arbitrary p-gp covers, pt 1

as_covers/as_auxilliary.sage

@@ -24,3 +24,35 @@ def magma_module_decomposition(A, B, text = False, prefix="", sufix="", matrices
     if text:
         return result
     return(magma_free(result))
+
+def magma_is_isomorphic(A1, B1, A2, B2, text=0):
+    q = parent(A1).base_ring().order()
+    p = q.factor()[0][0]
+    n = A1.dimensions()[0]
+    A1 = str(list(A1))
+    B1 = str(list(B1))
+    A2 = str(list(A2))
+    B2 = str(list(B2))
+    A1 = A1.replace("(", "")
+    A1 = A1.replace(")", "")
+    A2 = A2.replace("(", "")
+    A2 = A2.replace(")", "")    
+    B1 = B1.replace("(", "")
+    B1 = B1.replace(")", "")
+    B2 = B2.replace("(", "")
+    B2 = B2.replace(")", "")
+    result = ""
+    if q != p:
+        result += "F<a> := GF(" + str(q) + ");"
+    result += "A1 := MatrixAlgebra<GF("+str(q) + "),"+ str(n) + "|"
+    result += A1 + "," + B1
+    result += ">;"
+    result += "M1 := RModule(RSpace(GF("+str(q)+")," + str(n) + "), A1);"
+    result += "A2 := MatrixAlgebra<GF("+str(q) + "),"+ str(n) + "|"
+    result += A2 + "," + B2
+    result += ">;"
+    result += "M2 := RModule(RSpace(GF("+str(q)+")," + str(n) + "), A2);"
+    result += "IsIsomorphic(M1, M2);"
+    if text:
+        return result
+    return(magma_free(result))

as_covers/as_cech_class.sage

@@ -47,6 +47,9 @@ class as_cech:
         f = self.f
         return as_cech(C, constant*omega, constant*f)
     
+    def reduce(self):
+        return as_cech(self.curve, self.omega0.reduce(), self.f.reduce())
+    
     def coordinates(self, threshold=10, basis = 0):
         '''Find coordinates of self in the de Rham cohomology basis. Threshold is an argument passed to AS.de_rham_basis().'''
         AS = self.curve
@@ -94,11 +97,16 @@ class as_cech:
                             return vector(coh_coordinates)+vector(self.coordinates(threshold=threshold, basis = basis))
         else:
             self.omega0 -= self.f.diffn()
-            return vector(coh_coordinates) + vector(list(self.omega0.coordinates())+AS.genus()*[0])
+            return vector(coh_coordinates) + vector(list(self.omega0.coordinates(basis=holo_diffs))+AS.genus()*[0])
+        
         raise ValueError("Increase threshold.")
         
     def group_action(self, g):
         AS = self.curve
         omega = self.omega0
         f = self.f
-        return as_cech(self.curve, omega.group_action(g), f.group_action(g))
+        return as_cech(self.curve, omega.group_action(g), f.group_action(g))
+    
+    def trace(self):
+        AS = self.curve
+        return as_cech(AS, self.omega0.trace(), self.f.trace())

as_covers/as_cover_class.sage

@@ -384,6 +384,50 @@ class as_cover:
                 if (a*omega + fct.diffn()).is_regular_on_U0():
                     return a*omega + fct.diffn()
         raise ValueError("Unknown.")
+        
+    def lift_to_de_rham_form(self, eta, threshold = 20):
+        '''Given form eta regular on affine part find fct such that eta - d(fct) is regular in infty. (Works for one place at infty now)'''
+        from itertools import product
+        x_series = self.x_series
+        y_series = self.y_series
+        z_series = self.z_series
+        dx_series = self.dx_series
+        delta = self.nb_of_pts_at_infty
+        p = self.characteristic
+        n = self.height
+        prec = self.prec
+        C = self.quotient
+        F = self.base_ring
+        m = C.exponent
+        r = C.polynomial.degree()
+        RxyzQ, Rxyz, x, y, z = self.fct_field
+        Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
+        #Tworzymy zbiór S form z^i x^j y^k dx/y o waluacji >= waluacja z^(p-1)*dx/y
+        S = [(eta, eta.expansion_at_infty())]
+        pr = [list(GF(p)) for _ in range(n)]
+        for i in range(0, threshold*r):
+            for j in range(0, m):
+                for k in product(*pr):
+                    ff = as_function(self, prod(z[i1]^(k[i1]) for i1 in range(n))/x^i*y^j)
+                    ff_exp = ff.diffn().expansion_at_infty()
+                    if ff_exp != 0*ff_exp:
+                        S += [(ff, ff_exp)]
+        for i in range(0, threshold*r):
+            for j in range(0, m):
+                for k in product(*pr):
+                    ff = as_function(self, prod(z[i1]^(k[i1]) for i1 in range(n))*x^i*y^j)
+                    ff_exp = ff.diffn().expansion_at_infty()
+                    if ff_exp != 0*ff_exp:
+                        S += [(ff, ff_exp)]              
+        forms = holomorphic_combinations_mixed(S)
+        if len(forms) <= self.genus():
+            raise ValueError("Increase threshold!")
+        result = []
+        for ff in forms:
+            if ff[0] != 0*self.dx:
+                result += [as_cech(self, ff[0], -ff[1])]
+        return result
+        raise ValueError("Unknown.")
 
     def de_rham_basis(self, holo_basis = 0, cohomology_basis = 0, threshold = 30):
         if holo_basis == 0:

as_covers/as_form_class.sage

@@ -15,6 +15,9 @@ class as_form:
     def __repr__(self):
         return "(" + str(self.form)+") * dx"
     
