baza crys dziala; poprawki w coordinates dr; wspolrzednie cris prawie dzialaja
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sage/.run.term-0.term
16036
sage/.run.term-0.term
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@ -0,0 +1,11 @@
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p = 5
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m = 2
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F = GF(p)
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Rx.<x> = PolynomialRing(F)
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f = x^3 + x + 1
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C = superelliptic(f, m)
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g = f(x^p - x)
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C1 = superelliptic(g, m)
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ff = superelliptic_function(C, x)
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AS = as_cover(C, [ff])
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print(C1.cartier_matrix().rank() == AS.cartier_matrix().rank())
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@ -15,8 +15,14 @@ C = superelliptic(f, m)
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#b = C.crystalline_cohomology_basis()
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#print(autom(b[0]).coordinates(basis = b))
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#eta1 = (dy + dV(2xy) + V(x^5 \, dy), V(y/x))
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eta1 = superelliptic_drw_cech(C.y.teichmuller().diffn() + (2*C.x*C.y).verschiebung().diffn() + (C.x^5*C.y.diffn()).verschiebung(), (C.y/C.x).verschiebung())
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#eta1 = superelliptic_drw_cech(C.y.teichmuller().diffn() + (2*C.x*C.y).verschiebung().diffn() + (C.x^5*C.y.diffn()).verschiebung(), (C.y/C.x).verschiebung())
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#eta2 = ( x \, dy + 3 x^3 \, dy + dV((2x^4 + 2x^2 + 2) y) + V( (x^4 + x^2 + 1) dy), -[y/x])
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eta2 = superelliptic_drw_cech(C.x.teichmuller()*(C.y.teichmuller()).diffn() + ((2*C.x^4 + 2*C.x^2 + 2*C.one) * C.y).verschiebung().diffn(), - (C.y/C.x).teichmuller())
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aux = de_rham_witt_lift(C.de_rham_basis()[1])
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print(aux)
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#eta2 = superelliptic_drw_cech(C.x.teichmuller()*(C.y.teichmuller()).diffn() + ((2*C.x^4 + 2*C.x^2 + 2*C.one) * C.y).verschiebung().diffn(), - (C.y/C.x).teichmuller())
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#omega8_lift0, compare = de_rham_witt_lift(C.de_rham_basis()[1])
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#omega8_lift = -(C.x^(-3)).teichmuller()*C.y.teichmuller().diffn() + 2*C.y.teichmuller()*(C.x^(-4)).teichmuller()*C.x.teichmuller().diffn()
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#eta2 = de_rham_witt_lift(C.de_rham_basis()[1])
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#b = autom(eta2)
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#print(autom(C.crystalline_cohomology_basis()[1]).coordinates())
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@ -18,6 +18,7 @@ load('superelliptic_drw/decomposition_into_g0_g8.sage')
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load('superelliptic_drw/superelliptic_witt.sage')
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load('superelliptic_drw/superelliptic_drw_form.sage')
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load('superelliptic_drw/superelliptic_drw_cech.sage')
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load('superelliptic_drw/regular_form.sage')
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load('superelliptic_drw/de_rham_witt_lift.sage')
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load('superelliptic_drw/automorphism.sage')
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load('auxilliaries/reverse.sage')
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@ -26,8 +27,6 @@ load('auxilliaries/linear_combination_polynomials.sage')
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##############
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##############
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load('drafty/convert_superelliptic_into_AS.sage')
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load('drafty/regular_on_U0.sage')
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load('drafty/lift_to_de_rham.sage')
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load('drafty/draft.sage')
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#load('drafty/draft_klein_covers.sage')
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#load('drafty/2gpcovers.sage')
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@ -46,52 +46,30 @@ class superelliptic_cech:
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Fx = FractionField(Rx)
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FxRy.<y> = PolynomialRing(Fx)
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g = C.genus()
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degrees_holo = C.degrees_holomorphic_differentials()
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degrees_holo_inv = {b:a for a, b in degrees_holo.items()}
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degrees0 = C.degrees_de_rham0()
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degrees0_inv = {b:a for a, b in degrees0.items()}
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degrees1 = C.degrees_de_rham1()
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degrees1_inv = {b:a for a, b in degrees1.items()}
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basis = C.de_rham_basis()
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omega = self.omega0
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fct = self.f
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if fct.function == Rx(0) and omega.form == Rx(0):
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return vector((2*g)*[0])
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if fct.function == Rx(0) and omega.form != Rx(0):
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for j in range(1, m):
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omega_j = Fx(omega.jth_component(j))
