de rham for AS working

as_covers/as_cech_class.sage

@@ -55,6 +55,7 @@ class as_cech:
         m = C.exponent
         r = C.polynomial.degree()
         n = AS.height
+        p = AS.characteristic
         RxyzQ, Rxyz, x, y, z = AS.fct_field
         if basis == 0:
             basis = [AS.holomorphic_differentials_basis(), AS.cohomology_of_structure_sheaf_basis(), AS.de_rham_basis(threshold=threshold)]

as_covers/as_cover_class.sage

@@ -373,7 +373,7 @@ class as_cover:
         for i in range(0, threshold*r):
             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 = as_form(self, x^i*prod(z[i1]^(k[i1]) for i1 in range(n))/y^j)
                     eta_exp = eta.expansion_at_infty()
                     S += [(eta, eta_exp)]
         forms = holomorphic_combinations(S)
@@ -381,7 +381,7 @@ class as_cover:
             raise ValueError("Increase threshold!")
         for omega in forms:
             for a in F:
-                if (a*omega + fct.diffn()).form in Rxyz:
+                if (a*omega + fct.diffn()).is_regular_on_U0():
                     return a*omega + fct.diffn()
         raise ValueError("Unknown.")
 

as_covers/as_form_class.sage

@@ -167,6 +167,15 @@ class as_form:
             return as_form(C, result)
         raise ValueError("Please present first your form as sum z^i omega_i, where omega_i are forms on quotient curve.")
         
+    def is_regular_on_U0(self):
+        AS = self.curve
+        C = AS.quotient
+        m = C.exponent
+        RxyzQ, Rxyz, x, y, z = AS.fct_field
+        if y^(m-1)*self.form in Rxyz:
+            return True
+        return False
+    
 def artin_schreier_transform(power_series, prec = 10):
     """Given a power_series, find correction such that power_series - (correction)^p +correction has valuation
        -jump non divisible by p. Also, express t (the variable) in terms of the uniformizer at infty on the curve