quaternion template; fiber fixed

as_covers/as_cover_class.sage

@@ -108,23 +108,32 @@ class as_cover:
         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)
+        return p^n*(gY-1) + 1 + 1/2*sum(self.exponent_of_different(place)*len(self.fiber(place)) for place in self.branch_points)
     
     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))
+        dd = [0]
+        for i in range(1, n+1):
+            if jumps[place][i-1] == 0:
+                dd += [dd[i-1]]
+            else:
+                dd += [(jumps[place][i-1]+1)*(p-1) + p*dd[i-1]]
+        return dd[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))
+        dd = [0]
+        for i in range(1, n+1):
+            if jumps[place][i-1] == 0:
+                dd += [dd[i-1]]
+            else:
+                dd += [jumps[place][i-1]*(p-1) + p*dd[i-1]]
+        return dd[n]
     
     def holomorphic_differentials_basis(self, threshold = 8):
         from itertools import product
@@ -160,7 +169,8 @@ class as_cover:
         
         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)
+            raise ValueError("Increase threshold.")
+            #return holomorphic_differentials_basis(self, threshold = threshold + 1)
         if len(forms) > self.genus():
             raise ValueError("Increase precision.")
         return forms
@@ -283,11 +293,9 @@ class as_cover:
         F = self.base_ring
         rr_space = self.at_most_poles(threshold)
         list_of_fcts = [ff for ff in rr_space if ff.valuation(place)%p != 0]
-        print(len(list_of_fcts))
         list_of_fcts2 = [len(str(ff)) for ff in list_of_fcts]
         i_min = list_of_fcts2.index(min(list_of_fcts2))
         result = list_of_fcts.pop(i_min)
-        print(result)
         flag = 1
         while flag == 1:
             flag = 0
@@ -298,7 +306,6 @@ class as_cover:
                 list_of_fcts2 = [len(str(ff)) for ff in list_of_fcts]
                 i_min = list_of_fcts2.index(min(list_of_fcts2))
                 result = list_of_fcts.pop(i_min)
-                print(result)
         return result
 
     def stabilizer(self, place = 0):
@@ -320,11 +327,12 @@ class as_cover:
         'Gives representatives for the quotient G/G_P for given place. Those are in bijection with the fiber.'
         result = [(0, 0, 0)]
         p = self.characteristic
+        G = self.group
         H = self.stabilizer(place = place)
         for g in self.group.elts:
             flag = 1
             for v in result:
-                if heisenberg_mult(g, heisenberg_inv(v, p), p) in H:
+                if (G.elt(g)*(-G.elt(v))).as_tuple in H:
                     flag = 0
             if flag:
                 result += [g]
@@ -376,6 +384,15 @@ class as_cover:
             i+=1
         return result
     
+    def upper_ramification_jumps(self, place = 0, quasiuniformizer = 0, threshold = 20):
+        lj = self.ramification_jumps(place = place, quasiuniformizer = quasiuniformizer, threshold = threshold)
+        result = []
+        result += [lj[0]]
+        for j in range(1, len(lj)):
+            aux = len(self.stabilizer(place=place))//len(self.ith_ramification_gp(lj[j], place = place, quasiuniformizer = quasiuniformizer, threshold = threshold))
+            result += [result[j-1] + (lj[j] - lj[j-1])//aux]
+        return result
+    
     def a_number(self):
         g = self.genus()
         return g - self.cartier_matrix().rank()

as_covers/group.sage

@@ -94,3 +94,28 @@ def heisenberg(p):
     gens = [(1, 0, 0), (0, 1, 0), (0, 0, 1)]
     gp = group(name, short_name, elts, one, mult, inv, gens)
     return gp
+
+def quaternion_mult(aa, bb):
+    result = [(aa[0] + bb[0] + 2*aa[1]*bb[0])%4, (aa[1]+bb[1])%4]
+    if result[1]%4 == 2 or result[1]%4 == 3:
+        result[0] = (result[0] + 2)%4
+        result[1] = (result[1] - 2)%4
+    return tuple(result)
+
+def quaternion_inv(aa):
+    result = [((-1)^(aa[0]*aa[1])*(-aa[0]))%4, (-aa[1])%4]
+    if result[1]%4 == 2 or result[1]%4 == 3:
+        result[0] = (result[0] + 2)%4
+        result[1] = (result[1] - 2)%4
+    return tuple(result)
+
+def quaternion_gp():
+    name = "Q8"
+    short_name = name
+    elts = [(i, j) for i in range(4) for j in range(2)]
+    mult = quaternion_mult
+    inv = quaternion_inv
+    gens = [(1, 0), (0, 1)]
+    one = (0, 0)
+    gp = group(name, short_name, elts, one, mult, inv, gens)
+    return gp

as_covers/template.sage

@@ -37,6 +37,11 @@ def elementary_template(p, n):
     gp_action = [[z[j] + (i == j) for j in range(n)] for i in range(n)]
     return template(height, field, group, fcts, gp_action)
 