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if omega_j != Fx(0):
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d = degree_of_rational_fctn(omega_j, F)
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index = degrees_holo_inv[(d, j)]
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a = coeff_of_rational_fctn(omega_j, F)
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a1 = coeff_of_rational_fctn(basis[index].omega0.jth_component(j), F)
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elt = self - (a/a1)*basis[index]
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return elt.coordinates() + a/a1*vector([F(i == index) for i in range(0, 2*g)])
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for j in range(1, m):
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fct_j = Fx(fct.jth_component(j))
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if (fct_j != Rx(0)):
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d = degree_of_rational_fctn(fct_j, F)
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if (d, j) in degrees1.values():
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index = degrees1_inv[(d, j)]
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a = coeff_of_rational_fctn(fct_j, F)
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elt = self - (a/m)*basis[index]
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return elt.coordinates() + a/m*vector([F(i == index) for i in range(0, 2*g)])
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if d<0:
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a = coeff_of_rational_fctn(fct_j, F)
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h = superelliptic_function(C, FxRy(a*y^j*x^d))
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elt = superelliptic_cech(C, self.omega0, self.f - h)
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return elt.coordinates()
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if (fct_j != Rx(0)):
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G = superelliptic_function(C, y^j*x^d)
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a = coeff_of_rational_fctn(fct_j, F)
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elt =self - a*superelliptic_cech(C, diffn(G), G)
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return elt.coordinates()
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return vector(2*g*[0])
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result = list(omega.coordinates()) + g*[0]
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result = vector([F(a) for a in result])
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return result
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coord = fct.coordinates()
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coord = g*[0] + list(coord)
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coord = vector([F(a) for a in coord])
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aux = self
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for i in range(g, 2*g):
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aux -= coord[i]*basis[i]
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aux_f = decomposition_g0_g8(aux.f)[0]
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aux.omega0 -= aux_f.diffn()
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aux.f = 0*C.x
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aux.omega8 = aux.omega0
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return coord + aux.coordinates()
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def is_cocycle(self):
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w0 = self.omega0
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@ -196,9 +196,9 @@ class superelliptic:
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Fxy = FractionField(Rxy)
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basis = []
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for j in range(1, m):
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for i in range(1, r):
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if (r*j - m*i >= delta):
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basis += [superelliptic_function(self, Fxy(m*y^(j)/x^i))]
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for i in range(1, r):
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if (r*(m-j) - m*i >= delta):
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basis += [superelliptic_function(self, Fxy(m*y^(m-j)/x^i))]
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return basis
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#Auxilliary. Given a superelliptic curve C : y^m = f(x) and a polynomial g(x, y)
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@ -6,7 +6,12 @@ class superelliptic_form:
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g = Fxy(reduction_form(C, g))
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self.form = g
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self.curve = C
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def __eq__(self, other):
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if self.reduce().form == other.reduce().form:
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return True
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return False
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def __add__(self, other):
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C = self.curve
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g1 = self.form
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@ -37,9 +42,6 @@ class superelliptic_form:
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omega = self.form
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return superelliptic_form(C, constant*omega)
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def __eq__(self, other):
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return self.form == other.form
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def cartier(self):
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'''Computes Cartier operator on the form. Idea: y^m = f(x) -> y^(p^r - 1) = f(x)^M, where r = ord_p(m),
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M = (p^r - 1)/m. Thus h(x)/y^j dx = h(x) f(x)^(M*j)/y^(p^r * j) dx. Thus C(h(x)/y^j dx) = 1/y^(p^(r-1)*j) C(h(x) f(x)^(M*j) dx).'''
