template - matrix for holo works

as_covers/as_cover_class.sage

@@ -39,12 +39,10 @@ class as_cover:
             list_of_power_series = [g.expansion(pt=pt, prec=prec) for g in list_of_fcts]
             for j in range(n):
                 ####
-                #TUTAJ WSTAWIĆ ZMIANĘ
+                #
                 print(cover_template.fcts[j], {zgen[i] : z_series[i] for i in range(j)} | {zgen[i] : 0 for i in range(j, n)} | {fgen[i] : list_of_power_series[i] for i in range(n)})
                 power_series = Rzf(cover_template.fcts[j]).subs({zgen[i] : z_series[i] for i in range(j)} | {zgen[i] : 0 for i in range(j, n)} | {fgen[i] : list_of_power_series[i] for i in range(n)})
-                
                 ####
-                #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)
@@ -467,3 +465,23 @@ class as_cover:
             result += [as_cech(self, omega, f)]
         return result
     
+    def group_action_matrices_holo(AS, basis=0, threshold=10):
+        n = AS.height
+        if basis == 0:
+            basis = AS.holomorphic_differentials_basis(threshold=threshold)
+        F = AS.base_ring
+        return as_group_action_matrices(F, basis, AS.group.gens, basis = basis)
+    
+    def group_action_matrices_dR(AS, basis = 0, threshold=8):
+        n = AS.height
+        if basis == 0:
+            holo_basis = AS.holomorphic_differentials_basis(threshold = threshold)
+            str_basis = AS.cohomology_of_structure_sheaf_basis(holo_basis = holo_basis, threshold = threshold)
+            dr_basis = AS.de_rham_basis(holo_basis = holo_basis, cohomology_basis = str_basis, threshold=threshold)
+            basis = [holo_basis, str_basis, dr_basis]
+        return as_group_action_matrices(F, basis[2], AS.group.gens, basis = basis)
+    
+    def group_action_matrices_log_holo(AS):
+        n = AS.height
+        F = AS.base_ring
+        return as_group_action_matrices(F, AS.at_most_poles_forms(1), AS.group.gens, basis = AS.at_most_poles_forms(1))

as_covers/as_form_class.sage

@@ -3,13 +3,7 @@ class as_form:
         self.curve = C
         n = C.height
         F = C.base_ring
-        variable_names = 'x, y'
+        RxyzQ, Rxyz, x, y, z = C.fct_field
-        for i in range(n):
-            variable_names += ', z' + str(i)
-        Rxyz = PolynomialRing(F, n+2, variable_names)
-        x, y = Rxyz.gens()[:2]
-        z = Rxyz.gens()[2:]
-        RxyzQ = FractionField(Rxyz)
         self.form = RxyzQ(g)
         
     def __repr__(self):
@@ -27,13 +21,7 @@ class as_form:
         z_series = C.z_series[place]
         dx_series = C.dx_series[place]
         n = C.height
-        variable_names = 'x, y'
+        RxyzQ, Rxyz, x, y, z = C.fct_field
-        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
@@ -49,13 +37,7 @@ class as_form:
         z_series = C.z_series[pt]
         dx_series = C.dx_series[pt]
         n = C.height
-        variable_names = 'x, y'
+        RxyzQ, Rxyz, x, y, z = C.fct_field
-        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
@@ -85,19 +67,21 @@ class as_form:
         return as_form(C, constant*omega)
     
     def reduce(self):
-        aux = as_reduction(self.curve, self.form)
+        RxyzQ, Rxyz, x, y, z = self.curve.fct_field
+        aux = as_reduction(self.curve, RxyzQ(self.form))
         return as_form(self.curve, aux)
 
-    def group_action(self, ZN_tuple):
+    def group_action(self, elt):
         C = self.curve
         n = C.height
-        RxyzQ, Rxyz, x, y, z = C.fct_field
+        aux = as_function(C, self.form)
-        sub_list = {x : x, y : y} | {z[j] : z[j]+ZN_tuple[j] for j in range(n)}
+        aux = aux.group_action(elt)
-        g = self.form
+        return as_form(C, aux.function)
-        return as_form(C, g.substitute(sub_list))
 
     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()
         C = self.curve
         if basis == 0:
             basis = C.holomorphic_differentials_basis()
@@ -109,27 +93,23 @@ class as_form:
         self_with_no_denominator = denom*self
         return linear_representation_polynomials(Rxyz(self_with_no_denominator.form), [Rxyz(omega.form) for omega in basis])
 
