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added possibility of substituting x, y and template of a wrong hypoelementary action

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This commit is contained in:
Jędrzej Garnek 2024-09-27 15:18:38 +02:00
parent 884881c8aa
commit adfe3aacf3
5 changed files with 80 additions and 32 deletions
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6
as_covers/as_cover_class.sage
View File

@ -21,7 +21,7 @@ class as_cover:
self.branch_points = list(range(delta)) + branch_points
Rxy.<x, y> = PolynomialRing(F, 2)
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
Rzf, zgen, fgen = cover_template.fct_field
Rzf, zgen, fgen, xgen, ygen = cover_template.fct_field
all_x_series = {}
all_y_series = {}
all_z_series = {}
@ -73,8 +73,8 @@ class as_cover:
self.z = [as_function(self, z[j]) for j in range(n)]
self.dx = as_form(self, 1)
self.one = as_function(self, 1)
Rzf, zgen, fgen = cover_template.fct_field
subs_fs = {zgen[i] : z[i]}| {fgen[i] : RxyzQ(list_of_fcts[i].function) for i in range(n)}
Rzf, zgen, fgen, xgen, ygen = cover_template.fct_field
subs_fs = {zgen[i] : z[i]}| {fgen[i] : RxyzQ(list_of_fcts[i].function) for i in range(n)}|{xgen:x, ygen:y}
self.rhs = [RxyzQ(cover_template.fcts[i].subs(subs_fs)) for i in range(n)]
#####
##### We compute now the differentials dz[i]

2
as_covers/as_form_class.sage
View File

@ -88,8 +88,10 @@ class as_form:
# 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.
denom = LCM([denominator(omega.form) for omega in basis])
print(denom, basis, '\n')
basis = [denom*omega for omega in basis]
self_with_no_denominator = denom*self
print(self_with_no_denominator.form, [omega.form for omega in basis])
return linear_representation_polynomials(Rxyz(self_with_no_denominator.form), [Rxyz(omega.form) for omega in basis])
def trace(self, super=True):

6
as_covers/as_function_class.sage
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@ -104,7 +104,7 @@ class as_function:
def group_action(self, elt):
C = self.curve
RxyzQ, Rxyz, x, y, z = C.fct_field
Rzf, zgen, fgen = C.cover_template.fct_field
Rzf, zgen, fgen, xgen, ygen = C.cover_template.fct_field
if isinstance(elt, group_elt):
elt = elt.as_tuple
AS = self.curve
@ -114,8 +114,8 @@ class as_function:
if elt in G.gens:
ind = G.gens.index(elt)
gp_action_list = C.cover_template.gp_action[ind]
sub_list_gen = {zgen[i] : RxyzQ(z[i]) for i in range(n)}|{fgen[i] : RxyzQ(AS.functions[i].function) for i in range(n)}
sub_list = {x : RxyzQ(x), y : RxyzQ(y)} | {z[j] : RxyzQ(gp_action_list[j].subs(sub_list_gen)) for j in range(n)}
sub_list_gen = {zgen[i] : RxyzQ(z[i]) for i in range(n)}|{fgen[i] : RxyzQ(AS.functions[i].function) for i in range(n)}|{xgen:x}|{ygen:y}
sub_list = {x : RxyzQ(gp_action_list[-2]), y : RxyzQ(gp_action_list[-1])} | {z[j] : RxyzQ(gp_action_list[j].subs(sub_list_gen)) for j in range(n)}
g = self.function
return as_function(C, g.substitute(sub_list))
result = self

18
as_covers/group.sage
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@ -122,4 +122,22 @@ def quaternion_gp():
gens = [(1, 0), (0, 1)]
one = (0, 0)
gp = group(name, short_name, elts, one, mult, inv, gens)
return gp
def hypoelementary_mult(p, m, b, A, B, C, D):
return ((A+C)%m, (b^C*B+D)%p)
def hypoelementary_inv(p, m, b, A, B):
return hypoelementary_mult(p, m, b, 0, p-B, m - A, 0)
def hypoelementary(p, m, b):
'''We want m | p-1 and b to be of order m in F_p.'''
name = "Hypoelementary group Z/"+str(p)+"⋊ Z/"+str(m)+", glued by character 1 -->" + str(b)
short_name = "Z/"+str(p)+"⋊ Z/"+str(m)
elts = [(i, j) for i in range(m) for j in range(p)]
mult = lambda elt1, elt2: hypoelementary_mult(p, m, b, elt1[0], elt1[1], elt2[0], elt2[1])
inv = lambda elt1 : hypoelementary_inv(p, m, b, elt1[0], elt1[1])
gens = [(1, 0), (0, 1)]
one = (0, 0)
gp = group(name, short_name, elts, one, mult, inv, gens)
return gp