+    def __eq__(self, other):
+        return self.expansion_at_infty() == other.expansion_at_infty()
+    
     def expansion_at_infty(self, place = 0):
         C = self.curve
         delta = C.nb_of_pts_at_infty
@@ -80,6 +83,10 @@ class as_form:
         C = self.curve
         omega = self.form
         return as_form(C, constant*omega)
+    
+    def reduce(self):
+        aux = as_reduction(self.curve, self.form)
+        return as_form(self.curve, aux)
 
     def group_action(self, ZN_tuple):
         C = self.curve
@@ -121,7 +128,7 @@ class as_form:
         result = result.form
         Rxy.<x, y> = PolynomialRing(F, 2)
         Qxy = FractionField(Rxy)
-        result = as_reduction(AS, result)
+        result = as_reduction(C, result)
         return superelliptic_form(C_super, Qxy(result))
 
     def residue(self, place=0):

as_covers/as_function_class.sage

@@ -132,6 +132,10 @@ class as_function:
         g = self.function
         return as_function(C, g.substitute(sub_list))
 
+    def reduce(self):
+        aux = as_reduction(self.curve, self.function)
+        return as_function(self.curve, aux)
+    
     def trace(self):
         C = self.curve
         C_super = C.quotient
@@ -151,7 +155,7 @@ class as_function:
         result = result.function
         Rxy.<x, y> = PolynomialRing(F, 2)
         Qxy = FractionField(Rxy)
-        result = as_reduction(AS, result)
+        result = as_reduction(C, result)
         return superelliptic_function(C_super, Qxy(result))
     
     def coordinates(self, prec = 100, basis = 0):

as_covers/as_polyforms.sage

@@ -11,8 +11,8 @@ class as_polyform:
     def __repr__(self):
         return '(' + str(self.form) + ') dx⊗' + str(self.mult)
     
-    def expansion_at_infty(self):
-        return self.form.expansion_at_infty()*(self.curve.dx.expansion_at_infty())^(self.mult)
+    def expansion_at_infty(self, place):
+        return self.form.expansion_at_infty(place=place)*(self.curve.dx.expansion_at_infty(place=place))^(self.mult)
     
     def coordinates(self, basis = 0):
         """Find coordinates of the given holomorphic form self in terms of the basis forms in a list holo."""
@@ -27,6 +27,9 @@ class as_polyform:
         self_with_no_denominator = denom*self.form.function
         return linear_representation_polynomials(Rxyz(self_with_no_denominator), [Rxyz(omega) for omega in basis])
     
+    def group_action(self, elt):
+        return as_polyform(self.form.group_action(elt), self.mult)
+    
     
 def as_holo_polydifferentials_basis(AS, mult, threshold = 8):
     '''Give the basis of H^0(Ω^⊗n) for n = mult.'''

as_covers/group_action_matrices.sage

@@ -23,18 +23,19 @@ def as_group_action_matrices_holo(AS, basis=0, threshold=10):
 
 as_cover.group_action_matrices_holo = as_group_action_matrices_holo
     
-def as_group_action_matrices_dR(AS, threshold=8):
+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]
-    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)
-    F = AS.base_ring
-    basis = [holo_basis, str_basis, dr_basis]
+    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

as_covers/holomorphic_combinations.sage

@@ -22,7 +22,7 @@ def holomorphic_combinations(S):
     # Sprawdzamy, jakim formom odpowiadają elementy V.
     forms = []
     for vec in V.basis():
-        forma_holo = 0*C_AS.dx
+        forma_holo = 0*S[0][0]
         forma_holo_power_series = Rt(0)
         for vec_wspolrzedna, elt_S in zip(vec, S):
             eta = elt_S[0]
@@ -32,6 +32,49 @@ def holomorphic_combinations(S):
         forms += [forma_holo]
     return forms
 
+def holomorphic_combinations_mixed(S):
+    """Given a list S of pairs (form, corresponding Laurent series at some pt), find their combinations holomorphic at that pt."""
+    C_AS = S[0][0].curve
+    p = C_AS.characteristic
+    F = C_AS.base_ring
+    prec = C_AS.prec
+    Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
+    RtQ = FractionField(Rt)
+    minimal_valuation = min([g[1].valuation() for g in S])
+    print(minimal_valuation)
+    if minimal_valuation >= 0:
+        return [s[0] for s in S]
+    list_of_lists = [] #to będzie lista złożona z list współczynników część nieholomorficznych rozwinięcia form z S
+    for eta, eta_exp in S:
+        a = -minimal_valuation + eta_exp.valuation()
+        list_coeffs = a*[0] + eta_exp.list() + (-minimal_valuation)*[0]
+        list_coeffs = list_coeffs[:-minimal_valuation]
+        list_of_lists += [list_coeffs]
+    M = matrix(F, list_of_lists)
+    V = M.kernel() #chcemy wyzerować części nieholomorficzne, biorąc kombinacje form z S
+
+
+    # Sprawdzamy, jakim formom odpowiadają elementy V.
+    forms = []
+    for vec in V.basis():
+        forma_holo = 0*S[0][0]
+        forma_holo_power_series = Rt(0)
+        res1 = 0*C_AS.dx
+        res2 = 0*C_AS.x
+        res = 0*C_AS.dx
+        for vec_wspolrzedna, elt_S in zip(vec, S):
+            eta = elt_S[0]
+            if isinstance(eta, as_form):
+                res += vec_wspolrzedna*eta
+                res1 += vec_wspolrzedna*eta
+            if isinstance(eta, as_function):
+                res += vec_wspolrzedna*eta.diffn()
+                res2 += vec_wspolrzedna*eta
+            #eta_exp = elt_S[1]
+            #forma_holo_power_series += vec_wspolrzedna*eta_exp
+        forms += [(res1, res2)]
+    return forms
+
 def holomorphic_combinations_fcts(S, pole_order):
     '''given a set S of (form, corresponding Laurent series at some pt), find their combinations holomorphic at that pt'''
     C_AS = S[0][0].curve

