przed zmiana w expansion at infty w superelliptic

sage/drafty/draft.sage

@@ -1,28 +1,25 @@
-p = 3
+p = 5
 m = 2
 F = GF(p)
 Rx.<x> = PolynomialRing(F)
-f = x^3 - x
+#f = (x^3 - x)^3 + x^3 - x
+f = x^3 + x
+f1 = f(x = x^5 - x)
 C = superelliptic(f, m)
-#C1 = patch(C)
+C1 = superelliptic(f1, m)
-#print(C1.crystalline_cohomology_basis())
+B = C.crystalline_cohomology_basis(prec = 100, info = 1)
-#g1 = C1.polynomial
+B1 = C1.crystalline_cohomology_basis(prec = 500, info = 1)
-#g_AS = g1(x^p - x)
-#C2 = superelliptic(g_AS, 2)
-#print(convert_super_into_AS(C2))
-#converted = (C2.x)^4 - (C2.x)^2
-#print(convert_super_fct_into_AS(converted))
-#b = C.crystalline_cohomology_basis()
-#print(autom(b[0]).coordinates(basis = b))
-#eta1 = (dy + dV(2xy) + V(x^5 \, dy), V(y/x))
-#eta1 = superelliptic_drw_cech(C.y.teichmuller().diffn() + (2*C.x*C.y).verschiebung().diffn() + (C.x^5*C.y.diffn()).verschiebung(), (C.y/C.x).verschiebung())
-#eta2 =  (  x \, dy + 3 x^3 \, dy + dV((2x^4 + 2x^2 + 2) y) + V( (x^4 + x^2 + 1) dy), -[y/x])
-#eta2 = superelliptic_drw_cech(C.x.teichmuller()*(C.y.teichmuller()).diffn() + ((2*C.x^4 + 2*C.x^2 + 2*C.one) * C.y).verschiebung().diffn(), - (C.y/C.x).teichmuller())
-#omega8_lift0, compare = de_rham_witt_lift(C.de_rham_basis()[1])
-#omega8_lift = -(C.x^(-3)).teichmuller()*C.y.teichmuller().diffn() + 2*C.y.teichmuller()*(C.x^(-4)).teichmuller()*C.x.teichmuller().diffn()
-#eta2 = de_rham_witt_lift(C.de_rham_basis()[1])
-#b = autom(eta2)
-#print(autom(C.crystalline_cohomology_basis()[1]).coordinates())
-
 
+def crystalline_matrix(C, prec = 50):
+    B = C.crystalline_cohomology_basis(prec = prec)
+    g = C.genus()
+    p = C.characteristic
+    Zp2 = Integers(p^2)
+    M = matrix(Zp2, 2*g, 2*g)
+    for i, b in enumerate(B):
+        M[i, :] = vector(autom(b).coordinates(basis = B))
+    return M
 
+#M = crystalline_matrix(C, prec = 150)
+#print(M)
+#print(M^3)

sage/superelliptic_drw/de_rham_witt_lift.sage

@@ -22,9 +22,7 @@ def de_rham_witt_lift(cech_class, prec = 50):
     fct = cech_class.f
     omega0_lift = de_rham_witt_lift_form0(omega0)
     omega8_lift = de_rham_witt_lift_form8(omega8)
-    print('omega0_lift, omega8_lift', omega0_lift, omega8_lift)
     aux = omega0_lift - omega8_lift - fct.teichmuller().diffn() # now aux is of the form (V(smth) + dV(smth), V(smth))
-    #return aux
     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.
@@ -32,24 +30,21 @@ def de_rham_witt_lift(cech_class, prec = 50):
     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())
-    compare = omega8_lift-decom_aux_h2[1].verschiebung().diffn() - decomposition_omega0_omega8(aux.omega, prec=prec)[1].verschiebung()
+    return result
-    print("result.omega8 == compare", result.omega8 == compare)
-    print("result.omega8 - compare", result.omega8 - compare)
 
-    #print('test:', omega0_lift - omega8_lift - fct.teichmuller().diffn() == decom_aux_h2[0].verschiebung().diffn() - decom_aux_h2[1].verschiebung().diffn() + decomposition_omega0_omega8(aux.omega, prec=prec)[0].verschiebung() - decomposition_omega0_omega8(aux.omega, prec=prec)[1].verschiebung())
+def crystalline_cohomology_basis(self, prec = 50, info = 0):
-    #print('test 1:', omega0_lift - decom_aux_h2[0].verschiebung().diffn() - decomposition_omega0_omega8(aux.omega, prec=prec)[0].verschiebung() - fct.teichmuller().diffn() == omega8_lift - decom_aux_h2[1].verschiebung().diffn() - decomposition_omega0_omega8(aux.omega, prec=prec)[1].verschiebung())
-    #A = omega0_lift - decomposition_omega0_omega8(aux.omega, prec=prec)[0].verschiebung()
-    #B = decom_aux_h2[0].verschiebung() + fct.teichmuller()
-    #C = omega8_lift - decom_aux_h2[1].verschiebung().diffn() - decomposition_omega0_omega8(aux.omega, prec=prec)[1].verschiebung()
-    #print('test 2:', A - B.diffn() == C)
-    #print('test 3:', result.omega0 == A, result.f == B, result.omega8 == C)
-    #print(result.omega8, '\n \n', compare, '\n \n', aux_f, '\n \n')
-    return result#.reduce()
-
-def crystalline_cohomology_basis(self, prec = 50):
     result = []
-    for a in self.de_rham_basis():
+    for i, a in enumerate(self.de_rham_basis()):
-        result += [de_rham_witt_lift(a, prec = prec)]
+        if info:
+            print("Computing " + str(i) +". basis element")
+        prec1 = prec
+        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

