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separate files for holomorphic combinations

This commit is contained in:
jgarnek 2024-02-12 19:06:26 +00:00
parent 780c1f03f2
commit 498e55d11b
14 changed files with 184 additions and 327 deletions
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39
as_covers/as_cover_class.sage
View File

@ -135,11 +135,11 @@ class as_cover:
eta_exp = eta.expansion(pt=self.branch_points[0])
S += [(eta, eta_exp)]
forms = as_holomorphic_combinations(S)
forms = holomorphic_combinations(S)
for pt in self.branch_points[1:]:
forms = [(omega, omega.expansion(pt=pt)) for omega in forms]
forms = as_holomorphic_combinations(forms)
forms = holomorphic_combinations(forms)
if len(forms) < self.genus():
print("I haven't found all forms, only ", len(forms), " of ", self.genus())
@ -376,7 +376,7 @@ class as_cover:
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 = as_holomorphic_combinations(S)
forms = holomorphic_combinations(S)
if len(forms) <= self.genus():
raise ValueError("Increase threshold!")
for omega in forms:
@ -398,36 +398,3 @@ class as_cover:
result += [as_cech(self, omega, f)]
return result
def as_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

69
as_covers/as_form_class.sage
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@ -188,75 +188,6 @@ def are_forms_linearly_dependent(set_of_forms):
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 holomorphic_combinations_fcts(S, pole_order):
'''given a set S of (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([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)

3
as_covers/as_transform.sage
View File

@ -2,7 +2,6 @@ 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."""
print('as transform for ', power_series)
correction = 0
F = power_series.parent().base()
p = F.characteristic()
@ -16,7 +15,6 @@ def artin_schreier_transform(power_series, prec = 10):
aux = t^p - t
z = new_reverse(aux, prec = prec)
z = z(t = power_series)
print("a")
return(0, 0, t, z)
while(power_series.valuation() % p == 0 and power_series.valuation() < 0):
@ -41,5 +39,4 @@ def artin_schreier_transform(power_series, prec = 10):
z = new_reverse(aux, prec = prec)
z = z(t = power_series)
z = z + correction
print('corr', correction, 'ps', power_series, 'z', z)
return(0, correction, t, z)

106
as_covers/holomorphic_combinations.sage Normal file
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@ -0,0 +1,106 @@
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 = 0*C_AS.dx
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
def holomorphic_combinations_fcts(S, pole_order):
'''given a set S of (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([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()
if eta_exp !=0:
list_coeffs = a*[0] + Rt(eta_exp).list() + (-minimal_valuation)*[0]
list_coeffs = list_coeffs[:-minimal_valuation - pole_order]
else:
list_coeffs = (-minimal_valuation - pole_order)*[0]
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 = 0*C_AS.x
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
def holomorphic_combinations_forms(S, pole_order):
'''given a set S of (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([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 = 0*C_AS.dx
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

28
heisenberg_covers/dual_element.sage Normal file
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@ -0,0 +1,28 @@
def heisenberg_dual_elt(AS, zmag):
'''Find the trace dual of a given elt zmag in the function field of an Artin-Schreier cover AS.'''
p = AS.characteristic
n = 3
G = AS.group
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):
print('creating matrix', i, j)
elt = tr2((zmag.group_action(G[i])*zmag.group_action(G[j]))).function
elt = Rxyz(elt.numerator())/Rxyz(elt.denominator())
M[i, j] = RxyzQ(elt)
main_det = -1 #M.determinant()
zvee = as_function(AS, 0)
for i in range(p^n):
print(i)
Mprim = matrix(RxyzQ, M)
Mprim[:, i] = vector([(j == 0) for j in range(p^n)])
fi = Mprim.determinant()/main_det
zvee += heisenberg_function(AS, fi*(zmag.group_action(G[i]).function))
return zvee

52
heisenberg_covers/heisenberg_covers.sage
View File

@ -137,14 +137,14 @@ class heisenberg_cover:
eta_exp = eta.expansion(pt=self.branch_points[0])
S += [(eta, eta_exp)]
forms = heisenberg_holomorphic_combinations(S)
forms = holomorphic_combinations(S)
print("next iteration")
for pt in self.branch_points:
for g in [(0, i, 0) for i in range(p)]:
for g in self.fiber(place = pt):
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)
forms = holomorphic_combinations(forms)
return forms
if len(forms) < self.genus():
@ -422,7 +422,8 @@ class heisenberg_cover:
result += [g]
return result
def stabilizer_coset_reps(self, place = 0):
def fiber(self, place = 0):
'Gives representatives for the quotient G/G_P for given place. Those are in bijection with the fiber.'
result = [(0, 0, 0)]
p = self.characteristic
H = self.stabilizer(place = place)
@ -466,46 +467,11 @@ def at_most_poles2(self, pole_orders, threshold = 8):
S += [(eta, eta_exp)]
forms = holomorphic_combinations_fcts(S, pole_orders[(0, (0, 0, 0))])
print('iteration')
for i in range(delta):
for g in self.stabilizer_coset_reps(place = i):
for g in self.fiber(place = i):
if i!=0 or g != (0, 0, 0):
forms = [(omega, omega.expansion_at_infty(place = i)) for omega in forms]
print('iteration')
forms = [(omega, omega.group_action(g).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