auxilliaries/hensel.sage

@@ -10,7 +10,7 @@ def naive_hensel(fct, F, start = 1, prec=10):
     #while fct not in RptW:
     #    print(fct)
     #    fct *= W
-    alpha = (fct.derivative())(W = start)
+    alpha = (fct.derivative())(W = RtQ(start))
     w0 = Rt(start)
     i = 1
     while(i < prec):

elementary_covers/as_auxilliary.sage

@@ -0,0 +1,58 @@
+def magma_module_decomposition(A, B, text = False, prefix="", sufix="", matrices=True):
+    """Find decomposition of Z/p^2-module given by matrices A, B into indecomposables using magma.
+    If text = True, print the command for Magma. Else - return the output of Magma free."""
+    q = parent(A).base_ring().order()
+    p = q.factor()[0][0]
+    n = A.dimensions()[0]
+    A = str(list(A))
+    B = str(list(B))
+    A = A.replace("(", "")
+    A = A.replace(")", "")
+    B = B.replace("(", "")
+    B = B.replace(")", "")
+    result = prefix
+    if q != p:
+        result += "F<a> := GF(" + str(q) + ");"
+    result += "A := MatrixAlgebra<GF("+str(q) + "),"+ str(n) + "|"
+    result += A + "," + B
+    result += ">;"
+    result += "M := RModule(RSpace(GF("+str(q)+")," + str(n) + "), A);"
+    result += "L := IndecomposableSummands(M); L;"
+    if matrices:
+        result += "for i in [1 .. #L] do print(Generators(Action(L[i]))); end for;"
+    result += sufix
+    if text:
+        return result
+    return(magma_free(result))
+
+def magma_is_isomorphic(A1, B1, A2, B2, text=0):
+    q = parent(A1).base_ring().order()
+    p = q.factor()[0][0]
+    n = A1.dimensions()[0]
+    A1 = str(list(A1))
+    B1 = str(list(B1))
+    A2 = str(list(A2))
+    B2 = str(list(B2))
+    A1 = A1.replace("(", "")
+    A1 = A1.replace(")", "")
+    A2 = A2.replace("(", "")
+    A2 = A2.replace(")", "")    
+    B1 = B1.replace("(", "")
+    B1 = B1.replace(")", "")
+    B2 = B2.replace("(", "")
+    B2 = B2.replace(")", "")
+    result = ""
+    if q != p:
+        result += "F<a> := GF(" + str(q) + ");"
+    result += "A1 := MatrixAlgebra<GF("+str(q) + "),"+ str(n) + "|"
+    result += A1 + "," + B1
+    result += ">;"
+    result += "M1 := RModule(RSpace(GF("+str(q)+")," + str(n) + "), A1);"
+    result += "A2 := MatrixAlgebra<GF("+str(q) + "),"+ str(n) + "|"
+    result += A2 + "," + B2
+    result += ">;"
+    result += "M2 := RModule(RSpace(GF("+str(q)+")," + str(n) + "), A2);"
+    result += "IsIsomorphic(M1, M2);"
+    if text:
+        return result
+    return(magma_free(result))

elementary_covers/as_cech_class.sage

@@ -0,0 +1,112 @@
+class as_cech:
+    def __init__(self, C, omega, f):
+        self.curve = C
+        n = C.height
+        F = C.base_ring
+        variable_names = 'x, y'
+        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.omega0 = omega
+        self.f = f
+        self.omega8 = self.omega0 - self.f.diffn()
+        #if self.omega0.form not in Rxyz or self.omega8.valuation() < 0:
+        #    raise ValueError('cech cocycle not regular')
+
+    def __repr__(self):
+        return "( " + str(self.omega0)+", " + str(self.f) + " )"
+
+    def __add__(self, other):
+        C = self.curve
+        omega = self.omega0
+        f = self.f
+        omega1 = other.omega0
+        f1 = other.f
+        return as_cech(C, omega + omega1, f+f1)
+
+    def __sub__(self, other):
+        C = self.curve
+        omega = self.omega0
+        f = self.f
+        omega1 = other.omega0
+        f1 = other.f
+        return as_cech(C, omega - omega1, f - f1)
+    
+    def __neg__(self):
+        C = self.curve
+        omega = self.omega0
+        f = self.f
+        return as_cech(C, -omega, -f)
+    
+    def __rmul__(self, constant):
+        C = self.curve
+        omega = self.omega0
+        f = self.f
+        return as_cech(C, constant*omega, constant*f)
+    
+    def reduce(self):
+        return as_cech(self.curve, self.omega0.reduce(), self.f.reduce())
+    
+    def coordinates(self, threshold=10, basis = 0):
+        '''Find coordinates of self in the de Rham cohomology basis. Threshold is an argument passed to AS.de_rham_basis().'''
+        AS = self.curve
+        C = AS.quotient
+        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)]
+        holo_diffs = basis[0]
+        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]
+
+        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)
+        if self.f.function not in Rxyz:
+            #We remove now from f the summands which are obviously regular at infty
+            pr = [list(GF(p)) for _ in range(n)]
+            S = []
+            from itertools import product
+            for i in range(0, threshold*r):
+                for j in range(0, m):
+                    for k in product(*pr):
+                        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)
+            for g in fcts:
+                if g.function not in Rxyz:
+                    for a in F:
+                        if (self.f.function - a*g.function in Rxyz):
+                            self.f.function = self.f.function - a*g.function
+                            return vector(coh_coordinates)+vector(self.coordinates(threshold=threshold, basis = basis))
+        else:
+            self.omega0 -= self.f.diffn()
+            return vector(coh_coordinates) + vector(list(self.omega0.coordinates(basis=holo_diffs))+AS.genus()*[0])
+        
+        raise ValueError("Increase threshold.")
+        
+    def group_action(self, g):
+        AS = self.curve
+        omega = self.omega0
+        f = self.f
+        return as_cech(self.curve, omega.group_action(g), f.group_action(g))
+    
+    def trace(self):
+        AS = self.curve
+        return as_cech(AS, self.omega0.trace(), self.f.trace())

elementary_covers/as_cover_class.sage

@@ -0,0 +1,444 @@
+class as_cover:
+    def __init__(self, C, list_of_fcts, branch_points = [], prec = 10):
+        self.quotient = C
+        self.functions = list_of_fcts
+        self.height = len(list_of_fcts)
+        F = C.base_ring
+        self.base_ring = F
+        p = C.characteristic
+        self.characteristic = p
+        self.prec = prec
+        #group acting
+        n = self.height
+        from itertools import product
+        pr = [list(GF(p)) for _ in range(n)]
+        group = []
+        for a in product(*pr):
+            group += [a]
+        self.group = group
+        #########
+        f = C.polynomial
+        m = C.exponent
+        r = f.degree()
+        delta = GCD(m, r)
+        self.nb_of_pts_at_infty = delta
+        self.branch_points = list(range(delta)) + branch_points
+        Rxy.<x, y> = PolynomialRing(F, 2)
+        Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
+
+        all_x_series = {}
+        all_y_series = {}
+        all_z_series = {}
+        all_dx_series = {}
+        all_jumps = {}
+
+        for pt in self.branch_points:
+            x_series = superelliptic_function(C, x).expansion(pt=pt, prec=prec)
+            y_series = superelliptic_function(C, y).expansion(pt=pt, prec=prec)
+            z_series = []
+            jumps = []
+            n = len(list_of_fcts)
+            list_of_power_series = [g.expansion(pt=pt, prec=prec) for g in list_of_fcts]
+            for j 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)
+                z_series = [zi(t = t_old) for zi in z_series]
+                z_series += [z]
+                jumps += [jump]
+                list_of_power_series = [g(t = t_old) for g in list_of_power_series]
+            
+            all_jumps[pt] = jumps
+            all_x_series[pt] = x_series
+            all_y_series[pt] = y_series
+            all_z_series[pt] = z_series
+            all_dx_series[pt] = x_series.derivative()
+        self.jumps = all_jumps
+        self.x_series = all_x_series
+        self.y_series = all_y_series
+        self.z_series = all_z_series
+        self.dx_series = all_dx_series
+        ##############
+        #Function field
+        variable_names = 'x, y'
+        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.fct_field = (RxyzQ, Rxyz, x, y, z)
+        self.x = as_function(self, x)
+        self.y = as_function(self, y)
+        self.z = [as_function(self, z[j]) for j in range(n)]
+        self.dx = as_form(self, 1)
+        self.one = as_function(self, 1)
+        
+        
+    def __repr__(self):
+        n = self.height
+        p = self.characteristic
+        if n==1:
+            return "(Z/p)-cover of " + str(self.quotient)+" with the equation:\n z^" + str(p) + " - z = " + str(self.functions[0])
+        
+        result = "(Z/p)^"+str(self.height)+ "-cover of " + str(self.quotient)+" with the equations:\n"
+        for i in range(n):
+            result += 'z' + str(i) + "^" + str(p) + " - z" + str(i) + " = " + str(self.functions[i]) + "\n"
+        return result
+        
+    def genus(self):
+        jumps = self.jumps
+        gY = self.quotient.genus()
+        n = self.height
+        branch_pts = self.branch_points
+        p = self.characteristic
+        return p^n*gY + (p^n - 1)*(len(branch_pts) - 1) + sum(p^(n-j-1)*(jumps[pt][j]-1)*(p-1)/2 for j in range(n) for pt in branch_pts)
+    
+    def exponent_of_different(self, place = 0):
+        jumps = self.jumps
+        n = self.height
+        delta = self.nb_of_pts_at_infty
+        p = self.characteristic
+        return sum(p^(n-j-1)*(jumps[place][j]+1)*(p-1) for j in range(n))
+
+    def exponent_of_different_prim(self, place = 0):
+        jumps = self.jumps
+        n = self.height
+        delta = self.nb_of_pts_at_infty
+        p = self.characteristic
+        return sum(p^(n-j-1)*(jumps[place][j])*(p-1) for j in range(n))
+    
+    def holomorphic_differentials_basis(self, threshold = 8):
+        from itertools import product
+        x_series = self.x_series
+        y_series = self.y_series
+        z_series = self.z_series
+        dx_series = self.dx_series
+        delta = self.nb_of_pts_at_infty
+        p = self.characteristic
+        n = self.height
+        prec = self.prec
+        C = self.quotient
+        F = self.base_ring
+        m = C.exponent
+        r = C.polynomial.degree()
+        RxyzQ, Rxyz, x, y, z = self.fct_field
+        Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
+        #Tworzymy zbiór S form z^i x^j y^k dx/y o waluacji >= waluacja z^(p-1)*dx/y
+        S = []
+        pr = [list(GF(p)) for _ in range(n)]
+        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_exp = eta.expansion(pt=self.branch_points[0])
+                    S += [(eta, eta_exp)]
+
+        forms = holomorphic_combinations(S)
+        
+        for pt in self.branch_points[1:]:
+            forms = [(omega, omega.expansion(pt=pt)) for omega in forms]
+            forms = holomorphic_combinations(forms)
+        
+        if len(forms) < self.genus():
+            print("I haven't found all forms, only ", len(forms), " of ", self.genus())
+            return holomorphic_differentials_basis(self, threshold = threshold + 1)
+        if len(forms) > self.genus():
+            raise ValueError("Increase precision.")
+        return forms
+    
+    def cartier_matrix(self, prec=50):
+        g = self.genus()
+        F = self.base_ring
+        M = matrix(F, g, g)
+        for i, omega in enumerate(self.holomorphic_differentials_basis()):
+            M[:, i] = vector(omega.cartier().coordinates())
+        return M
+
+    def at_most_poles(self, pole_order, threshold = 8):
+        """ Find fcts with pole order in infty's at most pole_order. Threshold gives a bound on powers of x in the function.
+           If you suspect that you haven't found all the functions, you may increase it."""
+        from itertools import product
+        x_series = self.x_series
+        y_series = self.y_series
+        z_series = self.z_series
+        delta = self.nb_of_pts_at_infty
+        p = self.characteristic
+        n = self.height
+        prec = self.prec
+        C = self.quotient
+        F = self.base_ring
+        m = C.exponent
+        r = C.polynomial.degree()
+        RxyzQ, Rxyz, x, y, z = self.fct_field
+        F = C.base_ring
+        Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
+        #Tworzymy zbiór S form z^i x^j y^k dx/y o waluacji >= waluacja z^(p-1)*dx/y
+        S = []
+        RQxyz = FractionField(Rxyz)
+        pr = [list(GF(p)) for _ in range(n)]
+        for i in range(0, threshold*r):
+            for j in range(0, m):
+                for k in product(*pr):
+                    eta = as_function(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_fcts(S, pole_order)
+
+        for i in range(1, delta):
+            forms = [(omega, omega.expansion_at_infty(place = i)) for omega in forms]
+            forms = holomorphic_combinations_fcts(forms, pole_order)
+                  
+        return forms
+    
+    def magical_element(self, threshold = 8):
+        list_of_elts = self.at_most_poles(self.exponent_of_different_prim(), threshold)
+        result = []
+        for a in list_of_elts:
+            if a.trace().function != 0:
+                result += [a]
+        return result
+
+    def pseudo_magical_element(self, threshold = 8):
+        list_of_elts = self.at_most_poles(self.exponent_of_different(), threshold)
+        result = []
+        for a in list_of_elts:
+            if a.trace().function != 0:
+                result += [a]
+        return result
+
+    def at_most_poles_forms(self, pole_order, threshold = 8):
+        """Find forms with pole order in all the points at infty equat at most to pole_order. Threshold gives a bound on powers of x in the form.
+           If you suspect that you haven't found all the functions, you may increase it."""
+        from itertools import product
+        x_series = self.x_series
+        y_series = self.y_series
+        z_series = self.z_series
+        delta = self.nb_of_pts_at_infty
+        p = self.characteristic
+        n = self.height
+        prec = self.prec
+        C = self.quotient
+        F = self.base_ring
+        m = C.exponent
+        r = C.polynomial.degree()
+        RxyzQ, Rxyz, x, y, z = self.fct_field
+        Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
+        #Tworzymy zbiór S form z^i x^j y^k dx/y o waluacji >= waluacja z^(p-1)*dx/y
+        S = []
+        RQxyz = FractionField(Rxyz)
+        pr = [list(GF(p)) for _ in range(n)]
+        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_exp = eta.expansion_at_infty()
+                    S += [(eta, eta_exp)]
+
+        forms = holomorphic_combinations_forms(S, pole_order)
+
+        for pt in self.branch_points[1:]:
+            forms = [(omega, omega.expansion(pt=pt)) for omega in forms]
+            forms = holomorphic_combinations_forms(forms, pole_order)
+                  
+        return forms
+    
+    def uniformizer(self, place = 0):
+        '''Return uniformizer of curve self at place-th place at infinity.'''
+        p = self.characteristic
+        n = self.height
+        F = self.base_ring
+        RxyzQ, Rxyz, x, y, z = self.fct_field
+        fx = as_function(self, x)
+        z = [as_function(self, zi) for zi in z]
+        # We create a list of functions. We add there all variables...
+        list_of_fcts = [fx]+z
+        vfx = fx.valuation(place)
+        vz = [zi.valuation(place) for zi in z]
+        
+        # Then we subtract powers of variables with the same valuation (so that 1/t^(kp) cancels) and add to this list.
+        for  j1 in range(n):
+            for j2 in range(n):
+                if j1>j2:
+                    a = gcd(vz[j1] , vz[j2])
+                    vz1 = vz[j1]/a
+                    vz2 = vz[j2]/a
+                    for b in F:
+                        if (z[j1]^(vz2) - b*z[j2]^(vz1)).valuation(place) > (z[j2]^(vz1)).valuation(place):
+                            list_of_fcts += [z[j1]^(vz2) - b*z[j2]^(vz1)]
+        for j1 in range(n):
+            a = gcd(vz[j1], vfx)
+            vzj = vz[j1] /a
+            vfx = vfx/a
+            for b in F:
+                if (fx^(vzj) - b*z[j1]^(vfx)).valuation(place) > (z[j1]^(vfx)).valuation(place):
+                    list_of_fcts += [fx^(vzj) - b*z[j1]^(vfx)]
+        #Finally, we check if on the list there are two elements with the same valuation.
+        for f1 in list_of_fcts:
+            for f2 in list_of_fcts:
+                d, a, b = xgcd(f1.valuation(place), f2.valuation(place))
+                if d == 1:
+                    return f1^a*f2^b
+        raise ValueError("My method of generating fcts with relatively prime valuation failed.")
+        
+
+    def ith_ramification_gp(self, i, place = 0):
+        '''Find ith ramification group at place at infty of nb place.'''
+        G = self.group
+        t = self.uniformizer(place)
+        Gi = [G[0]]
+        for g in G:
+            if g != G[0]:
+                tg = t.group_action(g)
+                v = (tg - t).valuation(place)
+                if v >= i+1:
+                    Gi += [g]
+        return Gi
+    
+    def ramification_jumps(self, place = 0):
+        '''Return list of lower ramification jumps at at place at infty of nb place.'''
+        G = self.group
+        ramification_jps = []
+        i = 0
+        while len(G) > 1:
+            Gi = self.ith_ramification_gp(i+1, place)
+            if len(Gi) < len(G):
+                ramification_jps += [i]
+            G = Gi
+            i+=1
+        return ramification_jps
+    
+    def a_number(self):
+        g = self.genus()
+        return g - self.cartier_matrix().rank()
+
+    def cohomology_of_structure_sheaf_basis(self, holo_basis = 0, threshold = 8):
+        if holo_basis == 0:
+            holo_basis = self.holomorphic_differentials_basis(threshold = threshold)
+        from itertools import product
+        x_series = self.x_series
+        y_series = self.y_series
+        z_series = self.z_series
+        delta = self.nb_of_pts_at_infty
+        p = self.characteristic
+        n = self.height
+        prec = self.prec
+        C = self.quotient
+        F = self.base_ring
+        m = C.exponent
+        r = C.polynomial.degree()
+        RxyzQ, Rxyz, x, y, z = self.fct_field
+        Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
+        #Tworzymy zbiór S form z^i x^j y^k dx/y o waluacji >= waluacja z^(p-1)*dx/y
+        result_fcts = []
+        V = VectorSpace(F,self.genus())
+        S = V.subspace([])
+        RQxyz = FractionField(Rxyz)
+        pr = [list(GF(p)) for _ in range(n)]
+        i = 0
+        while len(result_fcts) < self.genus():
+            for j in range(0, m):
+                for k in product(*pr):
+                    f = as_function(self, prod(z[i1]^(k[i1]) for i1 in range(n))/x^i*y^j)
+                    f_products = [omega.serre_duality_pairing(f) for omega in holo_basis]
+                    if vector(f_products) not in S:
+                        S = S+V.subspace([V(f_products)])
+                        result_fcts += [f]
+            i += 1
+        return result_fcts
+
+    def lift_to_de_rham(self, fct, threshold = 30):
+        '''Given function fct, find form eta regular on affine part such that eta - d(fct) is regular in infty. (Works for one place at infty now)'''
+        from itertools import product
+        x_series = self.x_series
+        y_series = self.y_series
+        z_series = self.z_series
+        dx_series = self.dx_series
+        delta = self.nb_of_pts_at_infty
+        p = self.characteristic
+        n = self.height
+        prec = self.prec
+        C = self.quotient
+        F = self.base_ring
+        m = C.exponent
+        r = C.polynomial.degree()
+        RxyzQ, Rxyz, x, y, z = self.fct_field
+        Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
+        #Tworzymy zbiór S form z^i x^j y^k dx/y o waluacji >= waluacja z^(p-1)*dx/y
+        S = [(fct.diffn(), fct.diffn().expansion_at_infty())]
+        pr = [list(GF(p)) for _ in range(n)]
+        holo = self.holomorphic_differentials_basis(threshold = threshold)
+        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_exp = eta.expansion_at_infty()
+                    S += [(eta, eta_exp)]
+        forms = holomorphic_combinations(S)
+        if len(forms) <= self.genus():
+            raise ValueError("Increase threshold!")
+        for omega in forms:
+            for a in F:
+                if (a*omega + fct.diffn()).is_regular_on_U0():
+                    return a*omega + fct.diffn()
+        raise ValueError("Unknown.")
+        
+    def lift_to_de_rham_form(self, eta, threshold = 20):
+        '''Given form eta regular on affine part find fct such that eta - d(fct) is regular in infty. (Works for one place at infty now)'''
+        from itertools import product
+        x_series = self.x_series
+        y_series = self.y_series
+        z_series = self.z_series
+        dx_series = self.dx_series
+        delta = self.nb_of_pts_at_infty
+        p = self.characteristic
+        n = self.height
+        prec = self.prec
+        C = self.quotient
+        F = self.base_ring
+        m = C.exponent
+        r = C.polynomial.degree()
+        RxyzQ, Rxyz, x, y, z = self.fct_field
+        Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
+        #Tworzymy zbiór S form z^i x^j y^k dx/y o waluacji >= waluacja z^(p-1)*dx/y
+        S = [(eta, eta.expansion_at_infty())]
+        pr = [list(GF(p)) for _ in range(n)]
+        for i in range(0, threshold*r):
+            for j in range(0, m):
+                for k in product(*pr):
+                    ff = as_function(self, prod(z[i1]^(k[i1]) for i1 in range(n))/x^i*y^j)
+                    ff_exp = ff.diffn().expansion_at_infty()
+                    if ff_exp != 0*ff_exp:
+                        S += [(ff, ff_exp)]
+        for i in range(0, threshold*r):
+            for j in range(0, m):
+                for k in product(*pr):
+                    ff = as_function(self, prod(z[i1]^(k[i1]) for i1 in range(n))*x^i*y^j)
+                    ff_exp = ff.diffn().expansion_at_infty()
+                    if ff_exp != 0*ff_exp:
+                        S += [(ff, ff_exp)]              
+        forms = holomorphic_combinations_mixed(S)
+        if len(forms) <= self.genus():
+            raise ValueError("Increase threshold!")
+        result = []
+        for ff in forms:
+            if ff[0] != 0*self.dx:
+                result += [as_cech(self, ff[0], -ff[1])]
+        return result
+        raise ValueError("Unknown.")
+
+    def de_rham_basis(self, holo_basis = 0, cohomology_basis = 0, threshold = 30):
+        if holo_basis == 0:
+            holo_basis = self.holomorphic_differentials_basis(threshold = threshold)
+        if cohomology_basis == 0:
+            cohomology_basis = self.cohomology_of_structure_sheaf_basis(holo_basis = holo_basis, threshold = threshold)
+        result = []
+        for omega in holo_basis:
+            result += [as_cech(self, omega, as_function(self, 0))]
+        for f in cohomology_basis:
+            omega = self.lift_to_de_rham(f, threshold = threshold)
+            result += [as_cech(self, omega, f)]
+        return result
+    

