poprawiony rozklad na omega0 - omega8

sage/drafty/draft.sage

@@ -2,8 +2,7 @@ p = 3
 m = 2
 F = GF(p)
 Rx.<x> = PolynomialRing(F)
-f = x^3 - x
+f = x^3 - x + 1
 C = superelliptic(f, m)
-a = superelliptic_drw_form(C.one, 0*C.dx, 0*C.x)
+C1 = patch(C)
-b = a+a+a+a+a+a+a+a+a
+print(C1.crystalline_cohomology_basis())
-print(b)

sage/drafty/superelliptic_drw.sage

@@ -310,7 +310,26 @@ class superelliptic_drw_cech:
         f = self.f
         return superelliptic_drw_cech(other*omega0, other*f)
     
-def de_rham_witt_lift(cech_class):
+    def r(self):
+        omega0 = self.omega0
+        f = self.f
+        C = self.curve
+        return superelliptic_cech(C, omega0.h1*C.dx, f.t)
+    
+    def coordinates(self, basis = 0):
+        coord_mod_p = self.r().coordinates()
+        print(coord_mod_p)
+        coord_lifted = [lift(a) for a in coord_mod_p]
+        if basis == 0:
+            basis = self.curve().crystalline_cohomology_basis()
+        aux = self
+        for i, a in enumerate(basis):
+            aux -= coord_lifted[i]*a
+        aux = aux.reduce()
+        return aux
+        
+
+def de_rham_witt_lift(cech_class, prec = 50):
     C = cech_class.curve
     g = C.genus()
     omega0 = cech_class.omega0
@@ -324,7 +343,37 @@ def de_rham_witt_lift(cech_class):
     v = (C.y)/(C.x)^(g+1)
     omega8_lift = omega0_regular[0].teichmuller()*(u.teichmuller().diffn()) + omega0_regular[1].teichmuller()*(v.teichmuller().diffn())
     aux = omega0_lift - omega8_lift - fct.teichmuller().diffn()
-    aux_h2 = decomposition_g0_g8(aux.h2)[0]
+    decom_aux_h2 = decomposition_g0_g8(aux.h2, prec=prec)
-    aux_f = decomposition_g0_g8(aux.h2)[2] #do napisania - komponent od kohomologii
+    aux_h2 = decom_aux_h2[0]
-    aux_omega0 = decomposition_omega0_omega8(aux.omega)[0]
+    aux_f = decom_aux_h2[2]
+    aux_omega0 = decomposition_omega0_omega8(aux.omega, prec=prec)[0]
     return superelliptic_drw_cech(omega0_lift + aux_h2.verschiebung().diffn() + aux_omega0.verschiebung(), fct.teichmuller() + aux_f.verschiebung())
+
+def crystalline_cohomology_basis(self, prec = 50):
+    result = []
+    for a in self.de_rham_basis():
+        result += [de_rham_witt_lift(a, prec = prec)]
+    return result
+
+superelliptic.crystalline_cohomology_basis = crystalline_cohomology_basis
+
+def autom(self):
+    C = self.curve
+    F = C.base_ring
+    Rxy.<x, y> = PolynomialRing(F, 2)
+    Fxy = FractionField(Rxy)
+    if isinstance(self, superelliptic_function):
+        result = superelliptic_function(C, Fxy(self.function).subs({x:x+1, y:y}))
+        return result
+    if isinstance(self, superelliptic_form):
+        result = superelliptic_form(C, Fxy(self.form).subs({x:x+1, y:y}))
+        return result
+    if isinstance(self, superelliptic_witt):
+        result = superelliptic_witt(autom(self.t), autom(self.f))
+        return result
+    if isinstance(self, superelliptic_drw_form):
+        result = superelliptic_drw_form(autom(self.h1), autom(self.omega), autom(self.h2))
+        return result
+    if isinstance(self, superelliptic_drw_cech):
+        result = superelliptic_drw_cech(autom(self.omega0), autom(self.f))
+        return result

sage/superelliptic/decomposition_into_g0_g8.sage

@@ -1,4 +1,4 @@
-def decomposition_g0_g8(fct):
+def decomposition_g0_g8(fct, prec = 50):
     '''Writes fct as a difference g0 - g8, with g0 regular on the affine patch and g8 at the points in infinity.'''
     C = fct.curve
     g = C.genus()
@@ -7,7 +7,7 @@ def decomposition_g0_g8(fct):
     for i, a in enumerate(C.cohomology_of_structure_sheaf_basis()):
         nontrivial_part += coord[i]*a
     fct -= nontrivial_part
-    if fct.coordinates() != g*[0]:
+    if fct.coordinates(prec=prec) != g*[0]:
         raise ValueError("The given function cannot be written as g0 - g8.")
     
