naprawiony brak (1/p)-liniowosci cartiera

sage/drafty/draft.sage

@@ -1,13 +1,14 @@
-p = 5
+p = 3
 m = 2
-F = GF(p)
+F = GF(p^2, 'a')
+a = F.gens()[0]
 Rxx.<x> = PolynomialRing(F)
 #f = (x^3 - x)^3 + x^3 - x
-f = x^3 + x
-f1 = f(x = x^5 - x)
+f = x^3 + a*x + 1
+f1 = f(x = x^p - x)
 C = superelliptic(f, m)
-#C1 = superelliptic(f1, m, prec = 500)
-B = C.crystalline_cohomology_basis(prec = 100, info = 1)
+C1 = superelliptic(f1, m, prec = 500)
+#B = C.crystalline_cohomology_basis(prec = 100, info = 1)
 #B1 = C1.crystalline_cohomology_basis(prec = 100, info = 1)
 
 def crystalline_matrix(C, prec = 50):
@@ -20,8 +21,15 @@ def crystalline_matrix(C, prec = 50):
         M[i, :] = vector(autom(b).coordinates(basis = B))
     return M
 
-for b in B:
-    print(b.regular_form())
+#b0 = de_rham_witt_lift(C.de_rham_basis()[0], prec = 100)
+#b1 = de_rham_witt_lift(C1.de_rham_basis()[2], prec = 300)
+#print(b0.regular_form())
+#print(b1.regular_form())
+for b in C1.de_rham_basis():
+    print(mult_by_p(b.omega0).regular_form())
+
+#for b in B:
+#    print(b.regular_form())
     
 #for b in B1:
 #    print(b.regular_form())

sage/superelliptic/superelliptic_cech_class.sage

@@ -127,14 +127,14 @@ def cut(f, i):
     return sum(R(x^(j-i-1)) * coeff[j] for j in range(i+1, f.degree() + 1))
 
 def polynomial_part(p, h):
-    F = GF(p)
+    F = base_ring(parent(h))
     Rx.<x> = PolynomialRing(F)
     h = Rx(h)
     result = Rx(0)
     for i in range(0, h.degree()+1):
         if (i%p) == p-1:
             power = Integer((i-(p-1))/p)
-            result += Integer(h[i]) * x^(power)    
+            result += F(h[i]) * x^(power)
     return result
 
 #Find delta-th root of unity in field F

sage/superelliptic/superelliptic_form_class.sage

@@ -66,7 +66,11 @@ class superelliptic_form:
             h = Rx(h)
             j1 = (p^(mult_order-1)*j)%m
             B = floor(p^(mult_order-1)*j/m)
-            result += superelliptic_form(C, polynomial_part(p, h)/(f^B*y^(j1)*h_denom))
+            P = polynomial_part(p, h)
+            if F.cardinality() != p:
+                d = P.degree()
+                P = sum(P[i].nth_root(p)*x^i for i in range(0, d+1))
+            result += superelliptic_form(C, P/(f^B*y^(j1)*h_denom))
         return result   
 
     def serre_duality_pairing(self, fct, prec=20):

sage/superelliptic_drw/regular_form.sage

@@ -73,14 +73,22 @@ class superelliptic_regular_drw_form:
         return "[" + str(self.dx) + "] d[x] + [" + str(self.dy) + "] d[y] + V(" + str(self.omega) + ") + dV(" + str(self.h2) + ")"
         
 def regular_drw_form(omega):
+    print(omega.frobenius().is_regular_on_U0())
     C = omega.curve
     p = C.characteristic
     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()
+    print("aux == omega", aux == omega)
+    aux1 = aux.omega
     aux.omega, fct = decomposition_omega0_hpdh(aux.omega)
+    print('aux.omega, fct', aux.omega, fct, aux1.cartier() == fct.diffn(), aux1.verschiebung() == mult_by_p(fct.diffn()))
+    print('mult_by_p(fct.diffn()) == (fct^p).verschiebung().diffn()', mult_by_p(fct.diffn()) == (fct^p).verschiebung().diffn())
     aux.h2 += fct^p
+    print(aux.omega.is_regular_on_U0(), aux.frobenius().is_regular_on_U0())
+    print('aux - omega', aux - omega)
     aux.h2, A = decomposition_g0_pth_power(aux.h2)
+    print('A.diffn().is_regular_on_U0()', A.diffn().is_regular_on_U0())
     aux.omega += (A.diffn()).inv_cartier()
     result = superelliptic_regular_drw_form(omega_aux.dx, omega_aux.dy, aux.omega.regular_form(), aux.h2)
     return result
@@ -95,6 +103,8 @@ 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'''
+    if not omega.is_regular_on_U0():
+        raise ValueError("The form " + str(omega) + " is not regular on U0.")
     C = omega.curve
     f = C.polynomial
     Fxy, Rxy, x, y = C.fct_field

sage/superelliptic_drw/superelliptic_drw_cech.sage

@@ -13,7 +13,10 @@ class superelliptic_drw_cech:
         f_second_comp = fct.f
         decomp_first_comp = decomposition_g0_g8(f_first_comp)
         decomp_second_comp = decomposition_g0_g8(f_second_comp)
-        new = self
+        new = superelliptic_drw_cech(0*C.dx.verschiebung(), 0*C.x.verschiebung())
+        new.omega0 = self.omega0
+        new.omega8 = self.omega8
+        new.f = self.f
         new.omega0 -= decomposition_g0_g8(f_first_comp)[0].teichmuller().diffn()
         new.omega0 -= decomposition_g0_g8(f_second_comp)[0].verschiebung().diffn()
         new.f = decomposition_g0_g8(f_first_comp)[2].teichmuller() + decomposition_g0_g8(f_second_comp)[2].verschiebung()