cohomology of structure sheaf

sage/as_covers/as_auxilliary.sage

@@ -17,40 +17,3 @@ def magmathis(A, B, text = False):
     if text:
         return result
     print(magma_free(result))
-    
-    
-def as_reduction(AS, fct):
-    n = AS.height
-    F = AS.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)
-    ff = AS.functions
-    ff = [RxyzQ(F.function) for F in ff]
-    fct = RxyzQ(fct)
-    fct1 = numerator(fct)
-    fct2 = denominator(fct)
-    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.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(fct1.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)))
-        
-        
-    if change == 0:
-        return RxyzQ(result)
-    else:
-        print(fct, '\n')
-        return as_reduction(AS, RxyzQ(result))

sage/as_covers/as_cover_class.sage

@@ -58,6 +58,16 @@ class as_cover:
         self.y = all_y_series
         self.z = all_z_series
         self.dx = 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)
         
     def __repr__(self):
         n = self.height
@@ -106,17 +116,10 @@ class as_cover:
         F = self.base_ring
         m = C.exponent
         r = C.polynomial.degree()
-        variable_names = 'x, y'
+        RxyzQ, Rxyz, x, y, z = self.fct_field
-        for i in range(n):
-            variable_names += ', z' + str(i)
-        Rxyz = PolynomialRing(F, n+2, variable_names)
-        x, y = Rxyz.gens()[:2]
-        z = Rxyz.gens()[2:]
-        RxyzQ = FractionField(Rxyz)
         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):
@@ -153,13 +156,7 @@ class as_cover:
         F = self.base_ring
         m = C.exponent
         r = C.polynomial.degree()
-        variable_names = 'x, y'
+        RxyzQ, Rxyz, x, y, z = self.fct_field
-        for i in range(n):
-            variable_names += ', z' + str(i)
-        Rxyz = PolynomialRing(F, n+2, variable_names)
-        x, y = Rxyz.gens()[:2]
-        z = Rxyz.gens()[2:]
-        RxyzQ = FractionField(Rxyz)
         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 = []
@@ -211,13 +208,7 @@ class as_cover:
         F = self.base_ring
         m = C.exponent
         r = C.polynomial.degree()
-        variable_names = 'x, y'
+        RxyzQ, Rxyz, x, y, z = self.fct_field
-        for i in range(n):
-            variable_names += ', z' + str(i)
-        Rxyz = PolynomialRing(F, n+2, variable_names)
-        x, y = Rxyz.gens()[:2]
-        z = Rxyz.gens()[2:]
-        RxyzQ = FractionField(Rxyz)
         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 = []
@@ -242,12 +233,8 @@ class as_cover:
         '''Return uniformizer of curve self at i-th place at infinity.'''
         p = self.characteristic
         n = self.height
-        variable_names = 'x, y'
+        F = self.base_ring
-        for j in range(n):
+        RxyzQ, Rxyz, x, y, z = self.fct_field
-            variable_names += ', z' + str(j)
-        Rxyz = PolynomialRing(F, n+2, variable_names)
-        x, y = Rxyz.gens()[:2]
-        z = Rxyz.gens()[2:]
         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...
@@ -262,14 +249,14 @@ class as_cover:
                     a = gcd(vz[j1] , vz[j2])
                     vz1 = vz[j1]/a
                     vz2 = vz[j2]/a
-                    for b in GF(p):
+                    for b in F:
                         if (z[j1]^(vz2) - b*z[j2]^(vz1)).valuation(i) > (z[j2]^(vz1)).valuation(i):
                             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 GF(p):
+            for b in F:
                 if (fx^(vzj) - b*z[j1]^(vfx)).valuation(i) > (z[j1]^(vfx)).valuation(i):
                     list_of_fcts += [fx^(vzj) - b*z[j1]^(vfx)]
         #Finally, we check if on the list there are two elements with the same valuation.
@@ -306,7 +293,43 @@ class as_cover:
             G = Gi
             i+=1
         return ramification_jps
-            
+
+    def cohomology_of_structure_sheaf_basis(self, threshold = 8):
+        holo_diffs = self.holomorphic_differentials_basis(threshold = threshold)
+        from itertools import product
+        x_series = self.x
+        y_series = self.y
+        z_series = self.z
+        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 = []
+                    for omega in holo_diffs:
+                        f_products += [sum((f*omega).residue(place = _) for _ in range(self.nb_of_pts_at_infty))]
+                    if vector(f_products) not in S:
+                        S = S+V.subspace([V(f_products)])
+                        result_fcts += [f]
+            i += 1
+        return result_fcts
+    
 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

sage/as_covers/as_form_class.sage

@@ -85,31 +85,6 @@ class as_form:
         return linear_representation(Rxyz(self.form), holo)
 