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@ -52,7 +54,7 @@ class superelliptic_form:
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Fx = FractionField(Rx)
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FxRy.<y> = PolynomialRing(Fx)
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Fxy = FractionField(FxRy)
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result = superelliptic_form(C, FxRy(0))
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result = 0*C.dx
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mult_order = Integers(m)(p).multiplicative_order()
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M = Integer((p^(mult_order)-1)/m)
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@ -111,8 +113,8 @@ class superelliptic_form:
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Rx.<x> = PolynomialRing(F)
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for j in range(0, m):
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if self.jth_component(j) not in Rx:
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return 0
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return 1
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return False
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return True
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def is_regular_on_Uinfty(self):
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C = self.curve
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@ -128,8 +130,8 @@ class superelliptic_form:
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A = self.jth_component(j)
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d = degree_of_rational_fctn(A, F)
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if(-d*M + j*R -(M+1)<0):
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return 0
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return 1
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return False
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return True
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def expansion_at_infty(self, place = 0, prec=10):
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g = self.form
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@ -166,4 +168,38 @@ class superelliptic_form:
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Rxy.<x, y> = PolynomialRing(F, 2)
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Fxy = FractionField(Rxy)
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fct = reduction(C, Fxy(y^m*fct))
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return superelliptic_form(C, fct/y^m)
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return superelliptic_form(C, fct/y^m)
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def int(self):
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'''Computes an "integral" of a form dg. Idea: y^m = f(x) -> y^(p^r - 1) = f(x)^M, where r = ord_p(m),
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M = (p^r - 1)/m. Thus h(x)/y^j dx = h(x) f(x)^(M*j)/y^(p^r * j) dx. Thus int(h(x)/y^j dx) = 1/y^(p^(r-1)*j) int(h(x) f(x)^(M*j) dx).'''
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C = self.curve
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m = C.exponent
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p = C.characteristic
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f = C.polynomial
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F = C.base_ring
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Rx.<x> = PolynomialRing(F)
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Fx = FractionField(Rx)
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FxRy.<y> = PolynomialRing(Fx)
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Fxy = FractionField(FxRy)
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result = 0*C.x
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mult_order = Integers(m)(p).multiplicative_order()
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M = Integer((p^(mult_order)-1)/m)
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for j in range(0, m):
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fct_j = self.jth_component(j)
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h = Fx(fct_j*f^(M*j))
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h_denom = h.denominator()
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h *= (h_denom)^(p)
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h = Rx(h)
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j1 = (p^(mult_order)*j)%m
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B = floor(p^(mult_order)*j/m)
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result += superelliptic_function(C, h.integral()/(f^(B)*y^(j1)*h_denom^p))
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return result
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def inv_cartier(omega):
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'''If omega is regular, return form eta such that Cartier(eta) = omega'''
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omega_regular = omega.regular_form()
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C = omega.curve
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p = C.characteristic
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return (omega_regular.dx)^p*C.x^(p-1)*C.dx + (omega_regular.dy)^p*C.y^(p-1)*C.y.diffn()
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@ -0,0 +1,5 @@
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F = GF(67)
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P.<x>= PolynomialRing(F)
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X = HyperellipticCurve(x^7 + x^3 + x)
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C = superelliptic(x^7 + x^3 + x, 2)
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print(X.a_number() == C.a_number())
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@ -0,0 +1,12 @@
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p = 7
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m = 4
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F = GF(p)
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Rx.<x> = PolynomialRing(F)
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f = x^5 + x
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C = superelliptic(f, m)
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bbb = C.holomorphic_differentials_basis()
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v = [GF(p).random_element() for _ in range(C.genus())]
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aaa = 0*C.dx
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for i in range(C.genus()):
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aaa += v[i]*bbb[i]
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print(vector(aaa.coordinates()) == vector(v))
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print("Nie działa!")