elementary_covers/as_form_class.sage

@@ -0,0 +1,205 @@
+class as_form:
+    def __init__(self, C, g):
+        self.curve = C
+        n = C.height
+        F = C.base_ring
+        variable_names = 'x, y'
+        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):
+        return "(" + str(self.form)+") * dx"
+    
+    def __eq__(self, other):
+        return self.expansion_at_infty() == other.expansion_at_infty()
+    
+    def expansion_at_infty(self, place = 0):
+        C = self.curve
+        delta = C.nb_of_pts_at_infty
+        F = C.base_ring
+        x_series = C.x_series[place]
+        y_series = C.y_series[place]
+        z_series = C.z_series[place]
+        dx_series = C.dx_series[place]
+        n = C.height
+        variable_names = 'x, y'
+        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
+        sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}
+        return g.substitute(sub_list)*dx_series
+    
+    def expansion(self, pt = 0):
+        '''Same code as expansion_at_infty.'''
+        C = self.curve
+        F = C.base_ring
+        x_series = C.x_series[pt]
+        y_series = C.y_series[pt]
+        z_series = C.z_series[pt]
+        dx_series = C.dx_series[pt]
+        n = C.height
+        variable_names = 'x, y'
+        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
+        sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}
+        return g.substitute(sub_list)*dx_series
+    
+    def __add__(self, other):
+        C = self.curve
+        g1 = self.form
+        g2 = other.form
+        return as_form(C, g1 + g2)
+    
+    def __sub__(self, other):
+        C = self.curve
+        g1 = self.form
+        g2 = other.form
+        return as_form(C, g1 - g2)
+    
+    def __neg__(self):
+        C = self.curve
+        g = self.form
+        return as_form(C, -g)
+    
+    def __rmul__(self, constant):
+        C = self.curve
+        omega = self.form
+        return as_form(C, constant*omega)
+    
+    def reduce(self):
+        aux = as_reduction(self.curve, self.form)
+        return as_form(self.curve, aux)
+
+    def group_action(self, ZN_tuple):
+        C = self.curve
+        n = C.height
+        RxyzQ, Rxyz, x, y, z = C.fct_field
+        sub_list = {x : x, y : y} | {z[j] : z[j]+ZN_tuple[j] for j in range(n)}
+        g = self.form
+        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."""
+        C = self.curve
+        if basis == 0:
+            basis = C.holomorphic_differentials_basis()
+        RxyzQ, Rxyz, x, y, z = C.fct_field
+        # We need to have only polynomials to use monomial_coefficients in linear_representation_polynomials,
+        # and sometimes basis elements have denominators. Thus we multiply by them.
+        denom = LCM([denominator(omega.form) for omega in basis])
+        basis = [denom*omega for omega in basis]
+        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):
+        C = self.curve
+        C_super = C.quotient
+        n = C.height
+        F = C.base_ring
+        variable_names = 'x, y'
+        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
+        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))
+
+    def residue(self, place=0):
+        return self.expansion_at_infty(place = place).residue()
+
+    def valuation(self, place=0):
+        return self.expansion_at_infty(place = place).valuation()
+
+    def serre_duality_pairing(self, fct):
+        AS = self.curve
+        return sum((fct*self).residue(place = _) for _ in range(AS.nb_of_pts_at_infty))
+
+    def cartier(self):
+        C = self.curve
+        F = C.base_ring
+        n = C.height
+        ff = C.functions
+        p = F.characteristic()
+        C_super = C.quotient
+        (RxyzQ, Rxyz, x, y, z) = C.fct_field
+        fct = self.form
+        Rxy.<x, y> = PolynomialRing(F, 2)
+        RxyQ = FractionField(Rxy)
+        x, y = Rxyz.gens()[0], Rxyz.gens()[1]
+        z = Rxyz.gens()[2:]
+        num = Rxyz(fct.numerator())
+        den = Rxyz(fct.denominator())
+        result = RxyzQ(0)
+        #return (num, den, z, fct)
+        if den in Rxy:
+            sub_list = {x : x, y : y} | {z[j] : (z[j]^p - RxyzQ(ff[j].function)) for j in range(n)}
+            num = RxyzQ(num.substitute(sub_list))
+            den1 = Rxyz(num.denominator())
+            num = Rxyz(num*den1^p)
+            for monomial in Rxyz(num).monomials():
+                degrees = [monomial.degree(z[i]) for i in range(n)]
+                product_of_z = prod(z[i]^(degrees[i]) for i in range(n))
+                monomial_divided_by_z = monomial/product_of_z
+                product_of_z_no_p = prod(z[i]^(degrees[i]/p) for i in range(n))
+                aux_form = superelliptic_form(C_super, RxyQ(monomial_divided_by_z/den))
+                aux_form = aux_form.cartier()
+                result += product_of_z_no_p * Rxyz(num).monomial_coefficient(monomial) * aux_form.form/den1
+            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 are_forms_linearly_dependent(set_of_forms):
+    from sage.rings.polynomial.toy_variety import is_linearly_dependent
+    C = set_of_forms[0].curve
+    F = C.base_ring
+    n = C.height
+    variable_names = 'x, y'
+    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])
+
+def only_log_forms(C_AS):
+    list1 = AS.at_most_poles_forms(0)
+    list2 = AS.at_most_poles_forms(1)
+    result = []
+    for a in list2:
+        if not(are_forms_linearly_dependent(list1 + result + [a])):
+            result += [a]
+    return result

