DeRhamComputation/superelliptic_drw/de_rham_witt_lift.sage

def de_rham_witt_lift_form0(omega):
    C = omega.curve
    omega_regular = omega.regular_form() #Present omega0 in the form P dx + Q dy
    #Now the obvious lift of omega0 = P dx + Q dy to de Rham-Witt is [P] d[x] + [Q] d[y]
    return omega_regular.dx.teichmuller()*(C.x.teichmuller().diffn()) + omega_regular.dy.teichmuller()*(C.y.teichmuller().diffn())

def de_rham_witt_lift_form8(omega):
    C = omega.curve
    g = C.genus()
    omega_regular = second_patch(omega).regular_form()
    omega_regular = (second_patch(omega_regular.dx), second_patch(omega_regular.dy))
    u = (C.x)^(-1)
    v = (C.y)/(C.x)^(g+1)
    omega_lift = omega_regular[0].teichmuller()*(u.teichmuller().diffn()) + omega_regular[1].teichmuller()*(v.teichmuller().diffn())
    return omega_lift
    
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_lift = de_rham_witt_lift_form0(omega0)
    omega8_lift = de_rham_witt_lift_form8(omega8)
    aux = omega0_lift - omega8_lift - fct.teichmuller().diffn() # now aux is of the form (V(smth) + dV(smth), V(smth))
    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_omega0.verschiebung(), fct.teichmuller() + aux_h2.verschiebung() + aux_f.verschiebung())
    return result

def crystalline_cohomology_basis(self, prec = 50, info = 0):
    result = []
    prec1 = prec
    for i, a in enumerate(self.de_rham_basis()):
        if info:
            print("Computing " + str(i) +". basis element")
        while True:
            try:
                result += [de_rham_witt_lift(a, prec = prec1)]
                break
            except IndexError:
                prec1 += 100
                result += [de_rham_witt_lift(a, prec = prec1)]
    return result

superelliptic.crystalline_cohomology_basis = crystalline_cohomology_basis