Cohomology of structure sheaf of superelliptic

sage/as_covers/as_auxilliary.sage

@@ -1,7 +1,8 @@
-def magmathis(A, B, text = False):
+def magmathis(A, B, text = False, prefix="", sufix=""):
     """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))
@@ -9,11 +10,16 @@ def magmathis(A, B, text = False):
     A = A.replace(")", "")
     B = B.replace("(", "")
     B = B.replace(")", "")
-    result = "A := MatrixAlgebra<GF("+str(q) + "),"+ str(n) + "|"
+    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 += "IndecomposableSummands(M);"
+    result += "L := IndecomposableSummands(M); L;"
+    result += "for i in [1 .. #L] do print(Generators(Action(L[i]))); end for;"
+    result += sufix
     if text:
         return result
-    print(magma_free(result))
+    return(magma_free(result))

sage/as_covers/as_form_class.sage

@@ -73,7 +73,6 @@ class as_form:
         denom = LCM([denominator(omega.form) for omega in basis])
         basis = [denom*omega for omega in basis]
         self_with_no_denominator = denom*self
-        print(self_with_no_denominator, basis)
         return linear_representation_polynomials(Rxyz(self_with_no_denominator.form), [Rxyz(omega.form) for omega in basis])
 
     def trace(self):

sage/drafty/draft2.sage

@@ -1,14 +1,13 @@
 p = 3
-m = 1
-F = GF(p^2, 'a')
-a = F.gens()[0]
+m = 2
+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, a*x^7)
-AS = as_cover(C_super, [f1, f2], prec=1000)
-#print(AS.uniformizer())
-print(AS.ramification_jumps())
+f = x^3 - x
+C = superelliptic(f, m)
+x = C.x
+y = C.y
+dx = C.dx
+om1 = x^3*y*dx
+u = (C.one)/x
+v = y/x^2
+print(om1 + u^3*v*u.diffn() - (y/x)^2*(y/x).diffn())

sage/drafty/draft3.sage

@@ -1,13 +1,46 @@
-p = 2
-m = 1
-F = GF(p^(2), 'a')
-a = F.gens()[0]
+p = 3
+m = 2
+#F = GF(p)
+F = GF(p)
 Rx.<x> = PolynomialRing(F)
-f = x
-C_super = superelliptic(f, m)
+f = x^5 + 1
+g = f(x^p - x)
+Cf = superelliptic(f, m)
+C = superelliptic(g, m)
 
-Rxy.<x, y> = PolynomialRing(F, 2)
-f1 = superelliptic_function(C_super, x^3)
-f2 = superelliptic_function(C_super, a*x^3)
-AS = as_cover(C_super, [f1, f2], prec=1000)
-#print(AS.at_most_poles(7))
+#class superelliptic_automorphism:
+#    def __init__
+
+def autom(omega):
+    C1 = omega.curve
+    f = omega.form
+    F = C1.base_ring
+    Rxy.<x, y> = PolynomialRing(F, 2)
+    RxyQ = FractionField(Rxy)
+    f = RxyQ(f)
+    return superelliptic_form(C1, f(x=x+1, y=y))
+
+def automdR(omega):
+    C1 = omega.curve
+    om0 = omega.omega0.form
+    f = omega.f.function
+    F = C1.base_ring
+    Rxy.<x, y> = PolynomialRing(F, 2)
+    RxyQ = FractionField(Rxy)
+    f = RxyQ(f)
+    om0 = RxyQ(om0)
+    f = f(x=x+1, y=y)
+    om0 = om0(x=x+1, y=y)
+    f = superelliptic_function(C1, f)
+    om0 = superelliptic_form(C1, om0)
+    return superelliptic_cech(C1, om0, f)
+
+A = C.de_rham_basis()
+gC = C.genus()
+M = matrix(C.base_ring, 2*gC, 2*gC)
+for i in range(2*gC):
+    M[:, i] = vector(automdR(A[i]).coordinates())
+print(M, M^p == identity_matrix(2*gC))
+
+N = C.verschiebung_matrix().transpose()
+print(magmathis(M, N))

