nieudane proby coordinates dla cech_drw

sage/as_covers/as_cover_class.sage

@@ -1,5 +1,5 @@
 class as_cover:
-    def __init__(self, C, list_of_fcts, prec = 10):
+    def __init__(self, C, list_of_fcts, branch_points = [], prec = 10):
         self.quotient = C
         self.functions = list_of_fcts
         self.height = len(list_of_fcts)
@@ -22,24 +22,25 @@ class as_cover:
         r = f.degree()
         delta = GCD(m, r)
         self.nb_of_pts_at_infty = delta
+        self.branch_points = list(range(delta)) + branch_points
         Rxy.<x, y> = PolynomialRing(F, 2)
         Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
 
-        all_x_series = []
+        all_x_series = {}
-        all_y_series = []
+        all_y_series = {}
-        all_z_series = []
+        all_z_series = {}
-        all_dx_series = []
+        all_dx_series = {}
-        all_jumps = []
+        all_jumps = {}
 
-        for i in range(delta):
+        for pt in self.branch_points:
-            x_series = superelliptic_function(C, x).expansion_at_infty(place = i, prec=prec)
+            x_series = superelliptic_function(C, x).expansion(pt=pt, prec=prec)
-            y_series = superelliptic_function(C, y).expansion_at_infty(place = i, prec=prec)
+            y_series = superelliptic_function(C, y).expansion(pt=pt, prec=prec)
             z_series = []
             jumps = []
             n = len(list_of_fcts)
-            list_of_power_series = [g.expansion_at_infty(place = i, prec=prec) for g in list_of_fcts]
+            list_of_power_series = [g.expansion(pt=pt, prec=prec) for g in list_of_fcts]
-            for i in range(n):
+            for j in range(n):
-                power_series = list_of_power_series[i]
+                power_series = list_of_power_series[j]
                 jump, correction, t_old, z = artin_schreier_transform(power_series, prec = prec)
                 x_series = x_series(t = t_old)
                 y_series = y_series(t = t_old)
@@ -48,11 +49,11 @@ class as_cover:
                 jumps += [jump]
                 list_of_power_series = [g(t = t_old) for g in list_of_power_series]
             
-            all_jumps += [jumps]
+            all_jumps[pt] = jumps
-            all_x_series += [x_series]
+            all_x_series[pt] = x_series
-            all_y_series += [y_series]
+            all_y_series[pt] = y_series
-            all_z_series += [z_series]
+            all_z_series[pt] = z_series
-            all_dx_series += [x_series.derivative()]
+            all_dx_series[pt] = x_series.derivative()
         self.jumps = all_jumps
         self.x_series = all_x_series
         self.y_series = all_y_series
@@ -70,8 +71,9 @@ class as_cover:
         self.fct_field = (RxyzQ, Rxyz, x, y, z)
         self.x = as_function(self, x)
         self.y = as_function(self, y)
-        self.z = [as_function(self, z[i]) for i in range(n)]
+        self.z = [as_function(self, z[j]) for j in range(n)]
         self.dx = as_form(self, 1)
+        self.one = as_function(self, 1)
         
         
     def __repr__(self):
@@ -89,9 +91,9 @@ class as_cover:
         jumps = self.jumps
         gY = self.quotient.genus()
         n = self.height
-        delta = self.nb_of_pts_at_infty
+        branch_pts = self.branch_points
         p = self.characteristic
-        return p^n*gY + (p^n - 1)*(delta - 1) + sum(p^(n-j-1)*(jumps[i][j]-1)*(p-1)/2 for j in range(n) for i in range(delta))
+        return p^n*gY + (p^n - 1)*(len(branch_pts) - 1) + sum(p^(n-j-1)*(jumps[pt][j]-1)*(p-1)/2 for j in range(n) for pt in branch_pts)
     
     def exponent_of_different(self, place = 0):
         jumps = self.jumps
@@ -130,17 +132,17 @@ class as_cover:
             for j in range(0, m):
                 for k in product(*pr):
                     eta = as_form(self, x^i * prod(z[i1]^(k[i1]) for i1 in range(n))/y^j)
-                    eta_exp = eta.expansion_at_infty()
+                    eta_exp = eta.expansion(pt=self.branch_points[0])
                     S += [(eta, eta_exp)]
 
         forms = holomorphic_combinations(S)
         
-        for i in range(1, delta):
+        for pt in self.branch_points[1:]:
-            forms = [(omega, omega.expansion_at_infty(place = i)) for omega in forms]
+            forms = [(omega, omega.expansion(pt=pt)) for omega in forms]
             forms = holomorphic_combinations(forms)
         
         if len(forms) < self.genus():
-            print("I haven't found all forms.")
+            print("I haven't found all forms, only ", len(forms), " of ", self.genus())
             return holomorphic_differentials_basis(self, threshold = threshold + 1)
         if len(forms) > self.genus():
             print("Increase precision.")
@@ -236,8 +238,8 @@ class as_cover:
 
         forms = holomorphic_combinations_forms(S, pole_order)
 
