137 lines
5.4 KiB
Python
137 lines
5.4 KiB
Python
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 = superelliptic_drw_cech(0*C.dx.verschiebung(), 0*C.x.verschiebung())
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new.omega0 = self.omega0
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new.omega8 = self.omega8
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new.f = self.f
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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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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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def r(self):
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omega0 = self.omega0
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f = self.f
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C = self.curve
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return superelliptic_cech(C, omega0.h1*C.dx, f.t)
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def div_by_p(self, info = 0):
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'''Given a regular cocycle of the form (V(omega) + dV(h), [f] + V(t), ...), where [f] = 0 in H^1(X, OX),
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find de Rham cocycle (xi0, f, xi8) such that (V(omega) + dV(h), [f] + V(t), ...) = p*(xi0, f, xi8).'''
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#
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print('div by p', self)
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if info: print("Computing " + str(self) + " divided by p.")
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#
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C = self.curve
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F = C.base_ring
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p = F.characteristic()
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aux = self
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Fxy, Rxy, x, y = C.fct_field
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aux_f_t_0 = decomposition_g0_g8(aux.f.t, prec=50)[0]
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#
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if info: print("Computed decomposition_g0_g8 of self.f.t.")
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#
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aux.f.t = 0*C.x
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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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aux.omega0.omega, fct = decomposition_omega0_hpdh(aux.omega0.omega)
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aux.omega0.h2 += fct^p
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#
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if info: print("Computed decomposition_omega0_hpdh of self.omega0.omega.")
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#
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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.
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# Then we replace h2 by (regular on U0) and omega by omega + inverse Cartier(dA)
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aux.omega0.h2, A = decomposition_g0_pth_power(aux.omega0.h2)
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aux.omega0.omega += (A.diffn()).inv_cartier()
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#
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if info: print("Computed decomposition_g0_p2th_power(aux.omega0.h2).")
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#
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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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# Now the same for aux.omega8
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aux.omega8.omega, fct = decomposition_omega8_hpdh(aux.omega8.omega)
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aux.omega8.h2 += fct^p
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aux.omega8.h2 = decomposition_g8_p2th_power(aux.omega8.h2)[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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aux_divided_by_p = superelliptic_cech(C, aux.omega0.omega.cartier(), aux.f.f.pth_root())
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return aux_divided_by_p
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def coordinates(self, basis = 0, prec = 50, info = 0):
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if info: print("Computing coordinates of " + str(self))
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C = self.curve
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F = C.base_ring
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p = F.characteristic()
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g = C.genus()
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coord_mod_p = self.r().coordinates()
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if info: print("Computed coordinates mod p.")
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coord_lifted = [lift(a) for a in coord_mod_p]
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if basis == 0:
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basis = C.crystalline_cohomology_basis()
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aux = self
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for i, a in enumerate(basis):
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aux -= coord_lifted[i]*a
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print("aux!!!", aux)
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aux_divided_by_p = aux.div_by_p(info = info)
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print("aux by p!!!", aux_divided_by_p)
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print("check", mult_by_p(aux_divided_by_p)-aux)
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coord_aux_divided_by_p = aux_divided_by_p.coordinates()
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if info: print("Computed coordinates mod p of (self - lift)/p.")
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coord_aux_divided_by_p = [ZZ(a) for a in coord_aux_divided_by_p]
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coordinates = [ (coord_lifted[i] + p*coord_aux_divided_by_p[i])%p^2 for i in range(2*g)]
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return coordinates
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def is_regular(self):
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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 |