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trying to fix superelliptic cech coordinates

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This commit is contained in:
Jędrzej Garnek 2024-12-21 17:27:41 +01:00
parent fc5810d546
commit 85b0edd5d7
4 changed files with 195 additions and 10 deletions
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2
as_covers/as_form_class.sage
View File

@ -84,7 +84,6 @@ class as_form:
def coordinates(self, basis = 0):
"""Find coordinates of the given holomorphic form self in terms of the basis forms in a list holo."""
self = self.reduce()
print(self)
C = self.curve
if basis == 0:
basis = C.holomorphic_differentials_basis()
@ -94,7 +93,6 @@ class as_form:
denom = LCM([denominator(omega.form) for omega in basis])
basis = [denom*omega for omega in basis]
self_with_no_denominator = denom*self
print(denom, basis, self_with_no_denominator)
return linear_representation_polynomials(Rxyz(self_with_no_denominator.form), [Rxyz(omega.form) for omega in basis])
def trace(self, super=True):

166
drafty/mumford_curve.sage Normal file
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@ -0,0 +1,166 @@
def mumford_gp(p):
'''We want m | p-1 and b to be of order m in F_p.'''
name = "Mumford group (Z/"+str(p)+")^2⋊ D"+str(p-1)
short_name = "(Z/"+str(p)+")^2 ⋊ D"+str(p-1)
elts = [(i, j, s, t) for i in range(p) for j in range(p) for s in range(1) for t in range(p-1)]
mult = lambda elt1, elt2: (0, 0, 0, 0)
inv = lambda elt1 : (0, 0, 0, 0)
gens = [(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0,0,0,1)]
one = (0, 0, 0, 0)
gp = group(name, short_name, elts, one, mult, inv, gens)
return gp
def mumford_template(p):
group = mumford_gp(p)
field = GF(p)
n = 2
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]
height = n
fcts = [x + y, x - y]
gp_action = [[z[j] + (i == j) for j in range(n)]+[x, y] for i in range(n)]
gp_action += [[z[1], z[0], -x, y]]
d = field(primitive_root(p))
gp_action += [[d*z[0], d^(-1)*z[1], 1/d*(y+x) - d*(y-x), 1/d*(y+x) + d*(y-x)]]
return template(height, field, group, fcts, gp_action)
def mumford_cover(p, prec=10):
field = GF(p)
R.<x> = PolynomialRing(field)
C = superelliptic(x^2 + 1, 2)
return as_cover(C, mumford_template(p), [C.x, C.x], branch_points = [], prec = prec)
ASM = mumford_cover(3, prec = 400)
###############
###############
def metacyclic_gp(p, c):
name = "metacyclic group Z/"+str(p)+"⋊ Z/"+str(c)
short_name = "Z/"+str(p)+" ⋊ Z/"+str(c)
elts = [(i, j) for i in range(p) for j in range(c)]
mult = lambda elt1, elt2: (0, 0)
inv = lambda elt1 : (0, 0)
gens = [(1, 0), (0, 1)]
one = (0, 0)
gp = group(name, short_name, elts, one, mult, inv, gens)
return gp
def metacyclic_template(p, m, nn):
group = metacyclic_gp(p, m*(p-1))
N = Integers(m*(p-1))(p).multiplicative_order()
field.<a> = GF(p^N)
zeta = a^((p^N - 1)/((p-1)*m))
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]
height = n
fcts = [x]
gp_action = [[z[j] + (i == j) for j in range(n)]+[x, y] for i in range(n)]
gp_action += [[zeta^m * z[0], zeta^m * x, zeta * y]]
return template(height, field, group, fcts, gp_action)
def metacyclic_cover(p, m, nn, prec=10):
N = Integers(m*(p-1))(p).multiplicative_order()
field.<a> = GF(p^N)
R.<x> = PolynomialRing(field)
C = superelliptic(sum(x^(p^i) for i in range(0, nn)), m)
return as_cover(C, metacyclic_template(p, m, nn), [C.x], branch_points = [], prec = prec)
p = 3
m = 2
nn = 2
N = Integers(m*(p-1))(p).multiplicative_order()
field.<a> = GF(p^N)
zeta = a^((p^N - 1)/((p-1)*m))
AS = metacyclic_cover(p, m, nn, prec=100)
Bh = AS.holomorphic_differentials_basis(threshold = 30)
Bs = AS.cohomology_of_structure_sheaf_basis(threshold = 30)
BdR = AS.de_rham_basis(threshold = 30)
print(Bh[2].group_action((1, 0)).coordinates(basis = Bh))
#H = CyclicPermutationGroup(4)
#alpha = PermutationGroupMorphism(A,A,[A.gens()[0]^3*A.gens()[1],A.gens()[1]])
#phi = [[(1,2)],[alpha]]
p = 3
m = 2
nn = 2
N = Integers(m*(p-1))(p).multiplicative_order()
field.<a> = GF(p^N)
zeta = a^((p^N - 1)/((p-1)*m))
def gp_action(fct, elt):
if elt == (1, 0):
if isinstance(fct, superelliptic_function):
f = fct.function
C = fct.curve
Fxy, Rxy, x, y = C.fct_field
f = f.subs({x: x+1, y: y})
return superelliptic_function(C, f)
if isinstance(fct, superelliptic_form):
f = fct.form
C = fct.curve
Fxy, Rxy, x, y = C.fct_field
f = f.subs({x: x+1, y: y})
return superelliptic_form(C, f)
if isinstance(fct, superelliptic_cech):
omega = fct.omega0
ff = fct.f
C = fct.curve
return superelliptic_cech(C, gp_action(omega, elt), gp_action(ff, elt))
if elt == (0, 1):
if isinstance(fct, superelliptic_function):
f = fct.function
C = fct.curve
Fxy, Rxy, x, y = C.fct_field
f = f.subs({x : zeta^m*x, y:zeta*y})
return superelliptic_function(C, f)
if isinstance(fct, superelliptic_form):
f = fct.form
C = fct.curve
Fxy, Rxy, x, y = C.fct_field
f = f.subs({x : zeta^m*x, y:zeta*y})
return superelliptic_form(C, f*zeta^m)
if isinstance(fct, superelliptic_cech):
omega = fct.omega0
ff = fct.f
C = fct.curve
return superelliptic_cech(C, gp_action(omega, elt), gp_action(ff, elt))
superelliptic_function.group_action = gp_action
superelliptic_form.group_action = gp_action
superelliptic_cech.group_action = gp_action
R.<x> = PolynomialRing(field)
C = superelliptic(x^(p^nn) - x, m)
Bh = C.holomorphic_differentials_basis()
Bs = C.cohomology_of_structure_sheaf_basis()
BdR = C.de_rham_basis()
mat = as_group_action_matrices(field, BdR, [(1, 0), (0, 1)], BdR)