-        for i in range(1, delta):
+        for pt in self.branch_points[1:]:
-            forms = [(omega, omega.expansion_at_infty(place = i)) for omega in forms]
+            forms = [(omega, omega.expansion(pt=pt)) for omega in forms]
             forms = holomorphic_combinations_forms(forms, pole_order)
                   
         return forms

sage/as_covers/as_form_class.sage

@@ -37,6 +37,28 @@ class as_form:
         sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}
         return g.substitute(sub_list)*dx_series
     
+    def expansion(self, pt = 0):
+        '''Same code as expansion_at_infty.'''
+        C = self.curve
+        F = C.base_ring
+        x_series = C.x_series[pt]
+        y_series = C.y_series[pt]
+        z_series = C.z_series[pt]
+        dx_series = C.dx_series[pt]
+        n = C.height
+        variable_names = 'x, y'
+        for j in range(n):
+            variable_names += ', z' + str(j)
+        Rxyz = PolynomialRing(F, n+2, variable_names)
+        x, y = Rxyz.gens()[:2]
+        z = Rxyz.gens()[2:]
+        RxyzQ = FractionField(Rxyz)
+        prec = C.prec
+        Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
+        g = self.form
+        sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}
+        return g.substitute(sub_list)*dx_series
+    
     def __add__(self, other):
         C = self.curve
         g1 = self.form
@@ -152,6 +174,13 @@ def artin_schreier_transform(power_series, prec = 10):
     power_series = RtQ(power_series)
     if power_series.valuation() == +Infinity:
         raise ValueError("Precision is too low.")
+    if power_series.valuation() >= 0:
+        # THIS IS WRONG - THERE ARE SEVERAL PLACES OVER THIS PLACE, AND IT DEPENDS
+        aux = t^p - t
+        z = new_reverse(aux, prec = prec)
+        z = z(t = power_series)
+        return(0, 0, t, z)
+    
     while(power_series.valuation() % p == 0 and power_series.valuation() < 0):
         M = -power_series.valuation()/p
         coeff = power_series.list()[0]   #wspolczynnik a_(-p) w f_AS

sage/as_covers/as_function_class.sage

@@ -77,6 +77,28 @@ class as_function:
         g = RxyzQ(g)
         sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}
         return g.substitute(sub_list)
+    
+    def expansion(self, pt = 0):
+        C = self.curve
+        delta = C.nb_of_pts_at_infty
+        F = C.base_ring
+        x_series = C.x_series[pt]
+        y_series = C.y_series[pt]
+        z_series = C.z_series[pt]
+        n = C.height
+        variable_names = 'x, y'
+        for j in range(n):
+            variable_names += ', z' + str(j)
+        Rxyz = PolynomialRing(F, n+2, variable_names)
+        x, y = Rxyz.gens()[:2]
+        z = Rxyz.gens()[2:]
+        RxyzQ = FractionField(Rxyz)
+        prec = C.prec
+        Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
+        g = self.function
+        g = RxyzQ(g)
+        sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}
+        return g.substitute(sub_list)
 
     def group_action(self, ZN_tuple):
         C = self.curve

sage/drafty/draft.sage

@@ -2,13 +2,15 @@ p = 3
 m = 2
 F = GF(p)
 Rx.<x> = PolynomialRing(F)
-f = x^3 - x + 1
+f = x^3 - x
 C = superelliptic(f, m)
-C1 = patch(C)
+#C1 = patch(C)
 #print(C1.crystalline_cohomology_basis())
-g1 = C1.polynomial
+#g1 = C1.polynomial
-g_AS = g1(x^p - x)
+#g_AS = g1(x^p - x)
-C2 = superelliptic(g_AS, 2)
+#C2 = superelliptic(g_AS, 2)
-print(convert_super_into_AS(C2))
+#print(convert_super_into_AS(C2))
-converted = (C2.x)^4 - (C2.x)^2
+#converted = (C2.x)^4 - (C2.x)^2
-print(convert_super_fct_into_AS(converted))
+#print(convert_super_fct_into_AS(converted))
+b = C.crystalline_cohomology_basis()
+print(autom(b[0]).coordinates(basis = b))

sage/drafty/superelliptic_drw.sage

@@ -47,10 +47,11 @@ class superelliptic_witt:
             return other_integer*self
     
