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
@@ -94,7 +97,8 @@ 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):
@@ -102,3 +106,7 @@ class as_cech:
         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())

as_covers/as_cover_class.sage

@@ -385,6 +385,50 @@ class as_cover:
                     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)

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
@@ -81,6 +84,10 @@ class as_form:
         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
@@ -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):
+    def expansion_at_infty(self, place):
-        return self.form.expansion_at_infty()*(self.curve.dx.expansion_at_infty())^(self.mult)
+        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

@@ -23,17 +23,18 @@ 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]
+    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)
-    F = AS.base_ring
         basis = [holo_basis, str_basis, dr_basis]
     return as_group_action_matrices(F, basis[2], generators, basis = basis)
 

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 basis == 0:
+            basis = [AS.holomorphic_differentials_basis(), AS.cohomology_of_structure_sheaf_basis()]
+        holo_diffs = basis[0]
+        coh_basis =  basis[1]
+        f_products = []
+        for f in coh_basis:
+            f_products += [[omega.serre_duality_pairing(f) for omega in holo_diffs]]
+        product_of_fct_and_omegas = []
+        product_of_fct_and_omegas = [omega.serre_duality_pairing(self) 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)
+        return coh_coordinates
+    
+    def diffn(self):
+        C = self.curve
+        C_super = C.quotient
+        n = C.height
+        RxyzQ, Rxyz, x, y, z = C.fct_field
+        fcts = C.functions
+        f = self.function
+        y_super = superelliptic_function(C_super, y)
+        dy_super = y_super.diffn().form
+        dz = []
+        for i in range(n):
+            dfct = fcts[i].diffn().form
+            dz += [-dfct]
+        result = f.derivative(x)
+        result += f.derivative(y)*dy_super
+        for i in range(n):
+            result += f.derivative(z[i])*dz[i]
+        return as_form(C, result)
+        
+    def valuation(self, place = 0):
+        '''Return valuation at i-th place at infinity.'''
+        C = self.curve
+        F = C.base_ring
+        Rt.<t> = LaurentSeriesRing(F)
+        return Rt(self.expansion_at_infty(place = place)).valuation()