-    def trace(self):
+    def trace(self, super=True):
         C = self.curve
         C_super = C.quotient
         n = C.height
         F = C.base_ring
-        variable_names = 'x, y'
+        RxyzQ, Rxyz, x, y, z = C.fct_field
-        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)
         result = as_form(C, 0)
-        G = C.group
+        G = C.group.elts
         for a in G:
             result += self.group_action(a)
         result = result.form
         Rxy.<x, y> = PolynomialRing(F, 2)
         Qxy = FractionField(Rxy)
         result = as_reduction(C, result)
-        return superelliptic_form(C_super, Qxy(result))
+        if super:
+            return superelliptic_form(C_super, Qxy(result))
+        return as_form(C, Qxy(result))
 
     def residue(self, place=0):
         return self.expansion_at_infty(place = place).residue()
@@ -188,10 +168,7 @@ def are_forms_linearly_dependent(set_of_forms):
     C = set_of_forms[0].curve
     F = C.base_ring
     n = C.height
-    variable_names = 'x, y'
+    RxyzQ, Rxyz, x, y, z = C.fct_field
-    for i in range(n):
-        variable_names += ', z' + str(i)
-    Rxyz = PolynomialRing(F, n+2, variable_names)
     denominators = prod(denominator(omega.form) for omega in set_of_forms)
     return is_linearly_dependent([Rxyz(denominators*omega.form) for omega in set_of_forms])
 

as_covers/as_function_class.sage

@@ -116,8 +116,13 @@ class as_function:
             g = self.function
             return as_function(C, g.substitute(sub_list))
         result = self
-        for i in range(n):
+        for i in range(len(G.gens)):
-            for j in range(elt[i]):
+            if isinstance(elt, list): #elt can be a tuple...
+                range_limit = elt[i]
+            else: # ... or an integer.
+                range_limit = elt
+            for j in range(range_limit):
+                print(G.gens[i])
                 result = result.group_action(G.gens[i])
         return result
         
@@ -133,7 +138,7 @@ class as_function:
         F = C.base_ring
         RxyzQ, Rxyz, x, y, z = C.fct_field
         result = as_function(C, 0)
-        G = C.group
+        G = C.group.elts
         for a in G:
             result += self.group_action(a)
         result = result.function

as_covers/as_reduction.sage

@@ -1,5 +1,6 @@
 def as_reduction(AS, fct):
     '''Simplify rational function fct as a function in the function field of AS, so that z[i] appear in powers <p and only in numerator'''
+    print(fct, fct.parent())
     n = AS.height
     F = AS.base_ring
     RxyzQ, Rxyz, x, y, z = AS.fct_field
@@ -10,7 +11,7 @@ def as_reduction(AS, fct):
     fct1 = numerator(fct)
     fct2 = denominator(fct)
     denom = as_function(AS, fct2)
-    denom_norm = prod(as_function(AS, fct2).group_action(g) for g in AS.group if list(g) != n*[0])
+    denom_norm = prod(as_function(AS, fct2).group_action(g) for g in AS.group.elts if g != AS.group.one)
     fct1 = Rxyz(fct1*denom_norm.function)
     fct2 = Rxyz(fct2*denom_norm.function)
     if fct2 != 1:
@@ -24,7 +25,7 @@ def as_reduction(AS, fct):
         if d_div != n*[0]:
             change = 1
         d_rem = [a.degree(z[i])%p for i in range(n)]
-        monomial = fct1.monomial_coefficient(a)*x^(a.degree(x))*y^(a.degree(y))*prod(z[i]^(d_rem[i]) for i in range(n))*prod((z[i] + ff[i])^(d_div[i]) for i in range(n))
+        monomial = fct1.monomial_coefficient(a)*x^(a.degree(x))*y^(a.degree(y))*prod(z[i]^(d_rem[i]) for i in range(n))*prod((AS.rhs[i])^(d_div[i]) for i in range(n))
         result += RxyzQ(monomial)        
         
     if change == 0:

as_covers/group_action_matrices.sage

@@ -8,46 +8,3 @@ def as_group_action_matrices(F, space, list_of_group_elements, basis):
             v1 = omega1.coordinates(basis = basis)
             A[i][:, j] = vector(v1)
     return A
-
-def as_group_action_matrices_holo(AS, basis=0, threshold=10):
-    n = AS.height
-    generators = []
-    for i in range(n):
-        ei = n*[0]
-        ei[i] = 1
-        generators += [ei]
-    if basis == 0:
-        basis = AS.holomorphic_differentials_basis(threshold=threshold)
-    F = AS.base_ring
-    return as_group_action_matrices(F, basis, generators, basis = basis)
-
-as_cover.group_action_matrices_holo = as_group_action_matrices_holo
-    
-def as_group_action_matrices_dR(AS, basis = 0, threshold=8):
-    n = AS.height
-    generators = []
-    F = AS.base_ring
-    for i in range(n):
-        ei = n*[0]
-        ei[i] = 1
-        generators += [ei]
-    if basis == 0:
-        holo_basis = AS.holomorphic_differentials_basis(threshold = threshold)
-        str_basis = AS.cohomology_of_structure_sheaf_basis(holo_basis = holo_basis, threshold = threshold)
-        dr_basis = AS.de_rham_basis(holo_basis = holo_basis, cohomology_basis = str_basis, threshold=threshold)
-        basis = [holo_basis, str_basis, dr_basis]
-    return as_group_action_matrices(F, basis[2], generators, basis = basis)
-
-as_cover.group_action_matrices_dR = as_group_action_matrices_dR
-
-def as_group_action_matrices_log_holo(AS):
-    n = AS.height
-    generators = []
-    for i in range(n):
-        ei = n*[0]
-        ei[i] = 1
-        generators += [ei]
-    F = AS.base_ring
-    return as_group_action_matrices(F, AS.at_most_poles_forms(1), generators, basis = AS.at_most_poles_forms(1))
-
-as_cover.group_action_matrices_log_holo = as_group_action_matrices_log_holo

