433 lines
17 KiB
Python
433 lines
17 KiB
Python
class as_cover:
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def __init__(self, C, list_of_fcts, branch_points = [], prec = 10):
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self.quotient = C
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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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n = self.height
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from itertools import product
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pr = [list(GF(p)) for _ in range(n)]
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group = []
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for a in product(*pr):
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group += [a]
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self.group = 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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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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power_series = list_of_power_series[j]
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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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def __repr__(self):
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n = self.height
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p = self.characteristic
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if n==1:
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return "(Z/p)-cover of " + str(self.quotient)+" with the equation:\n z^" + str(p) + " - z = " + str(self.functions[0])
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result = "(Z/p)^"+str(self.height)+ "-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) + "^" + str(p) + " - z" + str(i) + " = " + str(self.functions[i]) + "\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):
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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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RxyzQ, Rxyz, x, y, z = self.fct_field
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fx = as_function(self, x)
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z = [as_function(self, zi) for zi in z]
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# We create a list of functions. We add there all variables...
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list_of_fcts = [fx]+z
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vfx = fx.valuation(place)
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vz = [zi.valuation(place) for zi in z]
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# Then we subtract powers of variables with the same valuation (so that 1/t^(kp) cancels) and add to this list.
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for j1 in range(n):
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for j2 in range(n):
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if j1>j2:
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a = gcd(vz[j1] , vz[j2])
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vz1 = vz[j1]/a
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vz2 = vz[j2]/a
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for b in F:
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if (z[j1]^(vz2) - b*z[j2]^(vz1)).valuation(place) > (z[j2]^(vz1)).valuation(place):
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list_of_fcts += [z[j1]^(vz2) - b*z[j2]^(vz1)]
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for j1 in range(n):
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a = gcd(vz[j1], vfx)
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vzj = vz[j1] /a
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vfx = vfx/a
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for b in F:
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if (fx^(vzj) - b*z[j1]^(vfx)).valuation(place) > (z[j1]^(vfx)).valuation(place):
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list_of_fcts += [fx^(vzj) - b*z[j1]^(vfx)]
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#Finally, we check if on the list there are two elements with the same valuation.
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for f1 in list_of_fcts:
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for f2 in list_of_fcts:
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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("My method of generating fcts with relatively prime valuation failed.")
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def ith_ramification_gp(self, i, place = 0):
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'''Find ith ramification group at place at infty of nb place.'''
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G = self.group
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t = self.uniformizer(place)
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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+1:
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Gi += [g]
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return Gi
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def ramification_jumps(self, place = 0):
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'''Return list of lower ramification jumps at at place at infty of nb place.'''
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G = self.group
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ramification_jps = []
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i = 0
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while len(G) > 1:
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Gi = self.ith_ramification_gp(i+1, place)
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if len(Gi) < len(G):
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ramification_jps += [i]
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G = Gi
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i+=1
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return ramification_jps
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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())]
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pr = [list(GF(p)) for _ in range(n)]
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holo = self.holomorphic_differentials_basis(threshold = threshold)
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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(S)
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if len(forms) <= self.genus():
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raise ValueError("Increase threshold!")
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for omega in forms:
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for a in F:
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if (a*omega + fct.diffn()).is_regular_on_U0():
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return a*omega + fct.diffn()
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raise ValueError("Unknown.")
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def de_rham_basis(self, holo_basis = 0, cohomology_basis = 0, threshold = 30):
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if holo_basis == 0:
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holo_basis = self.holomorphic_differentials_basis(threshold = threshold)
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if cohomology_basis == 0:
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cohomology_basis = self.cohomology_of_structure_sheaf_basis(holo_basis = holo_basis, threshold = threshold)
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result = []
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for omega in holo_basis:
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result += [as_cech(self, omega, as_function(self, 0))]
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for f in cohomology_basis:
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omega = self.lift_to_de_rham(f, threshold = threshold)
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result += [as_cech(self, omega, f)]
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return result
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def holomorphic_combinations(S):
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"""Given a list S of pairs (form, corresponding Laurent series at some pt), find their combinations holomorphic at that pt."""
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C_AS = S[0][0].curve
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p = C_AS.characteristic
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F = C_AS.base_ring
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prec = C_AS.prec
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Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
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RtQ = FractionField(Rt)
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minimal_valuation = min([g[1].valuation() for g in S])
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if minimal_valuation >= 0:
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return [s[0] for s in S]
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list_of_lists = [] #to będzie lista złożona z list współczynników część nieholomorficznych rozwinięcia form z S
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for eta, eta_exp in S:
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a = -minimal_valuation + eta_exp.valuation()
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list_coeffs = a*[0] + eta_exp.list() + (-minimal_valuation)*[0]
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list_coeffs = list_coeffs[:-minimal_valuation]
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list_of_lists += [list_coeffs]
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M = matrix(F, list_of_lists)
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V = M.kernel() #chcemy wyzerować części nieholomorficzne, biorąc kombinacje form z S
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# Sprawdzamy, jakim formom odpowiadają elementy V.
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forms = []
|
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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 |