     def __mul__(self, other):
+        C = self.curve
+        p = C.characteristic
         if isinstance(other, superelliptic_witt):
             t1 = self.t
             f1 = self.f
-            p = self.curve.characteristic
             t2 = other.t
             f2 = other.f
             return superelliptic_witt(t1*t2, t1^p*f2 + t2^p*f1)
@@ -58,7 +59,6 @@ class superelliptic_witt:
             h1 = other.h1
             h2 = other.h2
             omega = other.omega
-            p = other.curve.characteristic
             t = self.t
             f = self.f
             aux_form = t^p*omega - h2*t^(p-1)*t.diffn() + f*h1^p*(C.x)^(p-1)*C.dx
@@ -130,7 +130,7 @@ superelliptic_function.teichmuller = teichmuller
 #d[M] = a*[b x^(a-1)]
 
 def auxilliary_derivative(P):
-    '''Return "derivative" of P, where P depends only on x. '''
+    '''Return "derivative" of P, where P depends only on x. In other words d[P(x)].'''
     P0 = P.t.function
     P1 = P.f.function
     C = P.curve
@@ -156,6 +156,11 @@ class superelliptic_drw_form:
         self.omega = omega
         self.h2 = h2
         
+    def r(self):
+        C = self.curve
+        h1 = self.h1
+        return superelliptic_form(C, h1.function)
+        
     def __eq__(self, other):
         eq1 = (self.h1 == self.h1)
         try:
@@ -317,16 +322,29 @@ class superelliptic_drw_cech:
         return superelliptic_cech(C, omega0.h1*C.dx, f.t)
     
     def coordinates(self, basis = 0):
+        C = self.curve
+        g = C.genus()
         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()
+            basis = C.crystalline_cohomology_basis()
         aux = self
         for i, a in enumerate(basis):
             aux -= coord_lifted[i]*a
-        aux = aux.reduce()
+        print('aux before reduce', aux)
-        return aux
+        #aux = aux.reduce() # Now aux = p*cech class.
+        h02 = aux.omega0.h2
+        h82 = aux.omega8.h2
+        aux -= superelliptic_drw_cech(h02.verschiebung().diffn(), (h02 - h82).verschiebung())
+        # Now aux should be of the form (V(smth), V(smth), V(smth))
+        print('aux V(smth)', aux)
+        aux_divided_by_p = superelliptic_cech(C, aux.omega0.omega.cartier(), aux.f.f.pth_root())
+        print('aux.omega0.omega.cartier()', aux.omega0.omega.cartier())
+        coord_aux_divided_by_p = aux_divided_by_p.coordinates()
+        coord_aux_divided_by_p = [ZZ(a) for a in coord_aux_divided_by_p]
+        coordinates = [ (coord_lifted[i] + p*coord_aux_divided_by_p[i])%p^2 for i in range(2*g)]
+        return coordinates
         
 
 def de_rham_witt_lift(cech_class, prec = 50):
@@ -335,19 +353,24 @@ def de_rham_witt_lift(cech_class, prec = 50):
     omega0 = cech_class.omega0
     omega8 = cech_class.omega8
     fct = cech_class.f
-    omega0_regular = regular_form(omega0)
+    omega0_regular = regular_form(omega0) #Present omega0 in the form P dx + Q dy
+    print(omega0_regular)
     omega0_lift = omega0_regular[0].teichmuller()*(C.x.teichmuller().diffn()) + omega0_regular[1].teichmuller()*(C.y.teichmuller().diffn())
-    omega8_regular = regular_form(second_patch(omega8))
+    #Now the obvious lift of omega0 = P dx + Q dy to de Rham-Witt is [P] d[x] + [Q] d[y]
+    omega8_regular = regular_form(second_patch(omega8)) # The same for omega8.
     omega8_regular = (second_patch(omega8_regular[0]), second_patch(omega8_regular[1]))
     u = (C.x)^(-1)
     v = (C.y)/(C.x)^(g+1)
-    omega8_lift = omega0_regular[0].teichmuller()*(u.teichmuller().diffn()) + omega0_regular[1].teichmuller()*(v.teichmuller().diffn())
+    omega8_lift = omega8_regular[0].teichmuller()*(u.teichmuller().diffn()) + omega8_regular[1].teichmuller()*(v.teichmuller().diffn())
-    aux = omega0_lift - omega8_lift - fct.teichmuller().diffn()
+    print('omega8_lift.frobenius().expansion_at_infty()', omega8_lift.frobenius().expansion_at_infty())
-    decom_aux_h2 = decomposition_g0_g8(aux.h2, prec=prec)
+    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_h2.verschiebung().diffn() + aux_omega0.verschiebung(), fct.teichmuller() + aux_f.verschiebung())
+    result = superelliptic_drw_cech(omega0_lift - aux_h2.verschiebung().diffn() - aux_omega0.verschiebung(), fct.teichmuller() + aux_f.verschiebung())
     return result.reduce()
 
 def crystalline_cohomology_basis(self, prec = 50):