elementary_covers/as_polyforms.sage

@@ -0,0 +1,221 @@
+class as_polyform:
+    def __init__(self, form, mult):
+        '''Elements of H^0(Ω^⊗n). Usage: mult is n, form should be as_function.'''
+        self.form = form
+        self.curve = form.curve
+        self.mult =  mult
+        
+    def __add__(self, other):
+        return as_polyform(self.form + other.form, self.mult)
+    
+    def __repr__(self):
+        return '(' + str(self.form) + ') dx⊗' + str(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."""
+        AS = self.curve
+        if basis == 0:
+            basis = AS.holo_polydifferentials_basis(mult = self.mult)
+        RxyzQ, Rxyz, x, y, z = AS.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.function) for omega in basis])
+        basis = [denom*omega.form.function for omega in basis]
+        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.'''
+    v = AS.dx.valuation()
+    result = AS.at_most_poles(mult*v, threshold=threshold)
+    result = [as_polyform(omega, mult) for omega in result]
+    if mult == 1 and len(result) < AS.genus():
+        raise ValueError('Increase threshold, not all forms found.')
+    if mult > 1 and len(result) < (2*mult - 1)*(AS.genus() - 1):
+        raise ValueError('Increase threshold, not all forms found.')
+    return result
+
+as_cover.holo_polydifferentials_basis = as_holo_polydifferentials_basis
+
+def as_symmetric_power_basis(AS, n, threshold = 8):
+    '''Give the basis of H^0(Ω)^⊙n for n = mult.'''
+    g = AS.genus()
+    B0 = AS.holomorphic_differentials_basis(threshold=threshold)
+    from itertools import product
+    indices = [list(range(g)) for i in range(n)]
+    indices_nonrepeating = []
+    for i in product(*indices):
+        if non_decreasing(i):
+            indices_nonrepeating += [i]
+    result = []
+    for i in indices_nonrepeating:
+        tensor_form = [B0[i[j]] for j in range(n)]
+        tensor_form = [1] + tensor_form
+        result += [tensor_form]
+    result = [as_symmetric_product_forms([a]) for a in result]
+    return result
+
+def as_canonical_ideal(AS, n, threshold=8):
+    '''Return the n-th homogeneous part of the canonical ideal.'''
+    B0 = AS.holomorphic_differentials_basis(threshold=threshold)
+    F = AS.base_ring
+    g = AS.genus()
+    RxyzQ, Rxyz, x, y, z = AS.fct_field
+    from itertools import product
+    B1 = as_symmetric_power_basis(AS, n, threshold = 8)
+    B2 = AS.holo_polydifferentials_basis(n, threshold = threshold)
+    g = AS.genus()
+    r = len(B2)
+    M = matrix(F, len(B1), r)
+    for i in range(0, len(B1)):
+        c = B1[i].multiply()
+        #return M, c.coordinates(basis=B2)
+        M[i, :] = vector(c.coordinates(basis=B2))
+    K = M.kernel().basis()
+    result = []
+    for v in K:
+        kernel_vector = 0*B1[0]
+        for i in range(len(v)):
+            kernel_vector += v[i]*B1[i]
+        result += [kernel_vector]
+    return result
+
+as_cover.canonical_ideal = as_canonical_ideal
+
+def as_canonical_ideal_polynomials(AS, n, threshold=8):
+    '''Return the polynomials defining n-th homogeneous part of the canonical ideal.'''
+    return [a.polynomial() for a in AS.canonical_ideal(n, threshold=threshold)]
+
+as_cover.canonical_ideal_polynomials = as_canonical_ideal_polynomials
+
+class as_symmetric_product_forms:
+    def __init__(self, forms_and_coeffs):
+        '''Elements of forms_and_coeffs are of the form [coeff, form_1, ..., form_n]'''
+        self.n = len(forms_and_coeffs[0]) - 1
+        forms_and_coeffs1 = []
+        for atuple in forms_and_coeffs:
+            if atuple[0] != 0 and atuple[1:] not in [a[1:] for a in forms_and_coeffs1]:
+                forms_and_coeffs1 += [atuple]
+            elif atuple[1:] in [a[1:] for a in forms_and_coeffs1]:
+                i = [a[1:] for a in forms_and_coeffs1].index(atuple[1:])
+                forms_and_coeffs1[i][0] += atuple[0]
+        if len(forms_and_coeffs1) == 0:
+            forms_and_coeffs1 = [[0] + forms_and_coeffs[0][1:]]
+        self.tuples = forms_and_coeffs1
+        self.curve = forms_and_coeffs1[0][1].curve
