przeniesione pliki 2
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def decomposition_g0_g8(fct, prec = 50):
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'''Writes fct as a difference g0 - g8 + f, with g0 regular on the affine patch and g8 at the points in infinity
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and f is combination of basis of H^1(X, OX). Output is (g0, g8, f).'''
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C = fct.curve
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g = C.genus()
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coord = fct.coordinates()
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nontrivial_part = 0*C.x
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for i, a in enumerate(C.cohomology_of_structure_sheaf_basis()):
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nontrivial_part += coord[i]*a
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fct -= nontrivial_part
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Fxy, Rxy, x, y = C.fct_field
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fct = Fxy(fct.function)
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num = fct.numerator()
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den = fct.denominator()
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aux_den = superelliptic_function(C, Rxy(den))
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g0 = superelliptic_function(C, 0)
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g8 = superelliptic_function(C, 0)
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for monomial in num.monomials():
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aux = superelliptic_function(C, monomial)
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if aux.expansion_at_infty().valuation() >= aux_den.expansion_at_infty().valuation():
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g8 -= num.monomial_coefficient(monomial)*aux/aux_den
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else:
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g0 += num.monomial_coefficient(monomial)*aux/aux_den
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return (g0, g8, nontrivial_part)
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def decomposition_omega0_omega8(omega, prec=50):
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'''Writes omega as a difference omega0 - omega8, with omega0 regular on the affine patch and omega8 at the points in infinity.'''
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C = omega.curve
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omega.form = reduction(C, omega.form)
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F = C.base_ring
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delta = C.nb_of_pts_at_infty
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m = C.exponent
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if sum(omega.residue(place = i, prec = 50) for i in range(delta)) != 0:
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raise ValueError(str(omega) + " has non zero residue!")
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Fxy, Rxy, x, y = C.fct_field
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Rx.<x> = PolynomialRing(F)
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Fx = FractionField(Rx)
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fct = Fxy(omega.form)
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num = fct.numerator()
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den = fct.denominator()
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aux_den = superelliptic_function(C, Rxy(den))
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g0 = superelliptic_function(C, 0)
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g8 = superelliptic_function(C, 0)
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for j in range(0, m):
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component = Fx(omega.jth_component(j))
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q, r = component.numerator().quo_rem(component.denominator())
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g0 += (C.y)^(-j)*superelliptic_function(C, Rxy(q))
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if ((C.y)^(-j)*superelliptic_function(C, Fxy(r/component.denominator()))*C.dx).expansion_at_infty().valuation() < 0:
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raise ValueError("Something went wrong for "+str(omega))
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g8 -= (C.y)^(-j)*superelliptic_function(C, Fxy(r/component.denominator()))
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g0, g8 = g0*C.dx, g8*C.dx
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if g0.is_regular_on_U0():
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return (g0, g8)
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#Rx.<x> = PolynomialRing(F)
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#Rx.<x> = PolynomialRing(F)
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#aux_fct = (g0.form)*y
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else:
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raise ValueError("Something went wrong for "+str(omega) +". Result would be "+str(g0)+ " and " + str(g8))
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def decomposition_g0_g8_pth_power(fct):
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'''Decompose fct as g0 - g8 + A^p, if possible. Output: (g0, g8, A).'''
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coor = fct.coordinates()
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C = fct.curve
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return 0
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def patch(C):
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if C.exponent != 2:
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raise ValueError("Not implemented yet!")