     Fxy, Rxy, x, y = C.fct_field
@@ -28,22 +28,33 @@ def decomposition_g0_g8(fct):
 def decomposition_omega0_omega8(omega, prec=50):
     '''Writes omega as a difference omega0 - omega8, with omega0 regular on the affine patch and omega8 at the points in infinity.'''
     C = omega.curve
+    omega.form = reduction(C, omega.form)
+    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:
+        raise ValueError(str(omega) + " has non zero residue!")
     Fxy, Rxy, x, y = C.fct_field
+    Rx.<x> = PolynomialRing(F)
+    Fx = FractionField(Rx)
     fct = Fxy(omega.form)
     num = fct.numerator()
     den = fct.denominator()
     aux_den = superelliptic_function(C, Rxy(den))
     g0 = superelliptic_function(C, 0)
     g8 = superelliptic_function(C, 0)
-    dx_valuation = C.dx.expansion_at_infty(prec=prec).valuation()
+    for j in range(0, m):
-    for monomial in num.monomials():
+        component = Fx(omega.jth_component(j))
-        aux = superelliptic_function(C, monomial)
+        q, r = component.numerator().quo_rem(component.denominator())
-        if aux.expansion_at_infty(prec=prec).valuation() + dx_valuation >= aux_den.expansion_at_infty(prec=prec).valuation():
+        g0 += (C.y)^(-j)*superelliptic_function(C, Rxy(q))
-            g8 += num.monomial_coefficient(monomial)*aux/aux_den
+        if ((C.y)^(-j)*superelliptic_function(C, Fxy(r/component.denominator()))*C.dx).expansion_at_infty().valuation() < 0:
-        else:
+            raise ValueError("Something went wrong for "+str(omega))
-            g0 += num.monomial_coefficient(monomial)*aux/aux_den
+        g8 -= (C.y)^(-j)*superelliptic_function(C, Fxy(r/component.denominator()))
     g0, g8  = g0*C.dx, g8*C.dx
     if g0.is_regular_on_U0():
         return (g0, g8)
+    #Rx.<x> = PolynomialRing(F)
+    #Rx.<x> = PolynomialRing(F)
+    #aux_fct = (g0.form)*y
     else:
-        raise Error("Something went wrong.")
+        raise ValueError("Something went wrong for "+str(omega) +". Result would be "+str(g0)+ " and " + str(g8))

sage/superelliptic/superelliptic_cech_class.sage

@@ -71,7 +71,7 @@ class superelliptic_cech:
         for j in range(1, m):
             fct_j = Fx(fct.jth_component(j))
             if (fct_j != Rx(0)):
-                d = degree_of_rational_fctn(fct_j, p)
+                d = degree_of_rational_fctn(fct_j, F)
             
                 if (d, j) in degrees1.values():
                     index = degrees1_inv[(d, j)]

sage/superelliptic/superelliptic_form_class.sage

@@ -93,16 +93,14 @@ class superelliptic_form:
         '''If self = sum_j h_j(x)/y^j dx, output is h_j(x).'''
         g = self.form
         C = self.curve
+        m = C.exponent
         F = C.base_ring
         Rx.<x> = PolynomialRing(F)
         Fx = FractionField(Rx)
         FxRy.<y> = PolynomialRing(Fx)
-        Fxy = FractionField(FxRy)
+        g = reduction(C, y^m*g)
-        Ryinv.<y_inv> = PolynomialRing(Fx)
+        g = FxRy(g)
-        g = Fxy(g)
+        return g.monomial_coefficient(y^(m-j))
-        g = g(y = 1/y_inv)
-        g = Ryinv(g)
-        return coff(g, j)
     
     def is_regular_on_U0(self):
         C = self.curve
@@ -136,6 +134,9 @@ class superelliptic_form:
         C = self.curve
         g = superelliptic_function(C, g)
         g = g.expansion_at_infty(place = place, prec=prec)
-        x_series = superelliptic_function(C, x).expansion_at_infty(place = place, prec=prec)
+        x_series = C.x.expansion_at_infty(place = place, prec=prec)
         dx_series = x_series.derivative()
         return g*dx_series
+    
+    def residue(self, place = 0, prec=30):
+        return self.expansion_at_infty(place = place, prec=prec)[-1]