+        
+    def coordinates(self, basis = 0):
+        AS = self.curve
+        g = AS.genus()
+        n = self.n
+        F = AS.base_ring
+        if basis == 0:
+            basis = AS.holomorphic_differentials_basis()
+        from itertools import product
+        indices = [list(range(g)) for i in range(self.n)]
+        result = {i : 0 for i in product(*indices)}
+        for atuple in self.tuples:
+            coors = [omega.coordinates(basis = basis) for omega in atuple[1:]]
+            for i in product(*indices):
+                aux_product = 1
+                for j in range(n):
+                    aux_product *= coors[j][i[j]]
+                result[i] += atuple[0]*aux_product
+        return result
+    
+    def __repr__(self):
+        result = ''
+        for atuple in self.tuples:
+            if atuple[0] !=0:
+                result += str(atuple[0]) + ' * '
+                for j in range(1, self.n+1):
+                    result += "(" + str(atuple[j]) + ")"
+                    if j != self.n:
+                        result + '⊗'
+            result += ' + '
+        return result
+    
+    def __add__(self, other):
+        return as_symmetric_product_forms(self.tuples + other.tuples)
+    
+    def __rmul__(self, other):
+        AS = self.curve
+        F = AS.base_ring
+        if isinstance(other, int) or other in ZZ or other in F:
+            aux_tuples = []
+            for atuple in self.tuples:
+                atuple1 = [other * atuple[0]] + atuple[1:]
+                aux_tuples += [atuple1]
+            return as_symmetric_product_forms(aux_tuples)
+    
+    def multiply(self):
+        '''The map H^0(Ω)^⊙n ---> H^0(Ω^⊗n)'''
+        n = self.n
+        AS = self.curve
+        RxyzQ, Rxyz, x, y, z = AS.fct_field
+        result = as_polyform(0*AS.x, n)
+        for atuple in self.tuples:
+            aux_product = Rxyz(1)
+            for fct in atuple[1:]:
+                aux_product = aux_product * fct.form
+            aux_product = as_function(AS, aux_product)
+            aux_product = as_reduction(AS, aux_product)
+            aux_product = as_function(AS, aux_product)
+            result += as_polyform(atuple[0]*aux_product, n)
+        return result
+
+    def polynomial(self):
+        '''Return the associated polynomial.'''
+        AS = self.curve
+        F = AS.base_ring
+        g = AS.genus()
+        M = self.coordinates()
+        n = self.n
+        Rg = PolynomialRing(F, 'X', g)
+        X = Rg.gens()
+        from itertools import product
+        indices = [list(range(g)) for i in range(self.n)]
+        result = Rg(0)
+        for i in product(*indices):
+            aux_product = Rg(1)
+            for j in range(n):
+                aux_product *= X[i[j]]
+            result += M[i] * aux_product
+        return result
+    
+    def group_action(self, elt):
+        p = self.curve.base_ring.characteristic()
+        n = self.curve.height
+        aux_tuples = []
+        for atuple in self.tuples:
+            aux_tuple = [atuple[0]] + [a.group_action(elt) for a in atuple[1:]]
+            aux_tuples += [aux_tuple]
+        return as_symmetric_product_forms(aux_tuples)
+    
+def non_decreasing(L):
+    return all(x<=y for x, y in zip(L, L[1:]))
+
+def as_matrices_group_action_canonical_ideal(AS, mult, threshold = 8):
+    '''Return the group action matrices for the n-th homogeneous part of the canonical ideal.'''
+    K = as_canonical_ideal(AS, mult, threshold = threshold)
+    n = AS.height
+    F = AS.base_ring
+    K_polynomials = [a.polynomial() for a in K]
+    r = len(K)
+    matrices = []
+    for i in range(n):
+        M = matrix(F, r, r)
+        K_group_action_polynomials = [a.group_action([j == i for j in range(n)]).polynomial() for a in K]
+        for i in range(r):
+            M[i, :] = vector(linear_representation_polynomials(K_group_action_polynomials[i], K_polynomials))
+        matrices += [M]
+    return matrices
+
+as_cover.group_action_canonical_ideal = as_matrices_group_action_canonical_ideal