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Fxy, Rxy, x, y = C.fct_field
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F = C.base_ring
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Rx.<x> = PolynomialRing(F)
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f = C.polynomial
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g = C.genus()
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f_star = Rx(x^(2*g+2)*f(1/x))
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return superelliptic(f_star, 2)
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def second_patch(argument):
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C = argument.curve
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C1 = patch(C)
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Fxy, Rxy, x, y = C.fct_field
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g = C.genus()
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if isinstance(argument, superelliptic_function):
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fct = Fxy(argument.function)
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fct1 = Fxy(fct.subs({x : 1/x, y : y/x^(g+1)}))
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return superelliptic_function(C1, fct1)
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if isinstance(argument, superelliptic_form):
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fct = Fxy(argument.form)
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fct1 = Fxy(fct.subs({x : 1/x, y : y/x^(g+1)}))
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fct1 *= -x^(-2)
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return superelliptic_form(C1, fct1)
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def lift_form_to_drw(omega):
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A, B = regular_form(omega)
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A, B = A.change_ring(QQ), B.change_ring(QQ)
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print("%s dx + %s dy"%(A, B))
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class superelliptic_witt:
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def __init__(self, t, f):
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''' Define Witt function on C of the form [t] + V(f). '''
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self.curve = t.curve
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C = t.curve
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p = C.characteristic
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self.t = t #superelliptic_function
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self.f = f #superelliptic_function
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def __repr__(self):
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f = self.f
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t = self.t
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if f.function == 0:
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return "[" + str(t) + "]"
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if t.function == 0:
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return "V(" + str(f) + ")"
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return "[" + str(t) + "] + V(" + str(f) + ")"
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def __neg__(self):
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f = self.f
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t = self.t
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return superelliptic_witt(-t, -f)
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def __add__(self, other):
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C = self.curve
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second_coor = 0*C.x
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X = self.t
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Y = other.t
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for i in range(1, p):
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second_coor -= binomial_prim(p, i)*X^i*Y^(p-i)
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return superelliptic_witt(self.t + other.t, self.f + other.f + second_coor)
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def __sub__(self, other):
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return self + (-other)
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def __rmul__(self, other):
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p = self.curve.characteristic
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if other in ZZ:
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if other == 0:
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return superelliptic_witt(0*C.x, 0*C.x)
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if other > 0:
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return self + (other-1)*self
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if other < 0:
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return (-other)*(-self)
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if other in QQ:
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other_integer = Integers(p^2)(other)
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return other_integer*self
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def __mul__(self, other):
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C = self.curve
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p = C.characteristic
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if isinstance(other, superelliptic_witt):
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t1 = self.t
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f1 = self.f
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t2 = other.t
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f2 = other.f
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return superelliptic_witt(t1*t2, t1^p*f2 + t2^p*f1)
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if isinstance(other, superelliptic_drw_form):
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h1 = other.h1
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h2 = other.h2
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omega = other.omega
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t = self.t
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f = self.f
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aux_form = t^p*omega - h2*t^(p-1)*t.diffn() + f*h1^p*(C.x)^(p-1)*C.dx
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return superelliptic_drw_form(t*h1, aux_form, t^p*h2)
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def __eq__(self, other):
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return self.t == other.t and self.f == other.f
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def diffn(self, dy_w = 0):
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if dy_w == 0:
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dy_w = self.curve.dy_w()
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C = self.curve
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t = self.t
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f = self.f
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fC = C.polynomial
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F = C.base_ring
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Rxy.<x, y> = PolynomialRing(F, 2)
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if t.function == 0:
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return superelliptic_drw_form(0*C.x, 0*C.dx, f)
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t_polynomial = t.function
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num = t_polynomial.numerator()
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den = t_polynomial.denominator()
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num_t_fct = superelliptic_function(C, num)
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den_t_fct = superelliptic_function(C, den)
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inv_den_t_fct = superelliptic_function(C, 1/den)
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if den != 1:
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# d([N/D] + V(f)) = [1/D]*d([N]) - [N]*[D^(-2)]*d([D]) + dV(f)
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return ((den_t_fct)^(-1)).teichmuller()*num_t_fct.teichmuller().diffn() - ((den_t_fct)^(-2)).teichmuller()*num_t_fct.teichmuller()*den_t_fct.teichmuller().diffn() + superelliptic_drw_form(0*C.x, 0*C.dx, f)
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t_polynomial = Rxy(t_polynomial)
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M = t_polynomial.monomials()[0]
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a = t_polynomial.monomial_coefficient(M)
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#[P] = [aM] + Q, where Q = ([P] - [aM])
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aM_fct = superelliptic_function(C, a*M)
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Q = self - aM_fct.teichmuller()
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exp_x = M.exponents()[0][0]
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exp_y = M.exponents()[0][1]
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return Q.diffn() + exp_x*superelliptic_drw_form(aM_fct/C.x, 0*C.dx, 0*C.x) + exp_y*(aM_fct/C.y).teichmuller()*dy_w
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def binomial_prim(p, i):
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return binomial(p, i)/p
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def reduce_rational_fct(fct, p):
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Rxy.<x, y> = PolynomialRing(QQ)
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Fxy = FractionField(Rxy)
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fct = Fxy(fct)
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num = Rxy(fct.numerator())
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den = Rxy(fct.denominator())
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num1 = Rxy(0)
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for m in num.monomials():
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a = num.monomial_coefficient(m)
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num1 += (a%p^2)*m
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den1 = Rxy(0)
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for m in den.monomials():
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a = den.monomial_coefficient(m)
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den1 += (a%p^2)*m
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return num1/den1
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def teichmuller(fct):
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C = fct.curve
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return superelliptic_witt(fct, 0*C.x)
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superelliptic_function.teichmuller = teichmuller
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#dy = [f(x)]'/2*y dx
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#[f1 + M] = [f1] + [M] + V(cos)
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#d[f1 + M] = d[f1] + d[M] + dV(f1*M)
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#M = b x^a
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#d[M] = a*[b x^(a-1)]
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def auxilliary_derivative(P):
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'''Return "derivative" of P, where P depends only on x. In other words d[P(x)].'''