80
as_covers/template.sage
View File

@ -10,14 +10,15 @@ class template:
for i in range(n):
variable_names += 'z'+str(i)+','
for i in range(n):
variable_names += 'f'+str(i)
if i!=n-1:
variable_names += ','
Rzf = PolynomialRing(field, 2*n, variable_names)
variable_names += 'f'+str(i)+','
variable_names += 'x, y'
Rzf = PolynomialRing(field, 2*n+2, variable_names)
z = Rzf.gens()[:n]
f = Rzf.gens()[n:]
x = Rzf.gens()[-2]
y = Rzf.gens()[-1]
RzfQ = FractionField(Rzf) #nowa linijka
self.fct_field = RzfQ, z, f #Rzf zmienione na RzfQ
self.fct_field = RzfQ, z, f, x, y #Rzf zmienione na RzfQ
self.fcts = [RzfQ(ff) for ff in fcts] #RHSs of the Artin-Schreier equations
def elementary_template(p, n):
@ -27,15 +28,16 @@ def elementary_template(p, n):
for i in range(n):
variable_names += 'z'+str(i)+','
for i in range(n):
variable_names += 'f'+str(i)
if i!=n-1:
variable_names += ','
R = PolynomialRing(field, 2*n, variable_names)
variable_names += 'f'+str(i) + ','
variable_names += 'x, y'
R = PolynomialRing(field, 2*n+2, variable_names)
z = R.gens()[:n]
f = R.gens()[n:]
x = R.gens()[-2]
y = R.gens()[-1]
height = n
fcts = [f[i] for i in range(n)]
gp_action = [[z[j] + (i == j) for j in range(n)] for i in range(n)]
gp_action = [[z[j] + (i == j) for j in range(n)]+[x, y] for i in range(n)]
return template(height, field, group, fcts, gp_action)
def elementary_cover(list_of_fcts, prec=10):
@ -51,16 +53,17 @@ def heisenberg_template(p):
for i in range(n):
variable_names += 'z'+str(i)+','
for i in range(n):
variable_names += 'f'+str(i)
if i!=n-1:
variable_names += ','
R = PolynomialRing(field, 2*n, variable_names)
variable_names += 'f'+str(i)+','
variable_names += 'x, y'
R = PolynomialRing(field, 2*n+2, variable_names)
z = R.gens()[:n]
f = R.gens()[n:]
x = R.gens()[-2]
y = R.gens()[-1]
height = n
fcts = [f[i] for i in range(n)]
fcts[2] += (z[0] - z[1])*f[1]
gp_action = [[z[0] + 1, z[1], z[2] + z[1]], [z[0] + 1, z[1] + 1, z[2]], [z[0], z[1], z[2] - 1]]
gp_action = [[z[0] + 1, z[1], z[2] + z[1], x, y], [z[0] + 1, z[1] + 1, z[2], x, y], [z[0], z[1], z[2] - 1, x, y]]
return template(height, field, group, fcts, gp_action)
def heisenberg_cover(list_of_fcts, prec=10):
@ -121,12 +124,13 @@ def witt_template(p, n):
for i in range(n):
variable_names += 'z'+str(i)+','
for i in range(n):
variable_names += 'f'+str(i)
if i!=n-1:
variable_names += ','
R = PolynomialRing(field, 2*n, variable_names)
variable_names += 'f'+str(i)+','
variable_names += 'x, y'
R = PolynomialRing(field, 2*n+2, variable_names)
z = R.gens()[:n]
f = R.gens()[n:]
x = R.gens()[-2]
y = R.gens()[-1]
###########
rhs = []
gp_action = []
@ -142,7 +146,7 @@ def witt_template(p, n):
fcts = [-rhs[i] + z[i]^p - z[i] + f[i] for i in range(n)]
########
aux
gp_action = [gp_action]
gp_action = [gp_action+[x, y]]
return template(height, field, group, fcts, gp_action)
def witt_cover(list_of_fcts, prec=10):
@ -158,18 +162,42 @@ def quaternion_template():
for i in range(n):
variable_names += 'z'+str(i)+','
for i in range(n):
variable_names += 'f'+str(i)
if i!=n-1:
variable_names += ','
R = PolynomialRing(field, 2*n, variable_names)
variable_names += 'f'+str(i)+','
variable_names += 'x, y'
R = PolynomialRing(field, 2*n+2, variable_names)
z = R.gens()[:n]
f = R.gens()[n:]
x = R.gens()[-2]
y = R.gens()[-1]
group = quaternion_gp()
fcts = [f[0], f[1], f[2] + z[0]*f[0]+z[1]*(f[0] + f[1])]
gp_action = [[z[0]+1, z[1], z[2] + z[0]], [z[0], z[1] + 1, z[2] + z[1] + z[0]]]
gp_action = [[z[0]+1, z[1], z[2] + z[0], x, y], [z[0], z[1] + 1, z[2] + z[1] + z[0], x, y]]
return template(height, field, group, fcts, gp_action)
def quaternion_cover(list_of_fcts, prec=10):
n = len(list_of_fcts)
C = list_of_fcts[0].curve
return as_cover(C, quaternion_template(), list_of_fcts, branch_points = [], prec = prec)
return as_cover(C, quaternion_template(), list_of_fcts, branch_points = [], prec = prec)
def hypoelementary_template(p, m, b, zeta):
'''unfinished'''
field = GF(p)
height = 1
n = 1
variable_names = ''
for i in range(n):
variable_names += 'z'+str(i)+','
for i in range(n):
variable_names += 'f'+str(i)+','
variable_names += 'x, y'
R = PolynomialRing(field, 2*n+2, variable_names)
z = R.gens()[:n]
f = R.gens()[n:]
x = R.gens()[-2]
y = R.gens()[-1]
group = hypoelementary(p, m, b)
fcts = [1/(zeta - b)*f[0]^p*z[0]^p - 1/(zeta - b)*f[0]*z[0]
gp_action = []
gp_action += [b*z[0]+f[0]*y, x, zeta*y]
gp_action += [z[0]+1, x, y]
return template(height, field, group, fcts, gp_action)