naprawiony problem z uniformizatorem w superelliptic; decomposition_omega8_hpdh dziala
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@ -18,12 +18,14 @@ 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/superelliptic_drw_auxilliaries.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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load('auxilliaries/hensel.sage')
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load('auxilliaries/linear_combination_polynomials.sage')
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load('auxilliaries/laurent_analytic_part.sage')
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##############
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##############
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load('drafty/convert_superelliptic_into_AS.sage')
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@ -201,10 +201,22 @@ class superelliptic:
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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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#it replaces repeteadly all y^m's in g(x, y) by f(x). As a result
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#you obtain \sum_{i = 0}^{m-1} y^i g_i(x).
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def uniformizer(self):
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m = self.exponent
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r = self.polynomial.degree()
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delta, a, b = xgcd(m, r)
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a = -a
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M = m/delta
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R = r/delta
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while a<0:
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a += R
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b += M
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return (C.x)^a/(C.y)^b
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def reduction(C, g):
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'''Auxilliary. Given a superelliptic curve C : y^m = f(x) and a polynomial g(x, y)
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it replaces repeteadly all y^m's in g(x, y) by f(x). As a result
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you obtain \sum_{i = 0}^{m-1} y^i g_i(x).'''
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p = C.characteristic
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F = C.base_ring
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Rxy.<x, y> = PolynomialRing(F, 2)
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@ -122,13 +122,12 @@ class superelliptic_function:
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fct = RxyQ(fct)
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r = f.degree()
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delta, a, b = xgcd(m, r)
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b = -b
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a = -a
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M = m/delta
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R = r/delta
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while a<0:
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a += R
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b += M
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g = (x^r*f(x = 1/x))
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gW = RptWQ(g(x = t^M * W^b)) - W^(delta)
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ww = naive_hensel(gW, F, start = root_of_unity(F, delta)^place, prec = prec)
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@ -57,11 +57,4 @@ def decomposition_omega0_omega8(omega, prec=50):
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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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raise ValueError("Something went wrong for "+str(omega) +". Result would be "+str(g0)+ " and " + str(g8))
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@ -0,0 +1,49 @@
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def decomposition_g0_pth_power(fct):
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'''Decompose fct as g0 + A^p, if possible. Output: (g0, A).'''
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omega = fct.diffn().regular_form()
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g0 = omega.int()
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A = (fct - g0).pth_root()
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return (g0, A)
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def decomposition_g0_p2th_power(fct):
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'''Decompose fct as g0 + A^(p^2), if possible. Output: (g0, A).'''
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g0, A = decomposition_g0_pth_power(fct)
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A0, A1 = decomposition_g0_pth_power(A)
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return (g0 + A0^p, A1)
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def decomposition_omega0_hpdh(omega):
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'''Decompose omega = (regular on U0) + h^(p-1) dh, so that Cartier(omega) = (regular on U0) + dh.
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Result: (regular on U0, h)'''
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omega1 = omega.cartier().cartier()
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omega1 = omega1.inv_cartier().inv_cartier()
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fct = (omega.cartier() - omega1.cartier()).int()
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return (omega1, fct)
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def decomposition_omega8_hpdh(omega, prec = 50):
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'''Decompose omega = (regular on U8) + h^(p-1) dh, so that Cartier(omega) = (regular on U8) + dh.
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Result: (regular on U8, h)'''
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C = omega.curve
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g = C.genus()
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Fxy, Rxy, x, y = C.fct_field
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F = C.base_ring
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p = C.characteristic
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Rt.<t> = LaurentSeriesRing(F)
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RT.<T> = PolynomialRing(F)
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FT = FractionField(RT)
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omega_analytic = FT(laurent_analytic_part(omega.expansion_at_infty(prec = prec))(t = T))
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print('omega_analytic', omega_analytic)
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Cv = C.uniformizer()
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v = Fxy(Cv.function)
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omega_analytic = Fxy(omega_analytic(T = v))
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print('expansions', superelliptic_function(C, omega_analytic).expansion_at_infty(prec = prec), '\n', Cv.diffn().expansion_at_infty(prec = prec),
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'\n', (superelliptic_function(C, omega_analytic)*Cv.diffn()).expansion_at_infty(prec = prec))
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omega_analytic = superelliptic_function(C, omega_analytic)*Cv.diffn()
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print('omega_analytic.expansion_at_infty()', omega_analytic.expansion_at_infty(prec = prec))
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print('omega_analytic', omega_analytic)
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omega8 = omega - omega_analytic
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print('omega8', omega8)
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dh = omega.cartier() - omega8.cartier()
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print('dh', dh)
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h = dh.int()
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print('omega8.expansion_at_infty()', omega8.expansion_at_infty(prec = prec))
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return (omega8, h)
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@ -67,30 +67,34 @@ class superelliptic_drw_cech:
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aux.omega0 -= aux_f_t_0.teichmuller().diffn()
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aux.omega8 = aux.omega0 - aux.f.diffn()
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#
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# Now omega = (regular on U0) + h^(p-1) dh, so that Cartier(omega) = (regular on U0) + dh.