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P0 = P.t.function
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P1 = P.f.function
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C = P.curve
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F = C.base_ring
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Rx.<x> = PolynomialRing(F)
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P0 = Rx(P0)
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P1 = Rx(P1)
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if P0 == 0:
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return superelliptic_drw_form(0*C.x, 0*C.dx, P.f)
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M = P0.monomials()[0]
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a = P0.monomial_coefficient(M)
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#[P] = [aM] + Q, where Q = ([P] - [aM])
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aM_fct = superelliptic_function(C, a*M)
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Q = P - aM_fct.teichmuller()
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exp = M.exponents()[0]
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return auxilliary_derivative(Q) + exp*superelliptic_drw_form(aM_fct/C.x, 0*C.dx, 0*C.x)
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class superelliptic_drw_form:
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def __init__(self, h1, omega, h2):
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'''Form [h1] d[x] + V(omega) + dV([h])'''
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self.curve = h1.curve
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self.h1 = h1
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self.omega = omega
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self.h2 = h2
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def r(self):
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C = self.curve
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h1 = self.h1
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return superelliptic_form(C, h1.function)
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def __eq__(self, other):
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eq1 = (self.h1 == self.h1)
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try:
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H = (self.h2 - other.h2).pth_root()
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except:
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return False
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eq2 = ((self.omega - other.omega).cartier() - H.diffn()) == 0*self.curve.dx
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if eq1 and eq2:
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return True
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return False
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def __repr__(self):
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C = self.curve
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h1 = self.h1
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omega = self.omega
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h2 = self.h2
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result = ""
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if h1.function != 0:
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result += "[" + str(h1) + "] d[x]"
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if h1.function !=0 and omega.form != 0:
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result += " + "
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if omega.form != 0:
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result += "V(" + str(omega) + ")"
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if h2.function !=0 and omega.form != 0:
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result += " + "
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if h2.function != 0:
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result += "dV([" + str(h2) +"])"
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if result == "":
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result += "0"
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return result
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def __rmul__(self, other):
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h1 = self.h1
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h2 = self.h2
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omega = self.omega
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p = self.curve.characteristic
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if other in ZZ:
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if other == 0:
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return superelliptic_drw_form(0*C.x, 0*C.dx, 0*C.x)
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if other > 0:
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return self + (other-1)*self
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if other < 0:
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return (-other)*(-self)
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if other in QQ:
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other_integer = Integers(p^2)(other)
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return other_integer*self
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t = other.t
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f = other.f
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aux_form = t^p*omega - h2*t^(p-1)*t.diffn() + f*h1^p*(C.x)^(p-1)*C.dx
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return superelliptic_drw_form(t*h1, aux_form, t^p*h2)
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def __neg__(self):
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C = self.curve
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h1 = self.h1
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h2 = self.h2
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omega = self.omega
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return superelliptic_drw_form(-h1, -omega, -h2)
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def __sub__(self, other):
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return self + (-other)
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def __add__(self, other):
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C = self.curve
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h1 = self.h1
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h2 = self.h2
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omega = self.omega
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H1 = other.h1
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H2 = other.h2
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OMEGA = other.omega
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aux = (teichmuller(h1) + teichmuller(H1))*superelliptic_drw_form(C.one, 0*C.dx, 0*C.x)
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h1_aux = aux.h1
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h2_aux = aux.h2
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omega_aux = aux.omega
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return superelliptic_drw_form(h1_aux, omega + OMEGA + omega_aux, h2 + H2 + h2_aux)
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def frobenius(self):
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C = self.curve
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h1 = self.h1
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h2 = self.h2
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p = C.characteristic
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return h1^p*C.x^(p-1)*C.dx + h2.diffn()