     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)
-        g = self.form
-        result = RxyzQ(0)
-        g_num = Rxyz(numerator(g))
-        g_den = Rxyz(denominator(g))
-        z = prod(z[i] for i in range(n))^(p-1)
-        for a in g_num.monomials():
-            if (z.divides(a)):
-                result += g_num.monomial_coefficient(a)*a/z
-        result /= g_den
-        Rxy.<x, y> = PolynomialRing(F, 2)
-        return superelliptic_form(C_super, Rxy(result))
-
-    
-    def trace2(self):
         C = self.curve
         C_super = C.quotient
         n = C.height
@@ -128,9 +103,11 @@ class as_form:
         result = result.form
         Rxy.<x, y> = PolynomialRing(F, 2)
         Qxy = FractionField(Rxy)
+        result = as_reduction(AS, result)
         return superelliptic_form(C_super, Qxy(result))
     
-
+    def residue(self, place=0):
+        return self.expansion_at_infty(i = place).residue()
 
 def artin_schreier_transform(power_series, prec = 10):
     """Given a power_series, find correction such that power_series - (correction)^p +correction has valuation

sage/as_covers/as_function_class.sage

@@ -34,10 +34,16 @@ class as_function:
         return as_function(C, constant*g)
     
     def __mul__(self, other):
-        C = self.curve
+        if isinstance(other, as_function):
-        g1 = self.function
+            C = self.curve
-        g2 = other.function
+            g1 = self.function
-        return as_function(C, g1*g2)
+            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
@@ -88,33 +94,6 @@ class as_function:
         return as_function(C, g.substitute(sub_list))
 
     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)
-        g = self.function
-        g = as_reduction(C, g)
-        result = RxyzQ(0)
-        g_num = Rxyz(numerator(g))
-        g_den = Rxyz(denominator(g))
-        z = prod(z[i] for i in range(n))^(p-1)
-        for a in g_num.monomials():
-            if (z.divides(a)):
-                result += g_num.monomial_coefficient(a)*a/z
-        result /= g_den
-        result = as_reduction(C, result)
-        Rxy.<x, y> = PolynomialRing(F, 2)
-        Qxy = FractionField(Rxy)
-        return superelliptic_function(C_super, Qxy(result))
-
-    def trace2(self):
         C = self.curve
         C_super = C.quotient
         n = C.height
@@ -133,9 +112,32 @@ class as_function:
         result = result.function
         Rxy.<x, y> = PolynomialRing(F, 2)
         Qxy = FractionField(Rxy)
+        result = as_reduction(AS, result)
         return superelliptic_function(C_super, Qxy(result))
     
     
+    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, i = 0):
         '''Return valuation at i-th place at infinity.'''
-        return self.expansion_at_infty(i).valuation()
+        C = self.curve
+        F = C.base_ring
+        Rt.<t> = LaurentSeriesRing(F)
+        return Rt(self.expansion_at_infty(i)).valuation()

sage/as_covers/combination_components.sage

@@ -6,9 +6,9 @@ def combination_components(omega, zmag, w):
     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 i in range(p^2):
+    for g in AS.group:
-        omegai = ith_magical_component(omega, zvee, i)
+        omegag = ith_magical_component(omega, zvee, g)
-        aux_fct1 = w.group_action(group_elts[i]).function
+        aux_fct1 = w.group_action(g).function
-        aux_fct2 = omegai.form
+        aux_fct2 = omegag.form
         result += as_form(AS, aux_fct1*aux_fct2)
     return result

sage/as_covers/dual_element.sage

@@ -13,12 +13,14 @@ def dual_elt(AS, zmag):
     M = matrix(RxyzQ, p^n, p^n)
     for i in range(p^n):
         for j in range(p^n):
-            M[i, j] = (zmag.group_action(G[i])*zmag.group_action(G[j])).trace2()
+            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 += fi*zmag.group_action(G[i])
+        zvee += as_function(AS, fi*(zmag.group_action(G[i]).function))
     return zvee