elementary_covers/as_reduction.sage

@@ -0,0 +1,33 @@
+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'''
+    n = AS.height
+    F = AS.base_ring
+    RxyzQ, Rxyz, x, y, z = AS.fct_field
+    p = F.characteristic()
+    ff = AS.functions
+    ff = [RxyzQ(F.function) for F in ff]
+    fct = RxyzQ(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])
+    fct1 = Rxyz(fct1*denom_norm.function)
+    fct2 = Rxyz(fct2*denom_norm.function)
+    if fct2 != 1:
+        return as_reduction(AS, fct1)/as_reduction(AS, fct2)
+    
+    result = RxyzQ(0)
+    change = 0
+    for a in fct1.monomials():
+        degrees_zi = [a.degree(z[i]) for i in range(n)]
+        d_div = [a.degree(z[i])//p for i in range(n)]
+        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))
+        result += RxyzQ(monomial)        
+        
+    if change == 0:
+        return RxyzQ(result)
+    else:
+        return as_reduction(AS, RxyzQ(result))

elementary_covers/as_transform.sage

@@ -0,0 +1,42 @@
+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
+       z^p - z = power_series, where z = 1/t_new^(jump) and express z in terms of the new uniformizer."""
+    correction = 0
+    F = power_series.parent().base()
+    p = F.characteristic()
+    Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
+    RtQ = FractionField(Rt)
+    power_series = RtQ(power_series)
+    if power_series.valuation() == +Infinity:
+        raise ValueError("Precision is too low.")
+    if power_series.valuation() >= 0:
+        # THIS IS WRONG - THERE ARE SEVERAL PLACES OVER THIS PLACE, AND IT DEPENDS
+        aux = t^p - t
+        z = new_reverse(aux, prec = prec)
+        z = z(t = power_series)
+        return(0, 0, t, z)
+    
+    while(power_series.valuation() % p == 0 and power_series.valuation() < 0):
+        M = -power_series.valuation()/p
+        coeff = power_series.list()[0]   #wspolczynnik a_(-p) w f_AS
+        correction += coeff.nth_root(p)*t^(-M)
+        power_series = power_series - (coeff*t^(-p*M) - coeff.nth_root(p)*t^(-M))
+    jump = max(-(power_series.valuation()), 0)
+    try:
+        if jump != 0:
+            T = nth_root2((power_series)^(-1), jump, prec=prec) #T is defined by power_series = 1/T^m
+    except:
+        print("no ", str(jump), "-th root; divide by", power_series.list()[0])
+        return (jump, power_series.list()[0])
+    if jump != 0:
+        T_rev = new_reverse(T, prec = prec)
+        t_old = T_rev(t^p/nth_root2(1 - t^((p-1)*jump), jump, prec=prec))
+        z = 1/t^(jump) + Rt(correction)(t = t_old)
+        return(jump, correction, t_old, z)
+    if jump == 0:
+        aux = t^p - t
+        z = new_reverse(aux, prec = prec)
+        z = z(t = power_series)
+        z = z + correction
+        return(0, correction, t, z)

elementary_covers/combination_components.sage

@@ -0,0 +1,14 @@
+def combination_components(omega, zmag, w):
+    '''Given a form omega on AS cover and normal basis element zmag, find the decomposition
+    sum_g g(zmag) omega_g and return sum_g g(w) omega_g.'''
+    AS = omega.curve
+    p = AS.characteristic
+    group_elts = [(j1, j2) for j1 in range(p) for j2 in range(p)]
+    zvee = dual_elt(AS, zmag)
+    result = as_form(AS, 0)
+    for g in AS.group:
+        omegag = ith_magical_component(omega, zvee, g)
+        aux_fct1 = w.group_action(g).function
+        aux_fct2 = omegag.form
+        result += as_form(AS, aux_fct1*aux_fct2)
+    return result

elementary_covers/dual_element.sage

@@ -0,0 +1,26 @@
+def dual_elt(AS, zmag):
+    '''Find the trace dual of a given elt zmag in the function field of an Artin-Schreier cover AS.'''
+    p = AS.characteristic
+    n = AS.height
+    G = AS.group
+    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)
+    M = matrix(RxyzQ, p^n, p^n)
+    for i in range(p^n):
+        for j in range(p^n):
+            elt = (zmag.group_action(G[i])*zmag.group_action(G[j])).trace().function
+            elt = Rxyz(elt.numerator())/Rxyz(elt.denominator())
+            M[i, j] = RxyzQ(elt)
+    main_det = M.determinant()
+    zvee = as_function(AS, 0)
+    for i in range(p^n):
+        Mprim = matrix(RxyzQ, M)
+        Mprim[:, i] = vector([(j == 0) for j in range(p^n)])
+        fi = Mprim.determinant()/main_det
+        zvee += as_function(AS, fi*(zmag.group_action(G[i]).function))
+    return zvee

