530 lines
21 KiB
Python
530 lines
21 KiB
Python
class as_cover:
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def __init__(self, C, cover_template, list_of_fcts, branch_points = [], prec = 10):
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self.quotient = C
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self.cover_template = cover_template
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self.functions = list_of_fcts
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self.height = len(list_of_fcts)
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F = C.base_ring
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self.base_ring = F
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p = C.characteristic
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self.characteristic = p
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self.prec = prec
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#group acting
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self.height = cover_template.height
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self.group = cover_template.group
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#########
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f = C.polynomial
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m = C.exponent
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r = f.degree()
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delta = GCD(m, r)
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self.nb_of_pts_at_infty = delta
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self.branch_points = list(range(delta)) + branch_points
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Rxy.<x, y> = PolynomialRing(F, 2)
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Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
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Rzf, zgen, fgen = cover_template.fct_field
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all_x_series = {}
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all_y_series = {}
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all_z_series = {}
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all_dx_series = {}
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all_jumps = {}
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for pt in self.branch_points:
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x_series = superelliptic_function(C, x).expansion(pt=pt, prec=prec)
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y_series = superelliptic_function(C, y).expansion(pt=pt, prec=prec)
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z_series = []
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jumps = []
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n = len(list_of_fcts)
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list_of_power_series = [g.expansion(pt=pt, prec=prec) for g in list_of_fcts]
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for j in range(n):
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####
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#
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power_series = Rzf(cover_template.fcts[j]).subs({zgen[i] : z_series[i] for i in range(j)} | {zgen[i] : 0 for i in range(j, n)} | {fgen[i] : list_of_power_series[i] for i in range(n)})
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####
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jump, correction, t_old, z = artin_schreier_transform(power_series, prec = prec)
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x_series = x_series(t = t_old)
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y_series = y_series(t = t_old)
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z_series = [zi(t = t_old) for zi in z_series]
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z_series += [z]
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jumps += [jump]
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list_of_power_series = [g(t = t_old) for g in list_of_power_series]
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all_jumps[pt] = jumps
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all_x_series[pt] = x_series
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all_y_series[pt] = y_series
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all_z_series[pt] = z_series
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all_dx_series[pt] = x_series.derivative()
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self.jumps = all_jumps
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self.x_series = all_x_series
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self.y_series = all_y_series
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self.z_series = all_z_series
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self.dx_series = all_dx_series
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##############
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#Function field
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variable_names = 'x, y'
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for i in range(n):
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variable_names += ', z' + str(i)
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Rxyz = PolynomialRing(F, n+2, variable_names)
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x, y = Rxyz.gens()[:2]
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z = Rxyz.gens()[2:]
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RxyzQ = FractionField(Rxyz)
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self.fct_field = (RxyzQ, Rxyz, x, y, z)
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self.x = as_function(self, x)
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self.y = as_function(self, y)
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self.z = [as_function(self, z[j]) for j in range(n)]
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self.dx = as_form(self, 1)
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self.one = as_function(self, 1)
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Rzf, zgen, fgen = cover_template.fct_field
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subs_fs = {zgen[i] : z[i]}| {fgen[i] : RxyzQ(list_of_fcts[i].function) for i in range(n)}
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self.rhs = [RxyzQ(cover_template.fcts[i].subs(subs_fs)) for i in range(n)]
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#####
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##### We compute now the differentials dz[i]
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y_super = superelliptic_function(C, y)
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dy_super = y_super.diffn().form
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dz = []
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for i in range(n):
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aux_fct = self.rhs[i]
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result = 0
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for j in range(i):
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result += aux_fct.derivative(z[j])*dz[j]
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result += aux_fct.derivative(x)
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result += aux_fct.derivative(y)*dy_super
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dz += [-result]
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self.dz = dz
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def __repr__(self):
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n = self.height
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p = self.characteristic
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result = "("+self.group.short_name+")"
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result += "-cover of " + str(self.quotient)+" with the equations: \n"
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for i in range(n):
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result += 'z'+str(i)+'^p - z'+str(i) + ' = '
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aux = str(self.cover_template.fcts[i])
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for t in range(n):
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aux = aux.replace("f"+str(t), "(" + str(self.functions[t]) + ")")
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result += aux + '\n'
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return result
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def genus(self):
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jumps = self.jumps
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gY = self.quotient.genus()
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n = self.height
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branch_pts = self.branch_points
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p = self.characteristic
