fct):
if d_div != n*[0]:
change = 1
d_rem = [a.degree(z[i])%p for i in range(n)]
monomial = fct1.monomial_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))
monomial = fct1.monomial_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-1))*(z[2] + ff[2] + z[0]*ff[0] + z[1] * (ff[0] + ff[1]))^(d_div[2])
result += RxyzQ(monomial)
if change == 0: