partially fixed issue with de Rham basis and coordinates for more than one place at infty

as_covers/as_cech_class.sage

@@ -58,6 +58,7 @@ class as_cech:
         r = C.polynomial.degree()
         n = AS.height
         p = AS.characteristic
+        delta = AS.nb_of_pts_at_infty
         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)]
@@ -65,15 +66,8 @@ class as_cech:
         coh_basis =  basis[1]
         dR = basis[2]
         F = AS.base_ring
-        f_products = []
-        for f in coh_basis:
-            f_products += [[omega.serre_duality_pairing(f) for omega in holo_diffs]]
-        product_of_fct_and_omegas = []
         fct = self.f
-        product_of_fct_and_omegas = [omega.serre_duality_pairing(fct) for omega in holo_diffs]
+        coh_coordinates = fct.coordinates(basis = [holo_diffs, coh_basis]) #coeficients of self in the basis elts coming from cohomology of OX
-
-        V = (F^(AS.genus())).span_of_basis([vector(a) for a in f_products])
-        coh_coordinates = V.coordinates(product_of_fct_and_omegas) #coeficients of self in the basis elts coming from cohomology of OX
         for i in range(AS.genus()):
             self -= coh_coordinates[i]*dR[i+AS.genus()]
         coh_coordinates = AS.genus()*[0] + list(coh_coordinates)
@@ -88,8 +82,15 @@ class as_cech:
                         g = (AS.x)^i*prod((AS.z[i1])^(k[i1]) for i1 in range(n))*(AS.y)^j
                         S += [(g, g.expansion_at_infty())]
             S += [(self.f, self.f.expansion_at_infty())]
-            fcts = holomorphic_combinations_fcts(S, 0)
+            S = holomorphic_combinations_fcts(S, 0)
-            for g in fcts:
+            ########
+            for i in range(delta):
+                for g in AS.fiber(place = i):
+                    if i!=0 or g != AS.group.one:
+                        S = [(omega, omega.group_action(g).expansion_at_infty(place = i)) for omega in S]
+                        S = holomorphic_combinations_fcts(S, 0)
+            ######
+            for g in S:
                 if g.function not in Rxyz:
                     for a in F:
                         if (self.f.function - a*g.function in Rxyz):

as_covers/as_cover_class.sage

@@ -188,6 +188,7 @@ class as_cover:
         if len(forms) > self.genus():
             print(len(forms), forms)
             raise ValueError("Increase precision.")
+        forms = [om.reduce() for om in forms]    
         return forms
     
     def cartier_matrix(self, prec=50):

as_covers/as_form_class.sage

@@ -69,10 +69,15 @@ class as_form:
     def reduce(self):
         RxyzQ, Rxyz, x, y, z = self.curve.fct_field
         m = self.curve.quotient.exponent
-        aux = as_reduction(self.curve, RxyzQ(y^m*self.form))
+        #!!!! Problem: how to reduce?
-        return as_form(self.curve, aux/y^m)
+        #####
-        #aux = as_reduction(self.curve, RxyzQ(self.form))
+        #####
-        #return as_form(self.curve, aux)
+        # OLD VERSION: 
+        #aux = as_reduction(self.curve, RxyzQ(y^m*self.form))
+        #return as_form(self.curve, aux/y^m)
+        # NEW VERSION: 
+        aux = as_reduction(self.curve, RxyzQ(self.form))
+        return as_form(self.curve, aux)
     
     def group_action(self, elt):
         C = self.curve
@@ -83,7 +88,9 @@ class as_form:
 
     def coordinates(self, basis = 0):
         """Find coordinates of the given holomorphic form self in terms of the basis forms in a list holo."""
+        print(self)
         self = self.reduce()
+        print('s reduce', self)
         C = self.curve
         if basis == 0:
             basis = C.holomorphic_differentials_basis()

as_covers/group_action_matrices.sage

@@ -1,9 +1,13 @@
-def as_group_action_matrices(F, space, list_of_group_elements, basis):
+def as_group_action_matrices(F, space, list_of_group_elements, basis, info = 0):
     n = len(list_of_group_elements)
     d = len(space)
     A = [matrix(F, d, d) for i in range(n)]
     for i, g in enumerate(list_of_group_elements):
+        if info:
+            print("Matrix for group elt", g)
         for j, omega in enumerate(space):
+            if info:
+                print("coordinates of element " + str(j+1)+" out of " + str(len(space)))
             omega1 = omega.group_action(g)
             v1 = omega1.coordinates(basis = basis)
             A[i][:, j] = vector(v1)

as_covers/template.sage

@@ -43,6 +43,7 @@ def elementary_template(p, n):
 def elementary_cover(list_of_fcts, prec=10):
     n = len(list_of_fcts)
     C = list_of_fcts[0].curve
+    p = C.characteristic
     return as_cover(C, elementary_template(p, n), list_of_fcts, branch_points = [], prec = prec)
 
 def heisenberg_template(p):

superelliptic/superelliptic_cech_class.sage

@@ -39,7 +39,6 @@ class superelliptic_cech:
         return superelliptic_cech(C, superelliptic_form(C, Rx(0)), superelliptic_function(C, fct^p))
 
     def coordinates(self, basis = 0):
-        print('coord', self, self.omega8.valuation())
         C = self.curve
         F = C.base_ring
         m = C.exponent
@@ -48,7 +47,11 @@ class superelliptic_cech:
         FxRy.<y> = PolynomialRing(Fx)
         g = C.genus()
         if basis == 0:
-            basis = C.de_rham_basis()
+            basis_dR = C.de_rham_basis()
+            basis_h = C.holomorphic_differentials_basis()
+            basis_s = C.cohomology_of_structure_sheaf_basis()
+        else:
+            basis_h, basis_s, basis_dR = basis
             
         omega = self.omega0
         fct = self.f
@@ -57,24 +60,21 @@ class superelliptic_cech:
             return vector((2*g)*[0])
         
         if fct.function == Rx(0) and omega.form != Rx(0):
-            print('A')
+            result = list(omega.coordinates(basis = Bh)) + g*[0]
-            result = list(omega.coordinates()) + g*[0]
             result = vector([F(a) for a in result])
             return result
         
-        coord = fct.coordinates()
+        coord = fct.coordinates(basis = basis_h)
         coord = g*[0] + list(coord)
         coord = vector([F(a) for a in coord])
         aux = self
         for i in range(g, 2*g):
-            aux -= coord[i]*basis[i]
+            aux -= coord[i]*basis_dR[i]
         aux_f = decomposition_g0_g8(aux.f)[0]
-        print('aux_f', aux_f, 'aux.f', aux.f)
         aux.omega0 -= aux_f.diffn()
         aux.f = 0*C.x
         aux.omega8 = aux.omega0
-        print('B', aux)
+        return coord + aux.coordinates(basis = basis)
-        return coord + aux.coordinates()
     
     def is_cocycle(self):
         w0 = self.omega0

superelliptic/superelliptic_form_class.sage

@@ -91,7 +91,6 @@ class superelliptic_form:
         return -result
     
     def coordinates(self, basis = 0):
-        print('start', self)
         """Find coordinates of the given holomorphic form self in terms of the basis forms in a list holo."""
         C = self.curve
         if basis == 0:
@@ -102,7 +101,6 @@ class superelliptic_form:
         denom = LCM([denominator(omega.form) for omega in basis])
         basis = [denom*omega.form for omega in basis]
         self_with_no_denominator = denom*self.form
-        print('stop', self_with_no_denominator)
         return linear_representation_polynomials(Rxy(self_with_no_denominator), [Rxy(omega) for omega in basis])
     
     def jth_component(self, j):

superelliptic/superelliptic_function_class.sage

@@ -94,16 +94,14 @@ class superelliptic_function:
         B = g.derivative(y)*f.derivative(x)/(m*y^(m-1))
         return superelliptic_form(C, A+B)
 
-    def coordinates(self, basis = 0, basis_holo = 0, prec=50):
+    def coordinates(self, basis = 0, prec=50):
-        '''Find coordinates in H1(X, OX) in given basis basis with dual basis basis_holo.'''
+        '''Find coordinates in H1(X, OX) in given basis dual to basis *basis*.'''
         C = self.curve
         if basis == 0:
-            basis = C.cohomology_of_structure_sheaf_basis()
+            basis = C.holomorphic_differentials_basis()
-        if basis_holo == 0:
-            basis_holo = C.holomorphic_differentials_basis()
         g = C.genus()
         coordinates = g*[0]
-        for i, omega in enumerate(basis_holo):
+        for i, omega in enumerate(basis):
             coordinates[i] = omega.serre_duality_pairing(self, prec=prec)
         return coordinates