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return p^n*gY + (p^n - 1)*(len(branch_pts) - 1) + sum(p^(n-j-1)*(jumps[pt][j]-1)*(p-1)/2 for j in range(n) for pt in branch_pts)
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def exponent_of_different(self, place = 0):
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jumps = self.jumps
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n = self.height
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delta = self.nb_of_pts_at_infty
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p = self.characteristic
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return sum(p^(n-j-1)*(jumps[place][j]+1)*(p-1) for j in range(n))
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def exponent_of_different_prim(self, place = 0):
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jumps = self.jumps
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n = self.height
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delta = self.nb_of_pts_at_infty
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p = self.characteristic
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return sum(p^(n-j-1)*(jumps[place][j])*(p-1) for j in range(n))
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def holomorphic_differentials_basis(self, threshold = 8):
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from itertools import product
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x_series = self.x_series
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y_series = self.y_series
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z_series = self.z_series
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dx_series = self.dx_series
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delta = self.nb_of_pts_at_infty
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p = self.characteristic
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n = self.height
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prec = self.prec
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C = self.quotient
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F = self.base_ring
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m = C.exponent
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r = C.polynomial.degree()
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RxyzQ, Rxyz, x, y, z = self.fct_field
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Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
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#Tworzymy zbiór S form z^i x^j y^k dx/y o waluacji >= waluacja z^(p-1)*dx/y
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S = []
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pr = [list(GF(p)) for _ in range(n)]
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for i in range(0, threshold*r):
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for j in range(0, m):
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for k in product(*pr):
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eta = as_form(self, x^i * prod(z[i1]^(k[i1]) for i1 in range(n))/y^j)
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eta_exp = eta.expansion(pt=self.branch_points[0])
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S += [(eta, eta_exp)]
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forms = holomorphic_combinations(S)
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for pt in self.branch_points[1:]:
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forms = [(omega, omega.expansion(pt=pt)) for omega in forms]
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forms = holomorphic_combinations(forms)
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if len(forms) < self.genus():
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print("I haven't found all forms, only ", len(forms), " of ", self.genus())
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return holomorphic_differentials_basis(self, threshold = threshold + 1)
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if len(forms) > self.genus():
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raise ValueError("Increase precision.")
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return forms
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def cartier_matrix(self, prec=50):
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g = self.genus()
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F = self.base_ring
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M = matrix(F, g, g)
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for i, omega in enumerate(self.holomorphic_differentials_basis()):
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M[:, i] = vector(omega.cartier().coordinates())
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return M
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def at_most_poles(self, pole_order, threshold = 8):
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""" Find fcts with pole order in infty's at most pole_order. Threshold gives a bound on powers of x in the function.
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If you suspect that you haven't found all the functions, you may increase it."""
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from itertools import product
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x_series = self.x_series
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y_series = self.y_series
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z_series = self.z_series
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delta = self.nb_of_pts_at_infty
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p = self.characteristic
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n = self.height
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prec = self.prec
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C = self.quotient
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F = self.base_ring
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m = C.exponent
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r = C.polynomial.degree()
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RxyzQ, Rxyz, x, y, z = self.fct_field
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F = C.base_ring
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Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
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#Tworzymy zbiór S form z^i x^j y^k dx/y o waluacji >= waluacja z^(p-1)*dx/y
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S = []
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RQxyz = FractionField(Rxyz)
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pr = [list(GF(p)) for _ in range(n)]
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for i in range(0, threshold*r):
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for j in range(0, m):
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for k in product(*pr):
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eta = as_function(self, x^i * prod(z[i1]^(k[i1]) for i1 in range(n))*y^j)
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eta_exp = eta.expansion_at_infty()
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S += [(eta, eta_exp)]
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forms = holomorphic_combinations_fcts(S, pole_order)
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for i in range(1, delta):
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forms = [(omega, omega.expansion_at_infty(place = i)) for omega in forms]
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forms = holomorphic_combinations_fcts(forms, pole_order)
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return forms
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def magical_element(self, threshold = 8):
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list_of_elts = self.at_most_poles(self.exponent_of_different_prim(), threshold)
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result = []
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for a in list_of_elts:
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if a.trace().function != 0:
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result += [a]
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return result
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def pseudo_magical_element(self, threshold = 8):
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list_of_elts = self.at_most_poles(self.exponent_of_different(), threshold)
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result = []
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for a in list_of_elts:
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if a.trace().function != 0:
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result += [a]
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return result
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def at_most_poles_forms(self, pole_order, threshold = 8):
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"""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.
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If you suspect that you haven't found all the functions, you may increase it."""
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from itertools import product
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x_series = self.x_series
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y_series = self.y_series
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z_series = self.z_series
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delta = self.nb_of_pts_at_infty
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p = self.characteristic
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n = self.height
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prec = self.prec
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C = self.quotient
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F = self.base_ring
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m = C.exponent
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r = C.polynomial.degree()
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RxyzQ, Rxyz, x, y, z = self.fct_field
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Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
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#Tworzymy zbiór S form z^i x^j y^k dx/y o waluacji >= waluacja z^(p-1)*dx/y
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S = []
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RQxyz = FractionField(Rxyz)
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pr = [list(GF(p)) for _ in range(n)]
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for i in range(0, threshold*r):
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for j in range(0, m):
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for k in product(*pr):
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eta = as_form(self, x^i * prod(z[i1]^(k[i1]) for i1 in range(n))/y^j)
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eta_exp = eta.expansion_at_infty()
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S += [(eta, eta_exp)]
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forms = holomorphic_combinations_forms(S, pole_order)
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for pt in self.branch_points[1:]:
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forms = [(omega, omega.expansion(pt=pt)) for omega in forms]
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forms = holomorphic_combinations_forms(forms, pole_order)
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return forms
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def uniformizer(self, place = 0, threshold = 10):
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'''Return uniformizer of curve self at place-th place at infinity.'''
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p = self.characteristic
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n = self.height
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F = self.base_ring
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list_of_fcts = self.at_most_poles(threshold)
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list_of_fcts2 = self.z + [self.x, self.y]
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for f1 in list_of_fcts:
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for f2 in list_of_fcts2:
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d, a, b = xgcd(f1.valuation(place), f2.valuation(place))
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if d == 1:
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return f1^a*f2^b
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raise ValueError("Increase threshold.")
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def quasiuniformizer(self, place = 0, threshold = 10):
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'''Return an element with valuation coprime to p.'''
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p = self.characteristic
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n = self.height
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F = self.base_ring
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rr_space = self.at_most_poles(threshold)
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list_of_fcts = [ff for ff in rr_space if ff.valuation(place)%p != 0]
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print(len(list_of_fcts))
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list_of_fcts2 = [len(str(ff)) for ff in list_of_fcts]
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i_min = list_of_fcts2.index(min(list_of_fcts2))
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result = list_of_fcts.pop(i_min)
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print(result)
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flag = 1
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while flag == 1:
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flag = 0
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for g in self.group.elts:
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if result.group_action(g) == result and g != self.group.one:
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flag = 1
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if flag == 1:
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list_of_fcts2 = [len(str(ff)) for ff in list_of_fcts]
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i_min = list_of_fcts2.index(min(list_of_fcts2))
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result = list_of_fcts.pop(i_min)
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print(result)
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return result
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def stabilizer(self, place = 0):
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result = []
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for g in self.group.elts:
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flag = 1
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for i in range(self.height):
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if self.z[i].valuation(place = place) > 0:
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fct = self.z[i]
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elif self.z[i].valuation(place = place) < 0:
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fct = self.one/self.z[i]
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if fct.group_action(g).valuation(place = place) <= 0:
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flag = 0
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if flag:
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result += [g]
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return result
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def fiber(self, place = 0):
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'Gives representatives for the quotient G/G_P for given place. Those are in bijection with the fiber.'
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result = [(0, 0, 0)]
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p = self.characteristic
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H = self.stabilizer(place = place)
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for g in self.group.elts:
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flag = 1
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for v in result:
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if heisenberg_mult(g, heisenberg_inv(v, p), p) in H:
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flag = 0
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if flag:
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result += [g]
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return result
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def ith_ramification_gp(self, i, place = 0, quasiuniformizer = 0, threshold = 20):
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'''Find ith ramification group at place at infty of nb place.'''
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G = self.group.elts
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p = self.characteristic
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Gi = [G[0]]
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# list_of_fcts = self.z #[self.one/zz for zz in AS.z if zz.valuation(place) < 0]
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# list_of_fcts += [zz^2 for zz in self.z]
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if isinstance(quasiuniformizer, int) or isinstance(quasiuniformizer, Integer):
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t = self.quasiuniformizer(place, threshold=threshold)
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else:
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t = quasiuniformizer
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for g in G:
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if g != G[0]:
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tg = t.group_action(g)
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v = (tg - t).valuation(place)
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if v >= i + t.valuation(place):
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Gi += [g]
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return Gi
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def ramification_jumps(self, place = 0, quasiuniformizer = 0, threshold = 20):
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'''Return list of lower ramification jumps at at place at infty of nb place.'''
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G = self.stabilizer(place=place)
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p = self.characteristic
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Gi = [G[0]]
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# list_of_fcts = self.z #[self.one/zz for zz in AS.z if zz.valuation(place) < 0]
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# list_of_fcts += [zz^2 for zz in self.z]
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if isinstance(quasiuniformizer, int) or isinstance(quasiuniformizer, Integer):
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t = self.quasiuniformizer(place, threshold=threshold)
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else:
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t = quasiuniformizer
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result = []
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i = 0
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while len(G) > 1:
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Gi = [G[0]]
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for g in G:
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if g != G[0]:
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tg = t.group_action(g)
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v = (tg - t).valuation(place)
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if v >= i + t.valuation(place):
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Gi += [g]
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if len(Gi) < len(G):
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result += [i-1]
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G = Gi
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i+=1
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return result
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def a_number(self):
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g = self.genus()
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return g - self.cartier_matrix().rank()
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def cohomology_of_structure_sheaf_basis(self, holo_basis = 0, threshold = 8):
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if holo_basis == 0:
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holo_basis = self.holomorphic_differentials_basis(threshold = threshold)
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from itertools import product
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x_series = self.x_series
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y_series = self.y_series
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z_series = self.z_series
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delta = self.nb_of_pts_at_infty
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p = self.characteristic
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n = self.height
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prec = self.prec
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C = self.quotient
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F = self.base_ring
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m = C.exponent
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r = C.polynomial.degree()
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RxyzQ, Rxyz, x, y, z = self.fct_field
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Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
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#Tworzymy zbiór S form z^i x^j y^k dx/y o waluacji >= waluacja z^(p-1)*dx/y
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result_fcts = []
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V = VectorSpace(F,self.genus())
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S = V.subspace([])
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RQxyz = FractionField(Rxyz)
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pr = [list(GF(p)) for _ in range(n)]
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i = 0
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while len(result_fcts) < self.genus():
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for j in range(0, m):
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for k in product(*pr):
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f = as_function(self, prod(z[i1]^(k[i1]) for i1 in range(n))/x^i*y^j)
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f_products = [omega.serre_duality_pairing(f) for omega in holo_basis]
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if vector(f_products) not in S:
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S = S+V.subspace([V(f_products)])
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result_fcts += [f]
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i += 1
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return result_fcts
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def lift_to_de_rham(self, fct, threshold = 30):
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'''Given function fct, find form eta regular on affine part such that eta - d(fct) is regular in infty. (Works for one place at infty now)'''
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from itertools import product
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x_series = self.x_series
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y_series = self.y_series
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z_series = self.z_series
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dx_series = self.dx_series
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delta = self.nb_of_pts_at_infty
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p = self.characteristic
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n = self.height
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prec = self.prec
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C = self.quotient
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F = self.base_ring
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m = C.exponent
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|
r = C.polynomial.degree()
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RxyzQ, Rxyz, x, y, z = self.fct_field
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Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
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#Tworzymy zbiór S form z^i x^j y^k dx/y o waluacji >= waluacja z^(p-1)*dx/y
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|
S = [(fct.diffn(), fct.diffn().expansion_at_infty())]
|
|
pr = [list(GF(p)) for _ in range(n)]
|
|
holo = self.holomorphic_differentials_basis(threshold = threshold)
|
|
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)
|
|
if len(forms) <= self.genus():
|
|
raise ValueError("Increase threshold!")
|
|
for omega in forms:
|
|
for a in F:
|
|
if (a*omega + fct.diffn()).is_regular_on_U0():
|
|
return a*omega + fct.diffn()
|
|
raise ValueError("Unknown.")
|
|
|
|
def lift_to_de_rham_form(self, eta, threshold = 20):
|
|
'''Given form eta regular on affine part find fct such that eta - d(fct) is regular in infty. (Works for one place at infty now)'''
|
|
from itertools import product
|
|
x_series = self.x_series
|
|
y_series = self.y_series
|
|
z_series = self.z_series
|
|
dx_series = self.dx_series
|
|
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()
|
|
RxyzQ, Rxyz, x, y, z = self.fct_field
|
|
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 = [(eta, eta.expansion_at_infty())]
|
|
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):
|
|
ff = as_function(self, prod(z[i1]^(k[i1]) for i1 in range(n))/x^i*y^j)
|
|
ff_exp = ff.diffn().expansion_at_infty()
|
|
if ff_exp != 0*ff_exp:
|
|
S += [(ff, ff_exp)]
|
|
for i in range(0, threshold*r):
|
|
for j in range(0, m):
|
|
for k in product(*pr):
|
|
ff = as_function(self, prod(z[i1]^(k[i1]) for i1 in range(n))*x^i*y^j)
|
|
ff_exp = ff.diffn().expansion_at_infty()
|
|
if ff_exp != 0*ff_exp:
|
|
S += [(ff, ff_exp)]
|
|
forms = holomorphic_combinations_mixed(S)
|
|
if len(forms) <= self.genus():
|
|
raise ValueError("Increase threshold!")
|
|
result = []
|
|
for ff in forms:
|
|
if ff[0] != 0*self.dx:
|
|
result += [as_cech(self, ff[0], -ff[1])]
|
|
return result
|
|
raise ValueError("Unknown.")
|
|
|
|
def de_rham_basis(self, holo_basis = 0, cohomology_basis = 0, threshold = 30):
|
|
if holo_basis == 0:
|
|
holo_basis = self.holomorphic_differentials_basis(threshold = threshold)
|
|
if cohomology_basis == 0:
|
|
cohomology_basis = self.cohomology_of_structure_sheaf_basis(holo_basis = holo_basis, threshold = threshold)
|
|
result = []
|
|
for omega in holo_basis:
|
|
result += [as_cech(self, omega, as_function(self, 0))]
|
|
for f in cohomology_basis:
|
|
omega = self.lift_to_de_rham(f, threshold = threshold)
|
|
result += [as_cech(self, omega, f)]
|
|
return result
|
|
|
|
def group_action_matrices_holo(AS, basis=0, threshold=10):
|
|
n = AS.height
|
|
if basis == 0:
|
|
basis = AS.holomorphic_differentials_basis(threshold=threshold)
|
|
F = AS.base_ring
|
|
return as_group_action_matrices(F, basis, AS.group.gens, basis = basis)
|
|
|
|
def group_action_matrices_dR(AS, basis = 0, threshold=8):
|
|
n = AS.height
|
|
if basis == 0:
|
|
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)
|
|
basis = [holo_basis, str_basis, dr_basis]
|
|
return as_group_action_matrices(F, basis[2], AS.group.gens, basis = basis)
|
|
|
|
def group_action_matrices_log_holo(AS):
|
|
n = AS.height
|
|
F = AS.base_ring
|
|
return as_group_action_matrices(F, AS.at_most_poles_forms(1), AS.group.gens, basis = AS.at_most_poles_forms(1)) |