DeRhamComputation/superelliptic_drw/superelliptic_drw_auxilliaries.sage

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def decomposition_g0_pth_power(fct):
C = fct.curve
Fxy, Rxy, xy, y = C.fct_field
if fct.function in Rxy:
return (fct, 0*C.x)
'''Decompose fct as g0 + A^p, if possible. Output: (g0, A).'''
omega = fct.diffn().regular_form()
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g0 = omega.integral()
A = (fct - g0).pth_root()
return (g0, A)
def decomposition_g0_p2th_power(fct):
'''Decompose fct as g0 + A^(p^2), if possible. Output: (g0, A).'''
C = fct.curve
p = C.characteristic
g0, A = decomposition_g0_pth_power(fct)
A0, A1 = decomposition_g0_pth_power(A)
return (g0 + A0^p, A1)
def decomposition_omega0_hpdh(omega):
'''Decompose omega = (regular on U0) + h^(p-1) dh + d(smth), so that Cartier(omega) = (regular on U0) + dh.
Result: (regular on U0, h)'''
print('decomp!!', omega)
C = omega.curve
F = C.base_ring
p = F.characteristic()
if omega.is_regular_on_U0():
return (omega, 0*C.x)
omega1 = omega.cartier().cartier()
omega1 = omega1.inv_cartier().inv_cartier()
print('do scalkowania', omega.cartier() - omega1.cartier())
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fct = (omega.cartier() - omega1.cartier()).integral()
#dfct = omega.cartier() - omega1.cartier() #this is dh
#print('dfct.cartier = 0?', dfct.cartier())
#omegas_by_dh = (omega - omega1)/dfct
#d_omegas_by_dh = omegas_by_dh.diffn()
#fct = - (omega - omega1)/d_omegas_by_dh
#fct = (omega.cartier() - omega1.cartier()).integral()
#h = (omega - omega0)/(d((omega - omega0)/dh))
print('decomposition_omega0_hpdh', omega, omega1, fct)
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print('??', omega.verschiebung() == omega1.verschiebung() + p*fct.teichmuller().diffn())
return (omega1, fct)
def decomposition_omega8_hpdh(omega, prec = 50):
'''Decompose omega = (regular on U8) + h^(p-1) dh, so that Cartier(omega) = (regular on U8) + dh.
Result: (regular on U8, h)'''
C = omega.curve
g = C.genus()
Fxy, Rxy, x, y = C.fct_field
F = C.base_ring
p = C.characteristic
if omega.is_regular_on_Uinfty():
return (omega, 0*C.x)
Rt.<t> = LaurentSeriesRing(F)
omega_analytic = Rt(laurent_analytic_part(omega.expansion_at_infty(prec = prec)))
Cv = C.uniformizer()
v = Fxy(Cv.function)
omega_analytic = Fxy(omega_analytic(t = v))
omega_analytic = superelliptic_function(C, omega_analytic)*Cv.diffn()
omega8 = omega - omega_analytic
dh = omega.cartier() - omega8.cartier()
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h = dh.integral()
return (omega8, h)
def decomposition_g8_pth_power(fct, prec = 50):
'''Decompose fct as g8 + A^p, if possible. Output: (g8, A).'''
C = fct.curve
F = C.base_ring
p = F.characteristic()
Rt.<t> = LaurentSeriesRing(F)
Fxy, Rxy, x, y = C.fct_field
if fct.expansion_at_infty().valuation() >= 0:
return (fct, 0*C.x)
A = laurent_analytic_part(fct.expansion_at_infty(prec = prec))
Cv = C.uniformizer()
v = Cv.function
A = A(t = v)
A = superelliptic_function(C, A)
A = A.pth_root()
g8 = fct - A^p
return (g8, A)
def decomposition_g8_p2th_power(fct):
'''Decompose fct as g8 + A^(p^2), if possible. Output: (g8, A).'''
C = fct.curve
F = C.base_ring
p = F.characteristic()
g0, A = decomposition_g8_pth_power(fct)
A0, A1 = decomposition_g8_pth_power(A)
return (g0 + A0^p, A1)