frobenius kernel

sage/as_covers/as_cover_class.sage

@@ -172,6 +172,7 @@ class as_cover:
         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 = []
@@ -367,7 +368,7 @@ class as_cover:
         #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()
+        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):
@@ -385,9 +386,9 @@ class as_cover:
 
     def de_rham_basis(self, threshold = 30):
         result = []
-        for omega in self.holomorphic_differentials_basis():
+        for omega in self.holomorphic_differentials_basis(threshold = threshold):
             result += [as_cech(self, omega, as_function(self, 0))]
-        for f in self.cohomology_of_structure_sheaf_basis():
+        for f in self.cohomology_of_structure_sheaf_basis(threshold = threshold):
             omega = self.lift_to_de_rham(f, threshold = threshold)
             result += [as_cech(self, omega, f)]
         return result

sage/as_covers/as_form_class.sage

@@ -210,8 +210,8 @@ def are_forms_linearly_dependent(set_of_forms):
     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])
 
-#given a set S of (form, corresponding Laurent series at some pt), find their combinations holomorphic at that pt
 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

sage/as_covers/as_function_class.sage

@@ -56,13 +56,13 @@ class as_function:
         g1 = self.function
         return as_function(C, g1^(exponent))
     
-    def expansion_at_infty(self, i = 0):
+    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[i]
+        x_series = C.x_series[place]
-        y_series = C.y_series[i]
+        y_series = C.y_series[place]
-        z_series = C.z_series[i]
+        z_series = C.z_series[place]
         n = C.height
         variable_names = 'x, y'
         for j in range(n):
@@ -137,6 +137,22 @@ class as_function:
         result = as_reduction(AS, 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
@@ -157,9 +173,9 @@ class as_function:
             result += f.derivative(z[i])*dz[i]
         return as_form(C, result)
         
-    def valuation(self, i = 0):
+    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(i)).valuation()
+        return Rt(self.expansion_at_infty(place = place)).valuation()

sage/as_covers/group_action_matrices.sage

@@ -55,11 +55,20 @@ def group_action_matrices_old(C_AS):
         A[i] = A[i].transpose()
     return A
 
-def group_action_matrices_log(C_AS):
+def group_action_matrices_log(AS):
+    n = AS.height
+    generators = []
+    for i in range(n):
+        ei = n*[0]
+        ei[i] = 1
+        generators += [ei]
+    return group_action_matrices(AS.at_most_poles_forms(1), generators, basis = AS.at_most_poles_forms(1))
+
+def group_action_matrices_log_old(C_AS):
     F = C_AS.base_ring
     n = C_AS.height
     holo = C_AS.at_most_poles_forms(1)
-    holo_forms = [omega.form for omega in holo]
+    holo_forms = [omega for omega in holo]
     denom = LCM([denominator(omega) for omega in holo_forms])
     variable_names = 'x, y'
     for j in range(n):

sage/drafty/draft.sage

@@ -1,39 +1,8 @@
-p = 3
+p = 5
-m = 2
+m = 1
-F = GF(p^2, 'a')
+F = GF(p)
-a = F.gens()[0]
+Rx.<x> = PolynomialRing(F)
-Rxx.<x> = PolynomialRing(F)
+f = x
-#f = (x^3 - x)^3 + x^3 - x
+C = superelliptic(f, 1)
-f = x^3 + a*x + 1
+AS1 = as_cover(C, [C.x^3], prec = 200)
-f1 = f(x = x^p - x)
+print(AS1.genus())
-C = superelliptic(f, m)
-C1 = superelliptic(f1, m, prec = 500)
-#B = C.crystalline_cohomology_basis(prec = 100, info = 1)
-#B1 = C1.crystalline_cohomology_basis(prec = 100, info = 1)
-
-def crystalline_matrix(C, prec = 50):
-    B = C.crystalline_cohomology_basis(prec = prec)
-    g = C.genus()
-    p = C.characteristic
-    Zp2 = Integers(p^2)
-    M = matrix(Zp2, 2*g, 2*g)
-    for i, b in enumerate(B):
-        M[i, :] = vector(autom(b).coordinates(basis = B))
-    return M
-
-#b0 = de_rham_witt_lift(C.de_rham_basis()[0], prec = 100)
-#b1 = de_rham_witt_lift(C1.de_rham_basis()[2], prec = 300)
-#print(b0.regular_form())
-#print(b1.regular_form())
-for b in C1.de_rham_basis():
-    print(mult_by_p(b.omega0).regular_form())
-
-#for b in B:
-#    print(b.regular_form())
-    
-#for b in B1:
-#    print(b.regular_form())
-
-#M = crystalline_matrix(C, prec = 150)
-#print(M)
-#print(M^3)

sage/init.sage

@@ -28,8 +28,9 @@ load('auxilliaries/linear_combination_polynomials.sage')
 load('auxilliaries/laurent_analytic_part.sage')
 ##############
 ##############
-load('drafty/convert_superelliptic_into_AS.sage')
+#load('drafty/convert_superelliptic_into_AS.sage')
 load('drafty/draft.sage')
 #load('drafty/draft_klein_covers.sage')
+#load('drafty/draft_klein_covers.sage')
 #load('drafty/2gpcovers.sage')
 load('drafty/pole_numbers.sage')

sage/superelliptic/frobenius_kernel.sage

@@ -0,0 +1,25 @@
+def frobenius_kernel(C, prec=50):
+    M = C.frobenius_matrix(prec=prec).transpose()
+    K = M.kernel().basis()
+    g = C.genus()
+    result = []
+    basis = C.cohomology_of_structure_sheaf_basis()
+    for v in K:
+        coh = 0*C.x
+        for i in range(g):
+            coh += v[i] * basis[i]
+        result += [coh]
+    return result
+
+def cartier_kernel(C, prec=50):
+    M = C.cartier_matrix(prec=prec).transpose()
+    K = M.kernel().basis()
+    g = C.genus()
+    result = []
+    basis = C.holomorphic_differentials_basis()
+    for v in K:
+        coh = 0*C.dx
+        for i in range(g):
+            coh += v[i] * basis[i]
+        result += [coh]
+    return result