elementary_covers/group_action_matrices.sage

@@ -0,0 +1,53 @@
+def as_group_action_matrices(F, space, list_of_group_elements, basis):
+    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):
+        for j, omega in enumerate(space):
+            omega1 = omega.group_action(g)
+            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

elementary_covers/holomorphic_combinations.sage

@@ -0,0 +1,149 @@
+def holomorphic_combinations(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])
+    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)
+        for vec_wspolrzedna, elt_S in zip(vec, S):
+            eta = elt_S[0]
+            #eta_exp = elt_S[1]
+            forma_holo += vec_wspolrzedna*eta
+            #forma_holo_power_series += vec_wspolrzedna*eta_exp
+        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
+    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([Rt(g[1]).valuation() for g in S])
+    if minimal_valuation >= -pole_order:
+        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 + Rt(eta_exp).valuation()
+            if eta_exp !=0:
+                list_coeffs = a*[0] + Rt(eta_exp).list() + (-minimal_valuation)*[0]
+                list_coeffs = list_coeffs[:-minimal_valuation - pole_order]
+            else:
+                list_coeffs = (-minimal_valuation - pole_order)*[0]
+            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*C_AS.x
+        forma_holo_power_series = Rt(0)
+        for vec_wspolrzedna, elt_S in zip(vec, S):
+            eta = elt_S[0]
+            #eta_exp = elt_S[1]
+            forma_holo += vec_wspolrzedna*eta
+            #forma_holo_power_series += vec_wspolrzedna*eta_exp
+        forms += [forma_holo]
+    return forms
+
+def holomorphic_combinations_forms(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
+    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([Rt(g[1]).valuation() for g in S])
+    if minimal_valuation >= -pole_order:
+        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 + Rt(eta_exp).valuation()
+        list_coeffs = a*[0] + Rt(eta_exp).list() + (-minimal_valuation)*[0]
+        list_coeffs = list_coeffs[:-minimal_valuation - pole_order]
+        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*C_AS.dx
+        forma_holo_power_series = Rt(0)
+        for vec_wspolrzedna, elt_S in zip(vec, S):
+            eta = elt_S[0]
+            #eta_exp = elt_S[1]
+            forma_holo += vec_wspolrzedna*eta
+            #forma_holo_power_series += vec_wspolrzedna*eta_exp
+        forms += [forma_holo]
+    return forms
+
+#print only forms that are log at the branch pts, but not holomorphic

elementary_covers/ith_magical_component.sage

@@ -0,0 +1,6 @@
+def ith_magical_component(omega, zvee, g):
+    '''Given a form omega on AS cover, element g of group AS.group and normal basis element zmag, find the decomposition
+    sum_g g(zmag) omega_g and return omega_g.'''
+    z_vee_g = zvee.group_action(g)
+    new_form = z_vee_g*omega
+    return new_form.trace()

elementary_covers/tests/as_cover_test.sage

@@ -0,0 +1,26 @@
+p = 5
+m = 2
+F = GF(p^2, 'a')
+a = F.gens()[0]
+Rx.<x> = PolynomialRing(F)
+f = x^3 + x^2 + 1
+C_super = superelliptic(f, m)
+Rxy.<x, y> = PolynomialRing(GF(p), 2)
+fArS1 = superelliptic_function(C_super, y*x)
+fArS2 = superelliptic_function(C_super, y*x^2)
+fArS3 = superelliptic_function(C_super, y + x)
+AS1 = as_cover(C_super, [fArS1, fArS2, fArS3], prec=150)
+AS2 = as_cover(C_super, [fArS2, fArS3, fArS1], prec=150)
+print(AS1.genus() == AS2.genus())
+##################
+p = 5
+m = 2
+Rx.<x> = PolynomialRing(GF(p))
+f = x^3 + x^2 + 1
+C_super = superelliptic(f, m)
+Rxy.<x, y> = PolynomialRing(GF(p), 2)
+fArS1 = superelliptic_function(C_super, y*x)
+fArS2 = superelliptic_function(C_super, y*x^2)
+AS1 = as_cover(C_super, [fArS1, fArS2], prec=1000)
+omega = as_form(AS1, 1/y)
+print(omega.expansion_at_infty().valuation()==AS1.exponent_of_different())