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F = GF(67)
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P.<x>= PolynomialRing(F)
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X = HyperellipticCurve(x^7 + x^3 + x)
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C = superelliptic(x^7 + x^3 + x, 2)
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print(X.p_rank() == C.p_rank())
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@ -0,0 +1,11 @@
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p = 3
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m = 4
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F = GF(p)
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Rx.<x> = PolynomialRing(F)
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f = x^5 + x
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C = superelliptic(f, m)
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g = (C.x)^5 * (C.y)^2 + 2*(C.x)^2 * (C.y)^3
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g = g^p
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print(g.pth_root()==(C.x)^5 * (C.y)^2 + 2*(C.x)^2 * (C.y)^3)
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g = C.x
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print(g.pth_root())
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@ -11,6 +11,9 @@ def autom(self):
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if isinstance(self, superelliptic_form):
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result = superelliptic_form(C, Fxy(self.form).subs({x:x+1, y:y}))
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return result
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if isinstance(self, superelliptic_cech):
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result = superelliptic_cech(C, autom(self.omega0), autom(self.f))
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return result
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if isinstance(self, superelliptic_witt):
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result = superelliptic_witt(autom(self.t), autom(self.f))
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return result
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@ -1,31 +1,50 @@
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def de_rham_witt_lift_form0(omega):
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C = omega.curve
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omega_regular = omega.regular_form() #Present omega0 in the form P dx + Q dy
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#Now the obvious lift of omega0 = P dx + Q dy to de Rham-Witt is [P] d[x] + [Q] d[y]
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return omega_regular.dx.teichmuller()*(C.x.teichmuller().diffn()) + omega_regular.dy.teichmuller()*(C.y.teichmuller().diffn())
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def de_rham_witt_lift_form8(omega):
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C = omega.curve
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g = C.genus()
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omega_regular = second_patch(omega).regular_form()
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omega_regular = (second_patch(omega_regular.dx), second_patch(omega_regular.dy))
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u = (C.x)^(-1)
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v = (C.y)/(C.x)^(g+1)
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omega_lift = omega_regular[0].teichmuller()*(u.teichmuller().diffn()) + omega_regular[1].teichmuller()*(v.teichmuller().diffn())
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return omega_lift
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def de_rham_witt_lift(cech_class, prec = 50):
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C = cech_class.curve
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g = C.genus()
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omega0 = cech_class.omega0
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omega8 = cech_class.omega8
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fct = cech_class.f
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omega0_regular = regular_form(omega0) #Present omega0 in the form P dx + Q dy
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print('omega0_regular', omega0_regular)
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omega0_lift = omega0_regular[0].teichmuller()*(C.x.teichmuller().diffn()) + omega0_regular[1].teichmuller()*(C.y.teichmuller().diffn())
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#Now the obvious lift of omega0 = P dx + Q dy to de Rham-Witt is [P] d[x] + [Q] d[y]
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print('omega8', omega8, 'second_patch(omega8)', second_patch(omega8))
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omega8_regular = regular_form(second_patch(omega8)) # The same for omega8.
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print('omega8_regular 1', omega8_regular)
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omega8_regular = (second_patch(omega8_regular[0]), second_patch(omega8_regular[1]))
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print('omega8_regular 2', omega8_regular)
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u = (C.x)^(-1)
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v = (C.y)/(C.x)^(g+1)
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omega8_lift = omega8_regular[0].teichmuller()*(u.teichmuller().diffn()) + omega8_regular[1].teichmuller()*(v.teichmuller().diffn())
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omega0_lift = de_rham_witt_lift_form0(omega0)
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omega8_lift = de_rham_witt_lift_form8(omega8)
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print('omega0_lift, omega8_lift', omega0_lift, omega8_lift)
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aux = omega0_lift - omega8_lift - fct.teichmuller().diffn() # now aux is of the form (V(smth) + dV(smth), V(smth))
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return aux
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#return aux
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if aux.h1.function != 0:
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raise ValueError('Something went wrong - aux is not of the form (V(smth) + dV(smth), V(smth)).')
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decom_aux_h2 = decomposition_g0_g8(aux.h2, prec=prec) #decompose dV(smth) in aux as smth regular on U0 - smth regular on U8.