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# We replace omega by regular on U0
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omega = aux.omega0.omega
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omega1 = omega.cartier().cartier()
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omega1 = omega1.inv_cartier().inv_cartier()
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fct = (omega.cartier() - omega1.cartier()).int()
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aux.omega0.omega, fct = decomposition_omega0_hpdh(aux.omega0.omega)
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aux.omega0.h2 += fct^p
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aux.omega0.omega = omega1
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if aux.omega0.h2.function in Rxy:
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aux.f -= aux.omega0.h2.verschiebung()
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aux.omega0.h2 = 0*C.x
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if aux.omega8.h2.expansion_at_infty().valuation() >= 0:
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aux.f += aux.omega8.h2.verschiebung()
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aux.omega8.h2 = 0*C.x
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print('aux', aux)
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# Now aux should be of the form (V(omega1), V(f), V(omega2))
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# Thus aux = p*(Cartier(omega1), p-th_root(f), Cartier(omega2))
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aux_divided_by_p = superelliptic_cech(C, aux.omega0.omega.cartier(), aux.f.f.pth_root())
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print('aux_divided_by_p', aux_divided_by_p)
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print('is regular', aux_divided_by_p.omega0.is_regular_on_U0(), aux_divided_by_p.omega8.is_regular_on_Uinfty())
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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())
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return aux_divided_by_p
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else:
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raise ValueError("aux.omega8.h2.expansion_at_infty().valuation() < 0:", aux.omega8.h2.expansion_at_infty())
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# Now we have to ensure that aux.omega0.h2.function in Rxy...
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# In other words, we decompose h2 = (regular on U0) + A^(p^2).
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aux.omega0.h2 = decomposition_g0_p2th_power(aux.omega0.h2)[0]
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# Now we can reduce: (... + dV(h2), V(f), ...) --> (..., V(f - h2), ...)
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aux.f -= aux.omega0.h2.verschiebung()
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aux.omega0.h2 = 0*C.x
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if aux.omega8.h2.expansion_at_infty().valuation() >= 0:
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aux.f += aux.omega8.h2.verschiebung()
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aux.omega8.h2 = 0*C.x
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print('aux', aux)
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# Now aux should be of the form (V(omega1), V(f), V(omega2))
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# Thus aux = p*(Cartier(omega1), p-th_root(f), Cartier(omega2))
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aux_divided_by_p = superelliptic_cech(C, aux.omega0.omega.cartier(), aux.f.f.pth_root())
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print('aux_divided_by_p', aux_divided_by_p)
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print('is regular', aux_divided_by_p.omega0.is_regular_on_U0(), aux_divided_by_p.omega8.is_regular_on_Uinfty())
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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())
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return aux_divided_by_p
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else:
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raise ValueError("aux.omega0.h2.function not in Rxy:", aux.omega0.h2.function)
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print('aux.omega8.omega', aux.omega8.omega)
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print('aux.omega8.h2', aux.omega8.h2)
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print('second_patch(aux.omega8.h2.diffn()).is_regular_on_U0()', second_patch(aux.omega8.h2.diffn()).is_regular_on_U0())
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raise ValueError("aux.omega8.h2.expansion_at_infty().valuation() < 0:", aux.omega8.h2.expansion_at_infty())
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def coordinates(self, basis = 0):
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C = self.curve
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@ -113,4 +117,4 @@ class superelliptic_drw_cech:
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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())
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eq1 = self.omega0.r().is_regular_on_U0() and self.omega8.r().is_regular_on_Uinfty()
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eq2 = self.omega0.frobenius().is_regular_on_U0() and self.omega8.frobenius().is_regular_on_Uinfty()
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return eq1 and eq2
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return eq1 and eq2
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