elementary_covers/tests/as_polyforms_test.sage

@@ -0,0 +1,27 @@
+p = 2
+F = GF(p)
+Rx.<x> = PolynomialRing(F)
+C = superelliptic(x, 1)
+AS = as_cover(C, [C.x^3, C.x^5], prec = 200)
+B = AS.holo_polydifferentials_basis(3)
+print(len(B) == (2 * 3 - 1) * (AS.genus() - 1)) #is the dimension as predicted by Riemann--Roch?
+I2 = AS.canonical_ideal_polynomials(2)
+R = I2[0].parent() #ring, to which the polynomials belong
+J2 = R.ideal(I2) #ideal defined by the set I
+print(J2.is_prime(), len(R.gens()) - J2.dimension() - 1)
+I3 = AS.canonical_ideal_polynomials(3)
+J3 = R.ideal(I2 + I3)
+print(J3.is_prime(), len(R.gens()) - J3.dimension() - 1)
+
+p = 5
+F = GF(p)
+Rx.<x> = PolynomialRing(F)
+C = superelliptic(x, 1)
+AS = as_cover(C, [C.x^8], prec = 200)
+I2 = AS.canonical_ideal_polynomials(2, threshold=15)
+R = I2[0].parent() #ring, to which the polynomials belong
+J2 = R.ideal(I2) #ideal defined by the set I
+print(J2.is_prime(), len(R.gens()) - J2.dimension() - 1)
+I3 = AS.canonical_ideal_polynomials(3, threshold=20)
+J3 = R.ideal(I2 + I3)
+print(J3.is_prime(), len(R.gens()) - J3.dimension() - 1)

elementary_covers/tests/cartier_test.sage

@@ -0,0 +1,11 @@
+p = 5
+m = 2
+F = GF(p)
+Rx.<x> = PolynomialRing(F)
+f = x^3 + x + 1
+C = superelliptic(f, m)
+g = f(x^p - x)
+C1 = superelliptic(g, m)
+ff = superelliptic_function(C, x)
+AS = as_cover(C, [ff])
+print(C1.cartier_matrix().rank() == AS.cartier_matrix().rank())

elementary_covers/tests/diffn_test.sage

@@ -0,0 +1,15 @@
+p = 3
+m = 1
+F = GF(p)
+Rx.<x> = PolynomialRing(F)
+f = x^2 + 1
+C_super = superelliptic(f, m)
+
+Rxy.<x, y> = PolynomialRing(F, 2)
+f1 = superelliptic_function(C_super, x^2)
+f2 = superelliptic_function(C_super, x^4)
+AS = as_cover(C_super, [f1, f2], prec=1000)
+RxyzQ, Rxyz, x, y, z = AS.fct_field
+f_z = as_function(AS, z[0]*z[1]*y)
+df_z = f_z.diffn()
+print(df_z.form == -z[0]*z[1]*x + y*z[1]*x - y*z[0]*x^3)

elementary_covers/tests/dual_element_test.sage

@@ -0,0 +1,20 @@
+p = 5
+m = 1
+F = GF(p)
+Rx.<x> = PolynomialRing(F)
+f = x
+C_super = superelliptic(f, m)
+
+Rxy.<x, y> = PolynomialRing(F, 2)
+f1 = superelliptic_function(C_super, x^2)
+f2 = superelliptic_function(C_super, x^3)
+AS = as_cover(C_super, [f1, f2], prec=500)
+zmag = (AS.magical_element())[0]
+zdual = dual_elt(AS, zmag)
+
+for i in range(p):
+    for j in range(p):
+        if (i, j) == (0, 0):
+            print((zmag*(zdual.group_action([i, j]))).trace().function == 1)
+        else:
+            print((zmag*(zdual.group_action([i, j]))).trace().function == 0)

