fix for algebraic closure

README.md

@@ -99,9 +99,9 @@ One can check valuation of form/function at given place at infinity, using *valu
 
 ## Abelian covers of superelliptic curves
 
-This module allows to define $(\mathbb Z/p)^n$-covers of superelliptic curves in characteristic $p$ that
+This module allows to define $(\mathbb Z/p)^n$\-covers of superelliptic curves in characteristic $p$ that
 are **ramified over the points of infinity**.
-We define now a $(\mathbb Z/3)^2$ cover of curve $C : y^2 = x^3 + x$, given by the equations $z_0^3 - z_0 = x^2 * y$,
+We define now a $(\mathbb Z/3)^2$ cover of curve $C : y^2 = x^3 + x$, given by the equations $z_0^3 - z_0 = x^2 y$,
 $z_1^3 - z_1 = x^3$.
 
 ```

as_covers/as_auxilliary.sage

@@ -1,4 +1,4 @@
-def magma_module_decomposition(A, B, text = False, prefix="", sufix=""):
+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()
@@ -18,7 +18,8 @@ def magma_module_decomposition(A, B, text = False, prefix="", sufix=""):
     result += ">;"
     result += "M := RModule(RSpace(GF("+str(q)+")," + str(n) + "), A);"
     result += "L := IndecomposableSummands(M); L;"
-    result += "for i in [1 .. #L] do print(Generators(Action(L[i]))); end for;"
+    if matrices:
+        result += "for i in [1 .. #L] do print(Generators(Action(L[i]))); end for;"
     result += sufix
     if text:
         return result

as_covers/as_cech_class.sage

@@ -35,6 +35,12 @@ class as_cech:
         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

as_covers/as_form_class.sage

@@ -71,6 +71,11 @@ class as_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

as_covers/as_function_class.sage

@@ -33,6 +33,11 @@ class as_function:
         g = self.function
         return as_function(C, constant*g)
     
+    def __neg__(self, other):
+        C = self.curve
+        g = self.function
+        return as_function(C, -g)
+    
     def __mul__(self, other):
         if isinstance(other, as_function):
             C = self.curve

as_covers/group_action_matrices.sage

@@ -9,14 +9,15 @@ def group_action_matrices(space, list_of_group_elements, basis):
             A[i][:, j] = vector(v1)
     return A
 
-def group_action_matrices_holo(AS):
+def 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]
-    basis = AS.holomorphic_differentials_basis()
+    if basis == 0:
+        basis = AS.holomorphic_differentials_basis(threshold=threshold)
     return group_action_matrices(basis, generators, basis = basis)
     
 def group_action_matrices_dR(AS, threshold=8):

drafty/draft.sage

@@ -1,8 +1,7 @@
-p = 5
+p = 3
-m = 1
+F = GF(p^2, 'a')
-F = GF(p)
+#F1 = F.algebraic_closure('t')
-Rx.<x> = PolynomialRing(F)
+a = F.gens()[0]
-f = x
+R.<x> = PolynomialRing(F)
-C = superelliptic(f, 1)
+P1 = superelliptic(x, 1)
-AS1 = as_cover(C, [C.x^3], prec = 200)
+AS = as_cover(P1, [P1.x^2, a*P1.x^2])
-print(AS1.genus())

drafty/draft2.sage

@@ -1,5 +1,18 @@
-F = GF(2)
+def reduction(g):
-A = matrix(F, [[1, 1, 1], [0, 0, 1], [0, 1, 0]])
+    F = g.parent().base()
-B = matrix(F, [[0, 0, 1], [1, 1, 1], [1, 0, 0]])
+    x, y = g.parent().gens()
-print(A^2 == identity_matrix(3), B^2 == identity_matrix(3), A*B == B*A)
+    Rxy.<x, y> = PolynomialRing(F, 2)
-print(magmathis(A, B))
+    Fxy = FractionField(Rxy)
+    Rx.<x> = PolynomialRing(F)
+    Fx = FractionField(Rx)
+    FxRy.<y> = PolynomialRing(Fx)
+    g = Fxy(g)
+    g1 = g.numerator()
+    g2 = g.denominator()
+    print('aa', FxRy(g2))
+    
+    
+F = GF(3).algebraic_closure()
+R.<x, y> = PolynomialRing(F, 2)
+g = x
+reduction(g)

superelliptic/superelliptic_class.sage

@@ -238,40 +238,44 @@ class superelliptic:
             b += M
         return (C.x)^a/(C.y)^b
 
-def reduction(C, g):
+def reduction(curve, g):
     '''Auxilliary. Given a superelliptic curve C : y^m = f(x) and a polynomial g(x, y)
     it replaces repeteadly all y^m's in g(x, y) by f(x). As a result
     you obtain sum_{i = 0}^{m-1} y^i g_i(x).'''
-    p = C.characteristic
+    p = curve.characteristic
-    F = C.base_ring
+    F = curve.base_ring
-    Rxy.<x, y> = PolynomialRing(F, 2)
+    Fxy, Rxy, x, y = curve.fct_field
-    Fxy = FractionField(Rxy)