sage/init.sage

@@ -25,6 +25,7 @@ load('drafty/regular_on_U0.sage')
 load('drafty/lift_to_de_rham.sage')
 #load('drafty/superelliptic_cohomology_class.sage')
 load('drafty/superelliptic_drw.sage')
+load('drafty/draft.sage')
 #load('drafty/draft_klein_covers.sage')
-load('drafty/2gpcovers.sage')
+#load('drafty/2gpcovers.sage')
 load('drafty/pole_numbers.sage')

sage/superelliptic/decomposition_into_g0_g8.sage

@@ -1,5 +1,6 @@
 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.'''
+    '''Writes fct as a difference g0 - g8 + f, with g0 regular on the affine patch and g8 at the points in infinity
+    and f is combination of basis of H^1(X, OX). Output is (g0, g8, f).'''
     C = fct.curve
     g = C.genus()
     coord = fct.coordinates()
@@ -7,8 +8,6 @@ def decomposition_g0_g8(fct, prec = 50):
     for i, a in enumerate(C.cohomology_of_structure_sheaf_basis()):
         nontrivial_part += coord[i]*a
     fct -= nontrivial_part
-    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
     fct = Fxy(fct.function)
@@ -20,7 +19,7 @@ def decomposition_g0_g8(fct, prec = 50):
     for monomial in num.monomials():
         aux = superelliptic_function(C, monomial)
         if aux.expansion_at_infty().valuation() >= aux_den.expansion_at_infty().valuation():
-            g8 += num.monomial_coefficient(monomial)*aux/aux_den
+            g8 -= num.monomial_coefficient(monomial)*aux/aux_den
         else:
             g0 += num.monomial_coefficient(monomial)*aux/aux_den
     return (g0, g8, nontrivial_part)

sage/superelliptic/superelliptic_form_class.sage

@@ -100,6 +100,8 @@ class superelliptic_form:
         FxRy.<y> = PolynomialRing(Fx)
         g = reduction(C, y^m*g)
         g = FxRy(g)
+        if j == 0:
+            return g.monomial_coefficient(y^(0))/C.polynomial
         return g.monomial_coefficient(y^(m-j))
     
     def is_regular_on_U0(self):
@@ -107,7 +109,7 @@ class superelliptic_form:
         F = C.base_ring
         m = C.exponent
         Rx.<x> = PolynomialRing(F)
-        for j in range(1, m):
+        for j in range(0, m):
             if self.jth_component(j) not in Rx:
                 return 0
         return 1
@@ -138,6 +140,15 @@ class superelliptic_form:
         dx_series = x_series.derivative()
         return g*dx_series
     
+    def expansion(self, pt, prec = 50):
+        '''Expansion in the completed ring of the point pt. If pt is an integer, it means the corresponding place at infinity.'''
+        if pt in ZZ:
+            return self.expansion_at_infty(place=pt, prec=prec)
+        C = self.curve
+        dx_series = C.x.expansion(pt = pt, prec=prec).derivative()
+        aux_fct = superelliptic_function(C, self.form)
+        return aux_fct.expansion(pt=pt, prec=prec)*dx_series
+        
     def residue(self, place = 0, prec=30):
         return self.expansion_at_infty(place = place, prec=prec)[-1]
     

sage/superelliptic/superelliptic_function_class.sage

@@ -135,6 +135,30 @@ class superelliptic_function:
         xx = Rt(1/(t^M*ww^b))
         yy = 1/(t^R*ww^a)
         return Rt(fct(x = Rt(xx), y = Rt(yy)))
+    
+    def expansion(self, pt, prec = 50):
+        '''Expansion in the completed ring of the point pt. If pt is an integer, it means the corresponding place at infinity.'''
+        if pt in ZZ:
+            return self.expansion_at_infty(place=pt, prec=prec)
+        x0, y0 = pt[0], pt[1]
+        C = self.curve
+        f = C.polynomial
+        F = C.base_ring
+        m = C.exponent
+        Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
+        Rxy.<x, y> = PolynomialRing(F, 2)
+        Fxy = FractionField(Rxy)
+        if y0 !=0 and f.derivative()(x0) != 0:
+            y_series = f(x = t + x0).nth_root(m)
+            return Rt(self.function(x = t + x0, y = y_series))
+        if f.derivative()(x0) == 0: # then x - x0 is a uniformizer
+            y_series = Rt(f(x = t+x0).nth_root(m))
+            return Rt(self.function(x = t + x0, y = y_series))
+        if y0 == 0: #then y is a uniformizer
+            f1 = f(x = x+x0) - y0
+            x_series = new_reverse(f1(x = t), prec = prec)
+            x_series = x_series(t = t^m - y0) + x0
+            return self.function(x = x_series,  y = t)
 
     def pth_root(self):
         '''Compute p-th root of given function. This uses the following fact: if h = H^p, then C(h*dx/x) = H*dx/x.'''