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def mult_by_p(omega):
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C = omega.curve
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fct = omega.form
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Fxy, Rxy, x, y = C.fct_field
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omega = superelliptic_form(C, fct^p * x^(p-1))
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result = superelliptic_drw_form(0*C.x, omega, 0*C.x)
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return result
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def verschiebung(elt):
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C = elt.curve
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if isinstance(elt, superelliptic_function):
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return superelliptic_witt(0*C.x, elt)
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if isinstance(elt, superelliptic_form):
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return superelliptic_drw_form(0*C.x, elt, 0*C.x)
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superelliptic_form.verschiebung = verschiebung
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superelliptic_function.verschiebung = verschiebung
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class superelliptic_drw_cech:
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def __init__(self, omega0, f):
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self.curve = omega0.curve
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self.omega0 = omega0
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self.omega8 = omega0 - f.diffn()
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self.f = f
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def reduce(self):
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C = self.curve
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fct = self.f
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f_first_comp = fct.t
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f_second_comp = fct.f
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decomp_first_comp = decomposition_g0_g8(f_first_comp)
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decomp_second_comp = decomposition_g0_g8(f_second_comp)
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new = self
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new.omega0 -= decomposition_g0_g8(f_first_comp)[0].teichmuller().diffn()
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new.omega0 -= decomposition_g0_g8(f_second_comp)[0].verschiebung().diffn()
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new.f = decomposition_g0_g8(f_first_comp)[2].teichmuller() + decomposition_g0_g8(f_second_comp)[2].verschiebung()
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new.omega8 = new.omega0 - new.f.diffn()
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return new
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def __repr__(self):
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return("(" + str(self.omega0) + ", "+ str(self.f) + ", " + str(self.omega8) + ")")
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def __add__(self, other):
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C = self.curve
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omega0 = self.omega0
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f = self.f
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omega0_1 = other.omega0
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f_1 = other.f
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return superelliptic_drw_cech(omega0 + omega0_1, f + f_1)
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def __sub__(self, other):
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C = self.curve
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omega0 = self.omega0
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f = self.f
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omega0_1 = other.omega0
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f_1 = other.f
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return superelliptic_drw_cech(omega0 - omega0_1, f - f_1)
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def __neg__(self):
|
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C = self.curve
|
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omega0 = self.omega0
|
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f = self.f
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return superelliptic_drw_cech(-omega0, -f)
|
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|
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def __rmul__(self, other):
|
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omega0 = self.omega0
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f = self.f
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return superelliptic_drw_cech(other*omega0, other*f)
|
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|
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def r(self):
|
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omega0 = self.omega0
|
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f = self.f
|
||||
C = self.curve
|
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return superelliptic_cech(C, omega0.h1*C.dx, f.t)
|
||||
|
||||
def coordinates(self, basis = 0):
|
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C = self.curve
|
||||
g = C.genus()
|
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coord_mod_p = self.r().coordinates()
|
||||
print(coord_mod_p)
|
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coord_lifted = [lift(a) for a in coord_mod_p]
|
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if basis == 0:
|
||||
basis = C.crystalline_cohomology_basis()
|
||||
aux = self
|
||||
for i, a in enumerate(basis):
|
||||
aux -= coord_lifted[i]*a
|
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print('aux before reduce', aux)
|
||||
#aux = aux.reduce() # Now aux = p*cech class.
|
||||
# Now aux should be of the form (V(smth), V(smth), V(smth))
|
||||
print('aux V(smth)', aux)
|
||||
aux_divided_by_p = superelliptic_cech(C, aux.omega0.omega.cartier(), aux.f.f.pth_root())
|
||||
print('aux.omega0.omega.cartier()', aux.omega0.omega.cartier())
|
||||
coord_aux_divided_by_p = aux_divided_by_p.coordinates()
|
||||
coord_aux_divided_by_p = [ZZ(a) for a in coord_aux_divided_by_p]
|
||||
coordinates = [ (coord_lifted[i] + p*coord_aux_divided_by_p[i])%p^2 for i in range(2*g)]
|
||||
return coordinates
|
||||
|
||||
def is_regular(self):
|
||||
print(self.omega0.r().is_regular_on_U0(), self.omega8.r().is_regular_on_Uinfty(), self.omega0.frobenius().is_regular_on_U0(), self.omega8.frobenius().is_regular_on_Uinfty())
|
||||
eq1 = self.omega0.r().is_regular_on_U0() and self.omega8.r().is_regular_on_Uinfty()
|
||||
eq2 = self.omega0.frobenius().is_regular_on_U0() and self.omega8.frobenius().is_regular_on_Uinfty()
|
||||
return eq1 and eq2
|
||||
|
||||
|
||||
def de_rham_witt_lift(cech_class, prec = 50):
|
||||
C = cech_class.curve
|
||||
g = C.genus()
|
||||
omega0 = cech_class.omega0
|
||||
omega8 = cech_class.omega8
|
||||
fct = cech_class.f
|
||||
omega0_regular = regular_form(omega0) #Present omega0 in the form P dx + Q dy
|
||||
print('omega0_regular', omega0_regular)
|
||||
omega0_lift = omega0_regular[0].teichmuller()*(C.x.teichmuller().diffn()) + omega0_regular[1].teichmuller()*(C.y.teichmuller().diffn())
|
||||
#Now the obvious lift of omega0 = P dx + Q dy to de Rham-Witt is [P] d[x] + [Q] d[y]
|
||||
print('omega8', omega8, 'second_patch(omega8)', second_patch(omega8))
|
||||
omega8_regular = regular_form(second_patch(omega8)) # The same for omega8.