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aux_h2 = decom_aux_h2[0]
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aux_f = decom_aux_h2[2]
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aux_omega0 = decomposition_omega0_omega8(aux.omega, prec=prec)[0]
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result = superelliptic_drw_cech(omega0_lift - aux_h2.verschiebung().diffn() - aux_omega0.verschiebung(), fct.teichmuller() + aux_f.verschiebung())
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return result.reduce()
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result = superelliptic_drw_cech(omega0_lift - aux_omega0.verschiebung(), fct.teichmuller() + aux_h2.verschiebung() + aux_f.verschiebung())
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compare = omega8_lift-decom_aux_h2[1].verschiebung().diffn() - decomposition_omega0_omega8(aux.omega, prec=prec)[1].verschiebung()
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print("result.omega8 == compare", result.omega8 == compare)
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print("result.omega8 - compare", result.omega8 - compare)
|
||||
|
||||
#print('test:', omega0_lift - omega8_lift - fct.teichmuller().diffn() == decom_aux_h2[0].verschiebung().diffn() - decom_aux_h2[1].verschiebung().diffn() + decomposition_omega0_omega8(aux.omega, prec=prec)[0].verschiebung() - decomposition_omega0_omega8(aux.omega, prec=prec)[1].verschiebung())
|
||||
#print('test 1:', omega0_lift - decom_aux_h2[0].verschiebung().diffn() - decomposition_omega0_omega8(aux.omega, prec=prec)[0].verschiebung() - fct.teichmuller().diffn() == omega8_lift - decom_aux_h2[1].verschiebung().diffn() - decomposition_omega0_omega8(aux.omega, prec=prec)[1].verschiebung())
|
||||
#A = omega0_lift - decomposition_omega0_omega8(aux.omega, prec=prec)[0].verschiebung()
|
||||
#B = decom_aux_h2[0].verschiebung() + fct.teichmuller()
|
||||
#C = omega8_lift - decom_aux_h2[1].verschiebung().diffn() - decomposition_omega0_omega8(aux.omega, prec=prec)[1].verschiebung()
|
||||
#print('test 2:', A - B.diffn() == C)
|
||||
#print('test 3:', result.omega0 == A, result.f == B, result.omega8 == C)
|
||||
#print(result.omega8, '\n \n', compare, '\n \n', aux_f, '\n \n')
|
||||
return result#.reduce()
|
||||
|
||||
def crystalline_cohomology_basis(self, prec = 50):
|
||||
result = []
|
||||
|
|
|
|||
|
|
@ -3,7 +3,7 @@ def decomposition_g0_g8(fct, prec = 50):
|
|||
and f is combination of basis of H^1(X, OX). Output is (g0, g8, f).'''
|
||||
C = fct.curve
|
||||
g = C.genus()
|
||||
coord = fct.coordinates()
|
||||
coord = fct.coordinates(prec=prec)
|
||||
nontrivial_part = 0*C.x
|
||||
for i, a in enumerate(C.cohomology_of_structure_sheaf_basis()):
|
||||
nontrivial_part += coord[i]*a
|
||||
|
|
@ -13,8 +13,9 @@ def decomposition_g0_g8(fct, prec = 50):
|
|||
fct = Fxy(fct.function)
|
||||
num = fct.numerator()
|
||||
den = fct.denominator()
|
||||
integral_part, num = num.quo_rem(den)
|
||||
aux_den = superelliptic_function(C, Rxy(den))
|
||||
g0 = superelliptic_function(C, 0)
|
||||
g0 = superelliptic_function(C, integral_part)
|
||||
g8 = superelliptic_function(C, 0)
|
||||
for monomial in num.monomials():
|
||||
aux = superelliptic_function(C, monomial)
|
||||
|
|
@ -31,7 +32,7 @@ def decomposition_omega0_omega8(omega, prec=50):
|
|||
F = C.base_ring
|
||||
delta = C.nb_of_pts_at_infty
|
||||
m = C.exponent
|
||||
if sum(omega.residue(place = i, prec = 50) for i in range(delta)) != 0:
|
||||
if sum(omega.residue(place = i, prec = prec) for i in range(delta)) != 0:
|
||||
raise ValueError(str(omega) + " has non zero residue!")