elementary_covers/tests/group_action_matrices_test.sage

@@ -0,0 +1,18 @@
+p = 3
+m = 2
+F = GF(p)
+Rx.<x> = PolynomialRing(F)
+f = x^3 + x
+C_super = superelliptic(f, m)
+
+f1 = C_super.x^2*C_super.y
+f2 = C_super.x^3
+AS = as_cover(C_super, [f1, f2], prec=1000)
+
+A, B = AS.group_action_matrices_holo()
+n = A.dimensions()[0]
+print(A*B == B*A)
+print(A^p == identity_matrix(n))
+print(B^p == identity_matrix(n))
+
+print(magma_module_decomposition(A, B))

elementary_covers/tests/ith_component_test.sage

@@ -0,0 +1,15 @@
+p = 5
+m = 1
+F = GF(p)
+Rx.<x> = PolynomialRing(F)
+f = x
+C_super = superelliptic(f, m)
+
+Rxy.<x, y> = PolynomialRing(F, 2)
+f1 = superelliptic_function(C_super, x^2)
+f2 = superelliptic_function(C_super, x^3)
+AS = as_cover(C_super, [f1, f2], prec=500)
+zmag = (AS.magical_element())[0]
+
+om = AS.holomorphic_differentials_basis()[4]
+print(combination_components(om, zmag, zmag).form == om.form)

elementary_covers/tests/ith_ramification_gp_test.sage

@@ -0,0 +1,20 @@
+p = 3
+m = 1
+F = GF(p)
+Rx.<x> = PolynomialRing(F)
+f = x
+C_super = superelliptic(f, m)
+
+Rxy.<x, y> = PolynomialRing(F, 2)
+f1 = superelliptic_function(C_super, x^7)
+f2 = superelliptic_function(C_super, x^4)
+AS = as_cover(C_super, [f1, f2], prec=1000)
+n = AS.height
+d_test = (p^n - 1)
+Gi = AS.group
+i = 1
+while(len(Gi) > 1):
+    Gi = AS.ith_ramification_gp(i)
+    d_test += len(Gi) - 1
+    i+=1
+print(d_test == AS.exponent_of_different())

elementary_covers/tests/ramification_jumps_test.sage

@@ -0,0 +1,25 @@
+p = 5
+m = 1
+F = GF(p)
+Rx.<x> = PolynomialRing(F)
+f = x
+C_super = superelliptic(f, m)
+
+Rxy.<x, y> = PolynomialRing(F, 2)
+f1 = superelliptic_function(C_super, x^2)
+f2 = superelliptic_function(C_super, x^3)
+AS = as_cover(C_super, [f1, f2], prec=500)
+m = AS.jumps[0]
+m1 = m[0]
+m2 = m[1]
+#We compute jumps from jumps in the tower of two Z/p-covers.
+if m1 >= m2:
+    M1 = m2
+    M2 = m2 + p*(m1 - m2)
+else:
+    M1 = m1
+    M2 = m2
+
+theoretical_jumps = [M1, M2]
+theoretical_jumps.sort()
+print(theoretical_jumps == AS.ramification_jumps())

elementary_covers/tests/uniformizer_test.sage

@@ -0,0 +1,12 @@
+p = 3
+m = 1
+F = GF(p)
+Rx.<x> = PolynomialRing(F)
+f = x
+C_super = superelliptic(f, m)
+
+Rxy.<x, y> = PolynomialRing(F, 2)
+f1 = superelliptic_function(C_super, x^7)
+f2 = superelliptic_function(C_super, x^4)
+AS = as_cover(C_super, [f1, f2], prec=1000)
+print(AS.uniformizer().valuation() == 1)