sage/drafty/superelliptic_cohomology_class.sage

@@ -0,0 +1,92 @@
+class superelliptic_cohomology:
+    '''Cohomology of structure sheaf.'''
+    def __init__(self, C, f):
+        self.curve = C
+        self.function = f
+        
+    def prod(self, form, prec=20):
+        result = 0
+        C = self.curve
+        delta = C.nb_of_pts_at_infty
+        fct = superelliptic_function(C, self.function)
+        for i in range(delta):
+            result += (fct*form).expansion_at_infty(place=i, prec=prec)[-1]
+        return -result
+    
+    def __repr__(self):
+        return "[" + str(self.function) + "]"
+    
+    def __add__(self, other):
+        C = self.curve
+        g1 = self.function
+        g2 = other.function
+        g = reduction(C, g1 + g2)
+        return superelliptic_cohomology(C, g)
+    
+    def __sub__(self, other):
+        C = self.curve
+        g1 = self.function
+        g2 = other.function
+        g = reduction(C, g1 - g2)
+        return superelliptic_cohomology(C, g)
+    
+    def __rmul__(self, constant):
+        C = self.curve
+        g = self.function
+        return superelliptic_cohomology(C, constant*g)
+
+    def coordinates(self, basis = 0, basis_holo = 0, prec=20):
+        C = self.curve
+        if basis == 0:
+            basis = basis_of_cohomology(C)
+        if basis_holo == 0:
+            basis_holo = C.holomorphic_differentials_basis()
+        g = C.genus()
+        coordinates = g*[0]
+        for i, omega in enumerate(basis_holo):
+            coordinates[i] = self.prod(omega, prec=prec)
+        return coordinates
+    
+    def frobenius(self):
+        C = self.curve
+        f = self.function
+        p = C.base_ring.characteristic()
+        return superelliptic_cohomology(C, f^p)
+    
+def basis_of_cohomology(C):
+    m = C.exponent
+    f = C.polynomial
+    r = f.degree()
+    F = C.base_ring
+    delta = C.nb_of_pts_at_infty
+    Rx.<x> = PolynomialRing(F)
+    Rxy.<x, y> = PolynomialRing(F, 2)
+    Fxy = FractionField(Rxy)
+    basis = []
+    for j in range(1, m):
+            for i in range(1, r):
+                if (r*j - m*i >= delta):
+                    basis += [superelliptic_cohomology(C, Fxy(m*y^(j)/x^i))]
+    return basis
+
+def frobenius_matrix(C, prec=20):
+    g = C.genus()
+    F = C.base_ring
+    M = matrix(F, g, g)
+    for i, f in enumerate(basis_of_cohomology(C)):
+        M[i, :] = vector(f.frobenius().coordinates(prec=prec))
+    return M
+
+
+def frobenius_kernel(C, prec=20):
+    M = frobenius_matrix(C, prec=prec)
+    K = M.kernel().basis()
+    g = C.genus()
+    result = []
+    basis = basis_of_cohomology(C)
+    for v in K:
+        coh = superelliptic_cohomology(C, 0)
+        for i in range(g):
+            coh += v[i] * basis[i]
+        result += [coh]
+    return result

sage/init.sage

@@ -18,6 +18,7 @@ load('auxilliaries/linear_combination_polynomials.sage')
 ##############
 ##############
 load('drafty/lift_to_de_rham.sage')
+load('drafty/superelliptic_cohomology_class.sage')
 load('drafty/draft5.sage')
 load('drafty/pole_numbers.sage')
 #load('drafty/draft4.sage')

sage/superelliptic/superelliptic_class.sage

@@ -14,6 +14,11 @@ class superelliptic:
         self.characteristic = F.characteristic()
         r = Rx(f).degree()
         delta = GCD(r, m)
+        self.nb_of_pts_at_infty = delta
+        self.x = superelliptic_function(self, Rxy(x))
+        self.y = superelliptic_function(self, Rxy(y))
+        self.dx = superelliptic_form(self, Rxy(1))
+        self.one = superelliptic_function(self, Rxy(1))
         
     def __repr__(self):
         f = self.polynomial

sage/superelliptic/superelliptic_form_class.sage

@@ -119,11 +119,11 @@ class superelliptic_form:
                 return 0
         return 1
 
-    def expansion_at_infty(self, i = 0, prec=10):
+    def expansion_at_infty(self, place = 0, prec=10):
         g = self.form
         C = self.curve
         g = superelliptic_function(C, g)
-        g = g.expansion_at_infty(i = i, prec=prec)
-        x_series = superelliptic_function(C, x).expansion_at_infty(i = i, prec=prec)
+        g = g.expansion_at_infty(place = place, prec=prec)
+        x_series = superelliptic_function(C, x).expansion_at_infty(place = place, prec=prec)
         dx_series = x_series.derivative()
         return g*dx_series

sage/superelliptic/superelliptic_function_class.sage

@@ -42,10 +42,21 @@ class superelliptic_function:
     
     def __mul__(self, other):
         C = self.curve
-        g1 = self.function
-        g2 = other.function
-        g = reduction(C, g1 * g2)
-        return superelliptic_function(C, g)
+        try:
+            g1 = self.function
+            g2 = other.function
+            g = reduction(C, g1 * g2)
+            return superelliptic_function(C, g)
+        except:
+            g1 = self.function
+            g2 = other.form
+            g = reduction(C, g1 * g2)
+            return superelliptic_form(C, g)
+        
+    def __rmul__(self, constant):
+        C = self.curve
+        g = self.function
+        return superelliptic_function(C, constant*g)
     
     def __truediv__(self, other):
         C = self.curve
@@ -73,7 +84,7 @@ class superelliptic_function:
         return superelliptic_form(C, A+B)
 
     
-    def expansion_at_infty(self, i = 0, prec=10):
+    def expansion_at_infty(self, place = 0, prec=10):
         C = self.curve
         f = C.polynomial
         m = C.exponent
@@ -98,7 +109,7 @@ class superelliptic_function:
         
         g = (x^r*f(x = 1/x))
         gW = RptWQ(g(x = t^M * W^b)) - W^(delta)
-        ww = naive_hensel(gW, F, start = root_of_unity(F, delta)^i, prec = prec)
+        ww = naive_hensel(gW, F, start = root_of_unity(F, delta)^place, prec = prec)
         xx = Rt(1/(t^M*ww^b))
         yy = 1/(t^R*ww^a)
         return Rt(fct(x = Rt(xx), y = Rt(yy)))