+    f = curve.polynomial
-    f = C.polynomial
     r = f.degree()
-    m = C.exponent
+    m = curve.exponent
     g = Fxy(g)
     g1 = g.numerator()
     g2 = g.denominator()
     
     Rx.<x> = PolynomialRing(F)
     Fx = FractionField(Rx)
-    FxRy.<y> = PolynomialRing(Fx)    
+    FxRy1.<y> = PolynomialRing(Fx, 1) #coercion problems from Rxy to FxRy
-    (A, B, C) = xgcd(FxRy(g2), FxRy(y^m - f))
+    FxRy.<y2> = PolynomialRing(Fx) #coercion problems from Rxy to FxRy
-    g = FxRy(g1*B/A)
+    (A, B, C) = xgcd(FxRy(FxRy1(g2)(y = y2)), FxRy(FxRy1(y^m - f)(y = y2))) #coercion problems from Rxy to FxRy
+    g = FxRy(FxRy1(g1)(y = y2))*B/A
+    g = FxRy(g)
     
-    while(g.degree(Rxy(y)) >= m):
+    while(g.degree(y2) >= m):
-        d = g.degree(Rxy(y))
+        d = g.degree(y2)
         G = coff(g, d)
         i = floor(d/m)
-        g = g - G*y^d + f^i * y^(d%m) *G
+        g = g - G*y2^d + f^i * y2^(d%m) *G
-    
+    Rxy1.<x3, y3> = PolynomialRing(F, 2)
-    return(FxRy(g))
+    Fxy1 = FractionField(Rxy1)
+    g = sum(Fxy(y3)^i*Fx(coff(g, i)) for i in range(0, m))
+    g = Fxy(g)
+    return(g)
 
-#Auxilliary. Given a superelliptic curve C : y^m = f(x) and a polynomial g(x, y)
-#it replaces repeteadly all y^m's in g(x, y) by f(x). As a result
-#you obtain \sum_{i = 0}^{m-1} g_i(x)/y^i. This is needed for reduction of
-#superelliptic forms.
 def reduction_form(C, g):
+    '''Auxilliary. Given a superelliptic curve C : y^m = f(x) and a polynomial g(x, y)
+    it replaces repeteadly all y^m's in g(x, y) by f(x). As a result
+    you obtain sum_{i = 0}^{m-1} g_i(x)/y^i. This is needed for reduction of
+    superelliptic forms.'''
     F = C.base_ring
     Rxy.<x, y> = PolynomialRing(F, 2)
     Fxy = FractionField(Rxy)
@@ -283,14 +287,15 @@ def reduction_form(C, g):
     g1 = Rxy(0)
     Rx.<x> = PolynomialRing(F)
     Fx = FractionField(Rx)
-    FxRy.<y> = PolynomialRing(Fx)
+    FxRy1.<y> = PolynomialRing(Fx, 1)
+    FxRy.<y2> = PolynomialRing(Fx)
     
-    g = FxRy(g)
+    g = FxRy(FxRy1(g)(y = y2))
     for j in range(0, m):
         if j==0:
             G = coff(g, 0)
-            g1 += FxRy(G)
+            g1 += Fxy(Fx(G))
         else:
             G = coff(g, j)
-            g1 += Fxy(y^(j-m)*f*G)
+            g1 += Fxy(y)^(j-m)*Fxy(Fx(f*G))
     return(g1)

superelliptic/superelliptic_form_class.sage

@@ -101,14 +101,17 @@ class superelliptic_form:
         C = self.curve
         m = C.exponent
         F = C.base_ring
+        Fxy, Rxy, x, y = C.fct_field
+        g = reduction(C, y^m*g)
         Rx.<x> = PolynomialRing(F)
         Fx = FractionField(Rx)
-        FxRy.<y> = PolynomialRing(Fx)
+        FxRy.<y1> = PolynomialRing(Fx, 1)
-        g = reduction(C, y^m*g)
+        print('a')
-        g = FxRy(g)
+        g = FxRy(g(x = x, y = y1))
+        print('b')
         if j == 0:
-            return g.monomial_coefficient(y^(0))/C.polynomial
+            return g.monomial_coefficient(y1^(0))/C.polynomial
-        return g.monomial_coefficient(y^(m-j))
+        return g.monomial_coefficient(y1^(m-j))
     
     def is_regular_on_U0(self):
         C = self.curve

tests.sage

@@ -1,16 +1,16 @@
 #load('init.sage')
 print("Expansion at infty test:")
 load('superelliptic/tests/expansion_at_infty.sage')
-#print("superelliptic form coordinates test:")
+print("superelliptic form coordinates test:")
-#load('superelliptic/tests/form_coordinates_test.sage')
+load('superelliptic/tests/form_coordinates_test.sage')
-#print("p-th root test:")
+print("p-th root test:")
-#load('superelliptic/tests/pth_root_test.sage')
+load('superelliptic/tests/pth_root_test.sage')
 #print("not working! superelliptic p rank test:")
 #load('superelliptic/tests/p_rank_test.sage')
-#print("a-number test:")
+print("a-number test:")
-#load('superelliptic/tests/a_number_test.sage')
+load('superelliptic/tests/a_number_test.sage')
-#print("as_cover_test:")
+print("as_cover_test:")
-#load('as_covers/tests/as_cover_test.sage')
+load('as_covers/tests/as_cover_test.sage')
 #print("group_action_matrices_test:")
 #load('as_covers/tests/group_action_matrices_test.sage')
 #print("dual_element_test:")