heisenberg_covers/heisenberg_covers.sage

@@ -475,3 +475,42 @@ def at_most_poles2(self, pole_orders, threshold = 8):
                 forms = [(omega, omega.group_action(g).expansion_at_infty(place = i)) for omega in forms]
                 forms = holomorphic_combinations_fcts(forms, pole_orders[(i, g)])
     return forms
+
+def at_most_poles_forms2(self, pole_orders, threshold = 8):
+    """ Find fcts with pole order in infty's at most pole_order from the given dictionary. The keys of the dictionary are pairs (place_at_infty, group element). The items are the poles orders at those places.
+    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 = heisenberg_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_orders[(0, (0, 0, 0))])
+    print('iteration')
+    for i in range(delta):
+        for g in self.fiber(place = i):
+            if i!=0 or g != (0, 0, 0):
+                print('iteration')
+                forms = [(omega, omega.group_action(g).expansion_at_infty(place = i)) for omega in forms]
+                forms = holomorphic_combinations_forms(forms, pole_orders[(i, g)])
+    return forms

heisenberg_covers/heisenberg_form_class.sage

@@ -15,6 +15,14 @@ class heisenberg_form:
     def __repr__(self):
         return "(" + str(self.form)+") * dx"
     
+    def __eq__(self, other):
+        AS = self.curve
+        for i in len(AS.nb_of_pts_at_infty):
+            for g in AS.fiber(place = i):
+                if (self - other).group_action(g).expansion_at_infty(place = i) != 0:
+                    return False
+        return True
+    
     def expansion_at_infty(self, place = 0):
         C = self.curve
         delta = C.nb_of_pts_at_infty
@@ -146,7 +154,8 @@ class heisenberg_form:
         result = result.form
         Rxy.<x, y> = PolynomialRing(F, 2)
         Qxy = FractionField(Rxy)
-        result = heisenberg_reduction(AS, result)
+        result = heisenberg_reduction(C, result)
+        #return heisenberg_form(C, result)
         return superelliptic_form(C_super, Qxy(result))
 
     def residue(self, place=0):

heisenberg_covers/heisenberg_function_class.sage

@@ -14,7 +14,13 @@ class heisenberg_function:
         #self.function = heisenberg_reduction(AS, RxyzQ(g))
         
     def __repr__(self):
-        return str(self.function)
+        result = str(self.function)
+        result = result.replace('x', 'H.x')
+        result = result.replace('y', 'H.y')
+        result = result.replace('z0', 'H.z[0]')
+        result = result.replace('z1', 'H.z[1]')
+        result = result.replace('z2', 'H.z[2]')
+        return result
     
     def __eq__(self, other):
         AS = self.curve
@@ -148,7 +154,6 @@ class heisenberg_function:
             return self.group_action(elt1).group_action((0, 0, 1))
 
     def trace(self):
-        print('trace ')
         C = self.curve
         C_super = C.quotient
         n = C.height

quaternion_covers/quaternion_form_class.sage

@@ -140,7 +140,7 @@ class quaternion_form:
         result = result.form
         Rxy.<x, y> = PolynomialRing(F, 2)
         Qxy = FractionField(Rxy)
-        result = quaternion_reduction(AS, result)
+        result = quaternion_reduction(C, result)
         return superelliptic_form(C_super, Qxy(result))
 
     def residue(self, place=0):

quaternion_covers/quaternion_function_class.sage

@@ -160,7 +160,7 @@ class quaternion_function:
         result = result.function
         Rxy.<x, y> = PolynomialRing(F, 2)
         Qxy = FractionField(Rxy)
-        result = quaternion_reduction(AS, result)
+        result = quaternion_reduction(C, result)
         return superelliptic_function(C_super, Qxy(result))
     
     def coordinates(self, prec = 100, basis = 0):