sage/as_covers/ith_magical_component.sage

@@ -1,8 +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.'''
-    AS = omega.curve
+    z_vee_g = zvee.group_action(g)
-    p = AS.characteristic
+    new_form = z_vee_g*omega
-    z_vee_fct = zvee.group_action(g).function
+    return new_form.trace()
-    new_form = as_form(AS, z_vee_fct*omega.form)
-    return new_form.trace2()

sage/as_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)

sage/as_covers/tests/dual_element_test.sage

@@ -15,6 +15,6 @@ 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]))).trace2().function == 1)
+            print((zmag*(zdual.group_action([i, j]))).trace().function == 1)
         else:
-            print((zmag*(zdual.group_action([i, j]))).trace2().function == 0)
+            print((zmag*(zdual.group_action([i, j]))).trace().function == 0)

sage/drafty/draft.sage

@@ -1,4 +1,4 @@
-p = 3
+p = 5
 m = 1
 F = GF(p)
 Rx.<x> = PolynomialRing(F)
@@ -6,51 +6,14 @@ f = x
 C_super = superelliptic(f, m)
 
 Rxy.<x, y> = PolynomialRing(F, 2)
-f1 = superelliptic_function(C_super, x^2)
+for a in range(3, 13):
-f2 = superelliptic_function(C_super, x^4)
+    for b in range(3, 13):
-f3 = superelliptic_function(C_super, x^5)
+        if a %p != 0 and b%p != 0 and a !=b:
-AS = as_cover(C_super, [f1, f2, f3], prec=1000)
+            try:
-print(AS)
+                f1 = superelliptic_function(C_super, x^a+x)
-zmag = AS.magical_element(threshold = 20)[0]
+                f2 = superelliptic_function(C_super, x^b)
-zvee = dual_elt(AS, zmag)
+                AS = as_cover(C_super, [f1, f2], prec=1000)
-
+                #print(AS.at_most_poles(AS.exponent_of_different_prim()))
-### DEFINE THE POLYNOMIALS
+                print(AS.magical_element(threshold = 20))
-n = AS.height
+            except:
-variable_names = 'x, y'
+                pass
-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:]
-
-###############
-def val_of_components(omega, zvee):
-    result = []
-    AS = omega.curve
-    for i in range(p^2):
-        omega_i = ith_magical_component(omega, zvee, i)
-        val = omega_i.expansion_at_infty().valuation()
-        val = val*p^2 + AS.exponent_of_different()
-        result += [val]
-    return result
-#############
-G = AS.group
-
-g = AS.genus()
-print(AS.exponent_of_different_prim())
-v_x = as_function(AS, x).expansion_at_infty().valuation()
-n = AS.height
-v_z = [as_function(AS, z[i]).expansion_at_infty().valuation() for i in range(n)]
-omega = AS.holomorphic_differentials_basis()[-16]
-from itertools import product
-pr = [list(range(p)) for _ in range(n)]
-for i in range(0, 30):
-    for k in product(*pr):
-        v_w = i*v_x+ sum(k[i]*v_z[i] for i in range(n))
-        if (v_w < - AS.exponent_of_different_prim() + 10 and v_w > - AS.exponent_of_different_prim()):
-            w = as_function(AS, x^i * prod(z[i1]^(k[i1]) for i1 in range(n)))
-            result = as_form(AS, 0)
-            for g in G:
-                component = ith_magical_component(omega, zvee, g).form
-                result += as_form(AS, w.group_action(g).function*component)
-            print(result.expansion_at_infty().valuation(), w.expansion_at_infty().valuation() - zmag.expansion_at_infty().valuation())