|
||||
Fxy, Rxy, x, y = C.fct_field
|
||||
Rx.<x> = PolynomialRing(F)
|
||||
|
|
|
|||
|
|
@ -23,6 +23,13 @@ def second_patch(argument):
|
|||
fct1 = Fxy(fct.subs({x : 1/x, y : y/x^(g+1)}))
|
||||
fct1 *= -x^(-2)
|
||||
return superelliptic_form(C1, fct1)
|
||||
if isinstance(argument, superelliptic_drw_form):
|
||||
h1 = argument.h1
|
||||
omega = argument.omega
|
||||
h2 = argument.h2
|
||||
C = h1.curve
|
||||
return superelliptic_drw_form(-second_patch(h1)*(C.x)^(-2), second_patch(omega), second_patch(h2))
|
||||
|
||||
|
||||
def lift_form_to_drw(omega):
|
||||
A, B = regular_form(omega)
|
||||
|
|
|
|||
|
|
@ -56,6 +56,42 @@ class superelliptic_drw_cech:
|
|||
C = self.curve
|
||||
return superelliptic_cech(C, omega0.h1*C.dx, f.t)
|
||||
|
||||
def div_by_p(self):
|
||||
'''Given a regular cocycle of the form (V(omega) + dV(h), [f] + V(t), ...), where [f] = 0 in H^1(X, OX),
|
||||
find de Rham cocycle (xi0, f, xi8) such that (V(omega) + dV(h), [f] + V(t), ...) = p*(xi0, f, xi8).'''
|
||||
C = self.curve
|
||||
aux = self
|
||||
Fxy, Rxy, x, y = C.fct_field
|
||||
aux_f_t_0 = decomposition_g0_g8(aux.f.t, prec=50)[0]
|
||||
aux.f.t = 0*C.x
|
||||
aux.omega0 -= aux_f_t_0.teichmuller().diffn()
|
||||
aux.omega8 = aux.omega0 - aux.f.diffn()
|
||||
#
|
||||
omega = aux.omega0.omega
|
||||
omega1 = omega.cartier().cartier()
|
||||
omega1 = omega1.inv_cartier().inv_cartier()
|
||||
fct = (omega.cartier() - omega1.cartier()).int()
|
||||
aux.omega0.h2 += fct^p
|
||||
aux.omega0.omega = omega1
|
||||
if aux.omega0.h2.function in Rxy:
|
||||
aux.f -= aux.omega0.h2.verschiebung()
|
||||
aux.omega0.h2 = 0*C.x
|
||||
if aux.omega8.h2.expansion_at_infty().valuation() >= 0:
|
||||
aux.f += aux.omega8.h2.verschiebung()
|
||||
aux.omega8.h2 = 0*C.x
|
||||
print('aux', aux)
|
||||
# Now aux should be of the form (V(omega1), V(f), V(omega2))
|
||||
# Thus aux = p*(Cartier(omega1), p-th_root(f), Cartier(omega2))
|
||||
aux_divided_by_p = superelliptic_cech(C, aux.omega0.omega.cartier(), aux.f.f.pth_root())
|
||||
print('aux_divided_by_p', aux_divided_by_p)
|
||||
print('is regular', aux_divided_by_p.omega0.is_regular_on_U0(), aux_divided_by_p.omega8.is_regular_on_Uinfty())
|
||||
print('aux.omega0.omega.cartier() - aux.f.f.pth_root().diffn() == aux.omega8.omega.cartier()', aux.omega0.omega.cartier() - aux.f.f.pth_root().diffn() == aux.omega8.omega.cartier())
|
||||
return aux_divided_by_p
|
||||
else:
|
||||
raise ValueError("aux.omega8.h2.expansion_at_infty().valuation() < 0:", aux.omega8.h2.expansion_at_infty())
|
||||
else:
|
||||
raise ValueError("aux.omega0.h2.function not in Rxy:", aux.omega0.h2.function)
|
||||
|
||||
def coordinates(self, basis = 0):
|
||||
C = self.curve
|
||||
g = C.genus()
|
||||
|
|
@ -67,12 +103,7 @@ class superelliptic_drw_cech:
|
|||
aux = self
|