|
||||
print('omega8_regular 1', omega8_regular)
|
||||
omega8_regular = (second_patch(omega8_regular[0]), second_patch(omega8_regular[1]))
|
||||
print('omega8_regular 2', omega8_regular)
|
||||
u = (C.x)^(-1)
|
||||
v = (C.y)/(C.x)^(g+1)
|
||||
omega8_lift = omega8_regular[0].teichmuller()*(u.teichmuller().diffn()) + omega8_regular[1].teichmuller()*(v.teichmuller().diffn())
|
||||
aux = omega0_lift - omega8_lift - fct.teichmuller().diffn() # now aux is of the form (V(smth) + dV(smth), V(smth))
|
||||
return aux
|
||||
if aux.h1.function != 0:
|
||||
raise ValueError('Something went wrong - aux is not of the form (V(smth) + dV(smth), V(smth)).')
|
||||
decom_aux_h2 = decomposition_g0_g8(aux.h2, prec=prec) #decompose dV(smth) in aux as smth regular on U0 - smth regular on U8.
|
||||
aux_h2 = decom_aux_h2[0]
|
||||
aux_f = decom_aux_h2[2]
|
||||
aux_omega0 = decomposition_omega0_omega8(aux.omega, prec=prec)[0]
|
||||
result = superelliptic_drw_cech(omega0_lift - aux_h2.verschiebung().diffn() - aux_omega0.verschiebung(), fct.teichmuller() + aux_f.verschiebung())
|
||||
return result.reduce()
|
||||
|
||||
def crystalline_cohomology_basis(self, prec = 50):
|
||||
result = []
|
||||
for a in self.de_rham_basis():
|
||||
result += [de_rham_witt_lift(a, prec = prec)]
|
||||
return result
|
||||
|
||||
superelliptic.crystalline_cohomology_basis = crystalline_cohomology_basis
|
||||
|
||||
def autom(self):
|
||||
C = self.curve
|
||||
F = C.base_ring
|
||||
Rxy.<x, y> = PolynomialRing(F, 2)
|
||||
Fxy = FractionField(Rxy)
|
||||
if isinstance(self, superelliptic_function):
|
||||
result = superelliptic_function(C, Fxy(self.function).subs({x:x+1, y:y}))
|
||||
return result
|
||||
if isinstance(self, superelliptic_form):
|
||||
result = superelliptic_form(C, Fxy(self.form).subs({x:x+1, y:y}))
|
||||
return result
|
||||
if isinstance(self, superelliptic_witt):
|
||||
result = superelliptic_witt(autom(self.t), autom(self.f))
|
||||
return result
|
||||
if isinstance(self, superelliptic_drw_form):
|
||||
result = superelliptic_drw_form(0*C.x, autom(self.omega), autom(self.h2))
|
||||
result += autom(self.h1).teichmuller()*(C.x + C.one).teichmuller().diffn()
|
||||
return result
|
||||
if isinstance(self, superelliptic_drw_cech):
|
||||
result = superelliptic_drw_cech(autom(self.omega0), autom(self.f))
|
||||
return result
|
||||
|
||||
|
||||
def dy_w(C):
|
||||
'''Return d[y].'''
|
||||
fC = C.polynomial
|
||||
fC = superelliptic_function(C, fC)
|
||||
fC = fC.teichmuller()
|
||||
dy_w = 1/2* ((C.y)^(-1)).teichmuller()*auxilliary_derivative(fC)
|
||||
return dy_w
|
||||
superelliptic.dy_w = dy_w
|
||||
Loading…
Reference in New Issue