9
superelliptic/superelliptic_cech_class.sage
View File

@ -38,7 +38,8 @@ class superelliptic_cech:
Rx.<x> = PolynomialRing(F)
return superelliptic_cech(C, superelliptic_form(C, Rx(0)), superelliptic_function(C, fct^p))
def coordinates(self):
def coordinates(self, basis = 0):
print('coord', self, self.omega8.valuation())
C = self.curve
F = C.base_ring
m = C.exponent
@ -46,7 +47,8 @@ class superelliptic_cech:
Fx = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)
g = C.genus()
basis = C.de_rham_basis()
if basis == 0:
basis = C.de_rham_basis()
omega = self.omega0
fct = self.f
@ -55,6 +57,7 @@ class superelliptic_cech:
return vector((2*g)*[0])
if fct.function == Rx(0) and omega.form != Rx(0):
print('A')
result = list(omega.coordinates()) + g*[0]
result = vector([F(a) for a in result])
return result
@ -66,9 +69,11 @@ class superelliptic_cech:
for i in range(g, 2*g):
aux -= coord[i]*basis[i]
aux_f = decomposition_g0_g8(aux.f)[0]
print('aux_f', aux_f, 'aux.f', aux.f)
aux.omega0 -= aux_f.diffn()
aux.f = 0*C.x
aux.omega8 = aux.omega0
print('B', aux)
return coord + aux.coordinates()
def is_cocycle(self):

28
superelliptic/superelliptic_class.sage
View File

@ -92,27 +92,37 @@ class superelliptic:
f = self.polynomial
m = self.exponent
r = f.degree()
genus = self.genus()
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 = []
basis = 2*genus*[0]
index = 0
#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))]
basis[index] = superelliptic_cech(self, basis_holo[k], superelliptic_function(self, 0))
index += 1
## Next elements do not come from holomorphic differentials.
t = len(basis)
degrees0 = {}
degrees1 = {}
degrees = self.degrees_holomorphic_differentials()
degrees_inv = {b:a for a, b in degrees.items()}
for j in range(1, m):
for i in range(1, r):
if (r*(m-j) - m*i >= delta):
#########
index = degrees_inv[(i-1, m-j)]
fct = superelliptic_function(self, Fxy(m*y^(m-j)/x^i))
constant = self.holomorphic_differentials_basis()[index].serre_duality_pairing(fct)
#######
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)))]
basis[index+genus] = 1/constant*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
@ -216,14 +226,20 @@ class superelliptic:
r = f.degree()
F = self.base_ring
delta = self.nb_of_pts_at_infty
g = self.genus()
Rx.<x> = PolynomialRing(F)
Rxy.<x, y> = PolynomialRing(F, 2)
Fxy = FractionField(Rxy)
basis = []
basis = g*[0]
degrees = self.degrees_holomorphic_differentials()
degrees_inv = {b:a for a, b in degrees.items()}
for j in range(1, m):
for i in range(1, r):
if (r*(m-j) - m*i >= delta):
basis += [superelliptic_function(self, Fxy(m*y^(m-j)/x^i))]
if (r*(m-j) - m*i >= delta):
index = degrees_inv[(i-1, m-j)]
fct = superelliptic_function(self, Fxy(m*y^(m-j)/x^i))
constant = self.holomorphic_differentials_basis()[index].serre_duality_pairing(fct)
basis[index] = 1/constant*fct
return basis
def uniformizer(self):