sage/superelliptic_drw/regular_form.sage

@@ -66,7 +66,7 @@ class superelliptic_regular_drw_form:
         B = self.dy
         h2 = self.h2
         omega = self.omega
-        form1 = superelliptic_drw_form(A, omega, h2)
+        form1 = superelliptic_drw_form(A, omega.form(), h2)
         form2 = B.teichmuller()*C.y.teichmuller().diffn()
         
     def __repr__(self):
@@ -77,11 +77,19 @@ def regular_drw_form(omega):
     omega_aux = omega.r()
     omega_aux = omega_aux.regular_form()
     aux = omega - omega_aux.dx.teichmuller()*C.x.teichmuller().diffn() - omega_aux.dy.teichmuller()*C.y.teichmuller().diffn()
-    result = superelliptic_regular_drw_form(omega_aux.dx, omega_aux.dy, aux.omega, aux.h2)
+    aux.omega, fct = decomposition_omega0_hpdh(aux.omega)
+    aux.h2 += fct^p
+    aux.h2 = decomposition_g0_p2th_power(aux.h2)[0]
+    result = superelliptic_regular_drw_form(omega_aux.dx, omega_aux.dy, aux.omega.regular_form(), aux.h2)
     return result
 
 superelliptic_drw_form.regular_form = regular_drw_form
 
+def regular_drw_cech(cocycle):
+    return("( " + str(cocycle.omega0.regular_form()) + ", " + str(cocycle.f) + " )")
+
+superelliptic_drw_cech.regular_form = regular_drw_cech
+    
 def regular_form(omega):
     '''Given a form omega regular on U0, present it as P(x, y) dx + Q(x, y) dy for some polynomial P, Q.
     The output is A(x)*y, B(x), where omega = A(x) y dx + B(x) dy'''

sage/superelliptic_drw/superelliptic_drw_auxilliaries.sage

@@ -20,7 +20,6 @@ def decomposition_omega0_hpdh(omega):
     return (omega1, fct)
 
 def decomposition_omega8_hpdh(omega, prec = 50):
-    print('decomposition_omega8_hpdh', omega)
     '''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
@@ -30,23 +29,16 @@ def decomposition_omega8_hpdh(omega, prec = 50):
     p = C.characteristic
     Rt.<t> = LaurentSeriesRing(F)
     omega_analytic = Rt(laurent_analytic_part(omega.expansion_at_infty(prec = prec)))
-    print('omega_analytic', omega_analytic)
     Cv = C.uniformizer()
     v = Fxy(Cv.function)
     omega_analytic = Fxy(omega_analytic(t = v))
     omega_analytic = superelliptic_function(C, omega_analytic)*Cv.diffn()
-    print('omega_analytic.expansion_at_infty()', omega_analytic.expansion_at_infty(prec = prec))
-    print('omega_analytic', omega_analytic)
     omega8 = omega - omega_analytic
-    print('omega8', omega8)
     dh = omega.cartier() - omega8.cartier()
-    print('dh', dh)
     h = dh.int()
-    print('omega8.expansion_at_infty()', omega8.expansion_at_infty(prec = prec))
     return (omega8, h)
 
 def decomposition_g8_pth_power(fct, prec = 50):
-    print('decomposition_g8_pth_power', fct)
     '''Decompose fct as g8 + A^p, if possible. Output: (g8, A).'''
     C = fct.curve
     F = C.base_ring
@@ -62,7 +54,6 @@ def decomposition_g8_pth_power(fct, prec = 50):
     return (g8, A)
 
 def decomposition_g8_p2th_power(fct):
-    print('decomposition_g8_p2th_power', fct)
     '''Decompose fct as g8 + A^(p^2), if possible. Output: (g8, A).'''
     g0, A = decomposition_g8_pth_power(fct)
     A0, A1 = decomposition_g8_pth_power(A)

sage/superelliptic_drw/superelliptic_drw_cech.sage

@@ -84,15 +84,13 @@ class superelliptic_drw_cech:
         aux.omega8.h2 = decomposition_g8_p2th_power(aux.omega8.h2)[0]
         aux.f += aux.omega8.h2.verschiebung()
         aux.omega8.h2 = 0*C.x
-        print('aux.omega0.omega.cartier() - aux.f.f.pth_root().diffn() == aux.omega8.omega.cartier()', aux.omega0.omega.cartier() - aux.f.f.pth_root().diffn() == aux.omega8.omega.cartier())
         aux_divided_by_p = superelliptic_cech(C, aux.omega0.omega.cartier(), aux.f.f.pth_root())
         return aux_divided_by_p
         
-    def coordinates(self, basis = 0):
+    def coordinates(self, basis = 0, prec = 50):
         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 = C.crystalline_cohomology_basis()
@@ -106,7 +104,6 @@ class superelliptic_drw_cech:
         return coordinates
     
     def is_regular(self):
-        print(self.omega0.r().is_regular_on_U0(), self.omega8.r().is_regular_on_Uinfty(), self.omega0.frobenius().is_regular_on_U0(),     self.omega8.frobenius().is_regular_on_Uinfty())
         eq1 = self.omega0.r().is_regular_on_U0() and self.omega8.r().is_regular_on_Uinfty()
         eq2 = self.omega0.frobenius().is_regular_on_U0() and self.omega8.frobenius().is_regular_on_Uinfty()
         return eq1 and eq2