elementary_covers/as_function_class.sage

@@ -0,0 +1,202 @@
+class as_function:
+    def __init__(self, C, g):
+        self.curve = C
+        F = C.base_ring
+        n = C.height
+        variable_names = 'x, y'
+        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.function = RxyzQ(g)
+        #self.function = as_reduction(AS, RxyzQ(g))
+        
+    def __repr__(self):
+        return str(self.function)
+    
+    def __eq__(self, other):
+        AS = self.curve
+        RxyzQ, Rxyz, x, y, z = AS.fct_field
+        aux = self - other
+        aux = RxyzQ(aux.function)
+        aux = aux.numerator()
+        aux = as_function(AS, aux)
+        aux = aux.expansion_at_infty()
+        if aux.valuation() >= 1:
+            return True
+        return False
+
+    def __add__(self, other):
+        C = self.curve
+        g1 = self.function
+        g2 = other.function
+        return as_function(C, g1 + g2)
+
+    def __sub__(self, other):
+        C = self.curve
+        g1 = self.function
+        g2 = other.function
+        return as_function(C, g1 - g2)
+
+    def __rmul__(self, constant):
+        C = self.curve
+        g = self.function
+        return as_function(C, constant*g)
+    
+    def __neg__(self):
+        C = self.curve
+        g = self.function
+        return as_function(C, -g)
+    
+    def __mul__(self, other):
+        if isinstance(other, as_function):
+            C = self.curve
+            g1 = self.function
+            g2 = other.function
+            return as_function(C, g1*g2)
+        if isinstance(other, as_form):
+            C = self.curve
+            g1 = self.function
+            g2 = other.form
+            return as_form(C, g1*g2)
+    
+    def __truediv__(self, other):
+        C = self.curve
+        g1 = self.function
+        g2 = other.function
+        return as_function(C, g1/g2)
+    
+    def __pow__(self, exponent):
+        C = self.curve
+        g1 = self.function
+        return as_function(C, g1^(exponent))
+    
+    def expansion_at_infty(self, place = 0):
+        C = self.curve
+        delta = C.nb_of_pts_at_infty
+        F = C.base_ring
+        x_series = C.x_series[place]
+        y_series = C.y_series[place]
+        z_series = C.z_series[place]
+        n = C.height
+        variable_names = 'x, y'
+        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.function
+        g = RxyzQ(g)
+        sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}
+        return g.substitute(sub_list)
+    
+    def expansion(self, pt = 0):
+        C = self.curve
+        delta = C.nb_of_pts_at_infty
+        F = C.base_ring
+        x_series = C.x_series[pt]
+        y_series = C.y_series[pt]
+        z_series = C.z_series[pt]
+        n = C.height
+        variable_names = 'x, y'
+        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.function
+        g = RxyzQ(g)
+        sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}
+        return g.substitute(sub_list)
+
+    def group_action(self, ZN_tuple):
+        C = self.curve
+        n = C.height
+        F = C.base_ring
+        variable_names = 'x, y'
+        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)
+        sub_list = {x : x, y : y} | {z[j] : z[j]+ZN_tuple[j] for j in range(n)}
+        g = self.function
+        return as_function(C, g.substitute(sub_list))
+
+    def reduce(self):
+        aux = as_reduction(self.curve, self.function)
+        return as_function(self.curve, aux)
+    
+    def trace(self):
+        C = self.curve
+        C_super = C.quotient
+        n = C.height
+        F = C.base_ring
+        variable_names = 'x, y'
+        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_function(C, 0)
+        G = C.group
+        for a in G:
+            result += self.group_action(a)
+        result = result.function
+        Rxy.<x, y> = PolynomialRing(F, 2)
+        Qxy = FractionField(Rxy)
+        result = as_reduction(C, result)
+        return superelliptic_function(C_super, Qxy(result))
+    
+    def coordinates(self, prec = 100, basis = 0):
+        "Return coordinates in H^1(X, OX)."
+        AS = self.curve
+        if