as_covers/template.sage

@@ -56,3 +56,81 @@ def heisenberg_template(p):
     fcts[2] += (z[0] - z[1])*f[1]
     gp_action = [[z[0] + 1, z[1], z[2] + z[1]], [z[0] + 1, z[1] + 1, z[2]], [z[0], z[1], z[2] - 1]]
     return template(height, field, group, fcts, gp_action)
+
+
+def witt_pol(X, p, n):
+    n = len(X)
+    return sum(p^i*X[i]^(p^(n-i-1)) for i in range(0, n))
+
+def witt_sum(p, n):
+    variables = ''
+    for i in range(0, n+1):
+        variables += 'X' + str(i) + ','
+    for i in range(0, n+1):
+        variables += 'Y' + str(i)
+        if i!=n:
+            variables += ','
+    RQ = PolynomialRing(QQ, variables, 2*(n+1))
+    X = RQ.gens()[:n+1]
+    Y = RQ.gens()[n+1:]
+    Rpx.<x> = PolynomialRing(GF(p), 1)
+    RQx.<x> = PolynomialRing(QQ, 1)
+    if n == 0:
+        return X[0] + Y[0]
+    WS = []
+    for k in range(0, n):
+        aux = witt_sum(p, k)
+        Rold = aux.parent()
+        Xold = Rold.gens()[:k+1]
+        Yold = Rold.gens()[k+1:]
+        WS+= [aux.subs({Xold[i] : X[i] for i in range(0, k)} | {Yold[i] : Y[i] for i in range(0, k)})]
+    return 1/p^n*(witt_pol(X[:n+1], p, n) + witt_pol(Y[:n+1], p, n) - sum(p^k*WS[k]^(p^(n-k)) for k in range(0, n)))
+
+def witt_sum_mod_p(p, n):
+    variables = ''
+    for i in range(0, n+1):
+        variables += 'X' + str(i) + ','
+    for i in range(0, n+1):
+        variables += 'Y' + str(i)
+        if i!=n:
+            variables += ','
+    RQ = PolynomialRing(QQ, variables, 2*(n+1))
+    X = RQ.gens()[:n+1]
+    Y = RQ.gens()[n+1:]
+    P = RQ(witt_sum(p, n))
+    RQp = PolynomialRing(GF(p), variables, 2*(n+1))
+    Xp = RQp.gens()[:n+1]
+    Yp = RQp.gens()[n+1:]
+    return RQp(P)
+
+def witt_template(p, n):
+    height = n
+    field = GF(p)
+    group = cyclic_gp(p, n)
+    variable_names = ''
+    for i in range(n):
+        variable_names += 'z'+str(i)+','
+    for i in range(n):
+        variable_names += 'f'+str(i)
+        if i!=n-1:
+            variable_names += ','
+    R = PolynomialRing(field, 2*n, variable_names)
+    z = R.gens()[:n]
+    f = R.gens()[n:]
+    ###########
+    rhs = []
+    gp_action = []
+    for i in range(0, n):
+        aux = witt_sum_mod_p(p, i)
+        Raux = aux.parent()
+        Xpn = Raux.gens()[:i+1]
+        Ypn = Raux.gens()[i+1:]
+        rhs_aux = aux.subs({Xpn[ii] : z[ii]^p for ii in range(i+1)}|{Ypn[ii] : -z[ii] for ii in range(i+1)})
+        rhs += [rhs_aux]
+        gp_action_aux = aux.subs({Xpn[ii] : z[ii] for ii in range(i+1)}|{Ypn[ii] : ii == 0 for ii in range(i+1)})
+        gp_action += [gp_action_aux]
+    fcts = [rhs[i] - z[i]^p + z[i] + f[i] for i in range(n)]
+    ########
+    aux
+    gp_action = [gp_action]
+    return template(height, field, group, fcts, gp_action)