68
heisenberg_covers/heisenberg_form_class.sage
View File

@ -213,74 +213,6 @@ def are_forms_linearly_dependent(set_of_forms):
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 holomorphic_combinations_fcts(S, pole_order):
'''given a set S of (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([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 = heisenberg_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 = 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
#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)

4
heisenberg_covers/heisenberg_function_class.sage
View File

@ -148,6 +148,7 @@ class heisenberg_function:
return self.group_action(elt1).group_action((0, 0, 1))
def trace(self):
print('trace ')
C = self.curve
C_super = C.quotient
n = C.height
@ -166,7 +167,8 @@ class heisenberg_function:
result = result.function
Rxy.<x, y> = PolynomialRing(F, 2)
Qxy = FractionField(Rxy)
result = heisenberg_reduction(AS, result)
result = heisenberg_reduction(C, result)
#print(result)
return superelliptic_function(C_super, Qxy(result))
def coordinates(self, prec = 100, basis = 0):

23
heisenberg_covers/heisenberg_group.sage Normal file
View File

@ -0,0 +1,23 @@
def heisenberg_mult(v1, v2, p):
i1, j1, k1 = v1
i2, j2, k2 = v2
i3 = (i1 + i2)%p
j3 = (j1 + j2)%p
k3 = (-i2*j1 + k1 + k2)%p
return(i3, j3, k3)
def heisenberg_inv(v1, p):
i1, j1, k1 = v1
i2 = p-i1
j2 = p-j1
k2 = p-k1
k2 = (k2 - i2*j2)%p
return (i2, j2, k2)
def heisenberg_power(v1, n, p):
i1, j1, k1 = v1
if n < 0:
return heisenberg_inv(heisenberg_power((i1, j1, k1), -n, p), p)
if n == 0:
return (0, 0, 0)
return heisenberg_mult((i1, j1, k1), heisenberg_power((i1, j1, k1), n-1, p), p)

2
heisenberg_covers/heisenberg_reduction.sage
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@ -10,7 +10,7 @@ def heisenberg_reduction(AS, fct):
fct1 = numerator(fct)
fct2 = denominator(fct)
denom = heisenberg_function(AS, fct2)
denom_norm = prod(heisenberg_function(AS, fct2).group_action(g) for g in AS.group if list(g) != n*[0])
denom_norm = prod(heisenberg_function(AS, fct2).group_action(g) for g in AS.group if g != (0, 0, 0))
fct1 = Rxyz(fct1*denom_norm.function)
fct2 = Rxyz(fct2*denom_norm.function)
if fct2 != 1:

6
heisenberg_covers/ith_magical_component.sage Normal file
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@ -0,0 +1,6 @@
def heisenberg_ith_magical_component(omega, zvee, g):
'''Given a form omega on AS cover, element g of group AS.group and normal basis element zmag, find the decomposition
sum_g g(zmag) omega_g and return omega_g.'''
z_vee_g = zvee.group_action(g)
new_form = z_vee_g*omega
return new_form.trace()

1
init.sage
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@ -4,6 +4,7 @@ load('superelliptic/superelliptic_form_class.sage')
load('superelliptic/superelliptic_cech_class.sage')
load('superelliptic/frobenius_kernel.sage')
load('as_covers/as_transform.sage')
load('as_covers/holomorphic_combinations.sage')
load('as_covers/as_cover_class.sage')
load('as_covers/as_function_class.sage')
load('as_covers/as_form_class.sage')

42
quaternion_covers/quaternion_covers.sage
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@ -137,11 +137,11 @@ class quaternion_cover:
eta_exp = eta.expansion(pt=self.branch_points[0])
S += [(eta, eta_exp)]
forms = quaternion_holomorphic_combinations(S)
forms = holomorphic_combinations(S)
for pt in self.branch_points[1:]:
forms = [(omega, omega.expansion(pt=pt)) for omega in forms]
forms = quaternion_holomorphic_combinations(forms)
forms = holomorphic_combinations(forms)
if len(forms) < self.genus():
print("I haven't found all forms, only ", len(forms), " of ", self.genus())
@ -378,7 +378,7 @@ class quaternion_cover:
eta = quaternion_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 = quaternion_holomorphic_combinations(S)
forms = holomorphic_combinations(S)
if len(forms) <= self.genus():
raise ValueError("Increase threshold!")
for omega in forms:
@ -398,38 +398,4 @@ class quaternion_cover:
for f in cohomology_basis:
omega = self.lift_to_de_rham(f, threshold = threshold)
result += [quaternion_cech(self, omega, f)]
return result
def quaternion_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 = quaternion_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
return result

68
quaternion_covers/quaternion_form_class.sage
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@ -207,74 +207,6 @@ def are_forms_linearly_dependent(set_of_forms):
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 holomorphic_combinations_fcts(S, pole_order):
'''given a set S of (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([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 = quaternion_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 = quaternion_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)