sage/drafty/draft2.sage

@@ -1,43 +1,14 @@
 p = 3
 m = 1
-F = GF(p)
+F = GF(p^2, 'a')
+a = F.gens()[0]
 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)
+f2 = superelliptic_function(C_super, a*x^7)
 AS = as_cover(C_super, [f1, f2], prec=1000)
-print(AS.uniformizer())
+#print(AS.uniformizer())
-
+print(AS.ramification_jumps())
-def rep_jumps(AS, threshold = 30):
-    ile = 1
-    i = 1
-    result = []
-    flag = 0
-    while(flag == 0):
-        if len(AS.at_most_poles(i, threshold = threshold)) > ile:
-            ile = len(AS.at_most_poles(i, threshold = threshold))
-            result += [i]
-            if i%p != 0:
-                flag = 1
-        i+=1
-    return result
-
-#print(rep_jumps(AS, threshold = 30))
-
-def uniformizer(AS, threshold = 30):
-    p = AS.characteristic
-    n = AS.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:]
-    zmag = AS.magical_element()[0]
-    v_x = as_function(AS, x).expansion_at_infty().valuation()
-    dprim = AS.exponent_of_different_prim()
-    d, a, b = xgcd(v_x, -dprim)
-    return as_function(AS, x^a*zmag.function^b)

sage/drafty/draft3.sage

@@ -1,7 +1,23 @@
-Rx.<x> = PolynomialRing(GF(5))
+p = 3
-C = superelliptic(x^3 + 1, 2)
+m = 1
-Rxy.<x, y> = PolynomialRing(GF(5), 2)
+F = GF(p)
-f1 = superelliptic_function(C, x*y)
+Rx.<x> = PolynomialRing(F)
-CAS = as_cover(C, [f1])
+f = x
-g = CAS.uniformizer()
+C_super = superelliptic(f, m)
-AA = g
+
+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)
+AS1 = as_cover(C_super, [f1], prec=1000)
+#print(AS.ramification_jumps())
+#print(pole_numbers(AS))
+RxyzQ, Rxyz, x, y, z = AS.fct_field
+zmag = (AS.magical_element())[0]
+zvee = dual_elt(AS, zmag)
+t = AS.uniformizer()
+omega1 = AS1.holomorphic_differentials_basis()[4]
+omega2 = as_form(AS, t.function*RxyzQ(omega1.form))
+
+for g in AS.group:
+    print(ith_magical_component(omega2, zvee, g).expansion_at_infty().valuation(), AS.jumps[0][1])

sage/init.sage

@@ -5,6 +5,7 @@ load('superelliptic/superelliptic_cech_class.sage')
 load('as_covers/as_cover_class.sage')
 load('as_covers/as_function_class.sage')
 load('as_covers/as_form_class.sage')
+load('as_covers/as_reduction.sage')
 load('as_covers/as_auxilliary.sage')
 load('as_covers/dual_element.sage')
 load('as_covers/ith_magical_component.sage')
@@ -14,5 +15,6 @@ load('auxilliaries/reverse.sage')
 load('auxilliaries/hensel.sage')
 ##############
 ##############
-load('drafty/draft2.sage')
+load('drafty/draft.sage')
-#load('drafty/draft3.sage')
+load('drafty/pole_numbers.sage')
+#load('drafty/draft4.sage')

sage/tests.sage

@@ -4,11 +4,13 @@
 #load('as_covers/tests/group_action_matrices_test.sage')
 #print("dual_element_test:")
 #load('as_covers/tests/dual_element_test.sage')
-#print("ith_component_test:")
+print("ith_component_test:")
-#load('as_covers/tests/ith_component_test.sage')
+load('as_covers/tests/ith_component_test.sage')
 #print("ith ramification group test:")
 #load('as_covers/tests/ith_ramification_gp_test.sage')
 #print("uniformizer test:")
 #load('as_covers/tests/uniformizer_test.sage')
-print("ramification jumps test:")
+#print("ramification jumps test:")
-load('as_covers/tests/ramification_jumps_test.sage')
+#load('as_covers/tests/ramification_jumps_test.sage')
+print("diffn_test:")
+load('as_covers/tests/diffn_test.sage')

superelliptic.ipynb

@@ -726,9 +726,6 @@
    "outputs": [
    ],
    "source": [
-    "Rx.<x> = PolynomialRing(GF(5))\n",
-    "f = x^7 + x + 1\n",
-    "C = superelliptic(f, 2)"
    ]
   },
   {
@@ -740,7 +737,6 @@
    "outputs": [
    ],
    "source": [
-    "omega = C.de_rham_basis()[3]"
    ]
   },
   {