||||
for i, a in enumerate(basis):
|
||||
aux -= coord_lifted[i]*a
|
||||
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())
|
||||
aux_divided_by_p = aux.div_by_p()
|
||||
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)]
|
||||
|
|
|
|||
|
|
@ -17,7 +17,7 @@ class superelliptic_drw_form:
|
|||
H = (self.h2 - other.h2).pth_root()
|
||||
except:
|
||||
return False
|
||||
eq2 = ((self.omega - other.omega).cartier() - H.diffn()) == 0*self.curve.dx
|
||||
eq2 = ((other.omega - self.omega).cartier() - H.diffn()) == 0*self.curve.dx
|
||||
if eq1 and eq2:
|
||||
return True
|
||||
return False
|
||||
|
|
@ -30,7 +30,7 @@ class superelliptic_drw_form:
|
|||
result = ""
|
||||
if h1.function != 0:
|
||||
result += "[" + str(h1) + "] d[x]"
|
||||
if h1.function !=0 and omega.form != 0:
|
||||
if (h1.function !=0 and omega.form != 0) or (h2.function !=0 and omega.form != 0):
|
||||
result += " + "
|
||||
if omega.form != 0:
|
||||
result += "V(" + str(omega) + ")"
|
||||
|
|
|
|||
|
|
@ -0,0 +1,11 @@
|
|||
p = 3
|
||||
m = 2
|
||||
F = GF(p)
|
||||
Rx.<x> = PolynomialRing(F)
|
||||
f = x^3 - x
|
||||
C = superelliptic(f, m)
|
||||
Rxy.<x, y> = PolynomialRing(F, 2)
|
||||
omega = (((2*C.x^18 + 2*C.x^16 + 2*C.x^14 + 2*C.x^10 + 2*C.x^8 + 2*C.x^4 + 2*C.x^2 + 2*C.one)/(C.x^13 + C.x^11 + C.x^9))*C.y) * C.dx
|
||||
print(decomposition_omega0_omega8(aux.omega)[0] - decomposition_omega0_omega8(aux.omega)[1] == omega and decomposition_omega0_omega8(aux.omega)[0].is_regular_on_U0() and decomposition_omega0_omega8(aux.omega)[1].is_regular_on_Uinfty())
|
||||
h = ((C.x^10 + C.x^8 + C.x^6 + 2*C.x^4 + 2*C.x^2 + 2*C.one)/C.x^6)*C.y
|
||||
print(decomposition_g0_g8(h)[0] - decomposition_g0_g8(h)[1] + decomposition_g0_g8(h)[2] == h and decomposition_g0_g8(h)[0].function in Rxy and decomposition_g0_g8(h)[1].expansion_at_infty().valuation() >= 0)
|
||||
|
|
@ -0,0 +1,10 @@
|
|||
p = 3
|
||||
m = 2
|
||||
F = GF(p)
|
||||
Rx.<x> = PolynomialRing(F)
|
||||
f = x^3 - x
|
||||
C = superelliptic(f, m)
|
||||
print(auxilliary_derivative((C.x^3 - C.x).teichmuller()))
|
||||
print('Result should be: [2] d[x] + V((x^8) dx) + dV([2*x^7 + x^5])')
|
||||
print(2*(C.y).teichmuller() * (C.y).teichmuller().diffn() == (C.x^3 - C.x).teichmuller().diffn())
|
||||
print(C.y.teichmuller().diffn().frobenius() == (C.y)^2 * C.y.diffn()) #F(d[y]) = y^2*dy
|
||||
|
|
@ -24,4 +24,6 @@ load('superelliptic/tests/a_number_test.sage')
|
|||
#print("diffn_test:")
|
||||
#load('as_covers/tests/diffn_test.sage')
|
||||
#print("Cartier test:")
|
||||
#load('as_covers/tests/cartier_test.sage')
|
||||
#load('as_covers/tests/cartier_test.sage')
|
||||
#print("Decomposition into g0, g8/ omega0, omega8 test:")
|
||||
#load('superelliptic_drw/tests/decomposition_into_g0_g8_tests.sage')
|
||||
Loading…
Reference in New Issue