+def elementary_cover(list_of_fcts, prec=10):
+    n = len(list_of_fcts)
+    C = list_of_fcts[0].curve
+    return as_cover(C, elementary_template(p, n), list_of_fcts, branch_points = [], prec = prec)
+
 def heisenberg_template(p):
     group = heisenberg(p)
     field = GF(p)
@@ -57,6 +62,10 @@ def heisenberg_template(p):
     gp_action = [[z[0] + 1, z[1], z[2] + z[1]], [z[0] + 1, z[1] + 1, z[2]], [z[0], z[1], z[2] - 1]]
     return template(height, field, group, fcts, gp_action)
 
+def heisenberg_cover(list_of_fcts, prec=10):
+    n = len(list_of_fcts)
+    C = list_of_fcts[0].curve
+    return as_cover(C, heisenberg_template(p), list_of_fcts, branch_points = [], prec = prec)
 
 def witt_pol(X, p, n):
     n = len(X)
@@ -134,3 +143,32 @@ def witt_template(p, n):
     aux
     gp_action = [gp_action]
     return template(height, field, group, fcts, gp_action)
+
+def witt_cover(list_of_fcts, prec=10):
+    n = len(list_of_fcts)
+    C = list_of_fcts[0].curve
+    return as_cover(C, witt_template(p, n), list_of_fcts, branch_points = [], prec = prec)
+
+def quaternion_template():
+    field = GF(2)
+    height = 3
+    n = 3
+    variable_names = ''
+    for i in range(n):
+        variable_names += 'z'+str(i)+','
+    for i in range(n):
+        variable_names += 'f'+str(i)
+        if i!=n-1:
+            variable_names += ','
+    R = PolynomialRing(field, 2*n, variable_names)
+    z = R.gens()[:n]
+    f = R.gens()[n:]
+    group = quaternion_gp()
+    fcts = [f[0], f[1], f[2] + z[0]*f[0]+z[1]*(f[0] + f[1])]
+    gp_action = [[z[0]+1, z[1], z[2] + z[0]], [z[0], z[1] + 1, z[2] + z[1] + z[0]]]
+    return template(height, field, group, fcts, gp_action)
+
+def quaternion_cover(list_of_fcts, prec=10):
+    n = len(list_of_fcts)
+    C = list_of_fcts[0].curve
+    return as_cover(C, quaternion_template(), list_of_fcts, branch_points = [], prec = prec)

init.sage

@@ -35,21 +35,21 @@ load('auxilliaries/laurent_analytic_part.sage')
 #load('as_drw/as_witt.sage')
 #load('as_drw/as_witt_form.sage')
 #load('as_drw/as_compability.sage')
-load('quaternion_covers/quaternion_covers.sage')
+#load('quaternion_covers/quaternion_covers.sage')
-load('quaternion_covers/quaternion_function_class.sage')
+#load('quaternion_covers/quaternion_function_class.sage')
-load('quaternion_covers/quaternion_form_class.sage')
+#load('quaternion_covers/quaternion_form_class.sage')
-load('quaternion_covers/quaternion_polyforms.sage')
+#load('quaternion_covers/quaternion_polyforms.sage')
-load('quaternion_covers/quaternion_reduction.sage')
+#load('quaternion_covers/quaternion_reduction.sage')
-load('quaternion_covers/quaternion_group_action_matrices.sage')
+#load('quaternion_covers/quaternion_group_action_matrices.sage')
-load('heisenberg_covers/heisenberg_covers.sage')
+#load('heisenberg_covers/heisenberg_covers.sage')
-load('heisenberg_covers/heisenberg_function_class.sage')
+#load('heisenberg_covers/heisenberg_function_class.sage')
-load('heisenberg_covers/heisenberg_form_class.sage')
+#load('heisenberg_covers/heisenberg_form_class.sage')
-load('heisenberg_covers/heisenberg_polyforms.sage')
+#load('heisenberg_covers/heisenberg_polyforms.sage')
-load('heisenberg_covers/heisenberg_reduction.sage')
+#load('heisenberg_covers/heisenberg_reduction.sage')
-load('heisenberg_covers/heisenberg_group_action_matrices.sage')
+#load('heisenberg_covers/heisenberg_group_action_matrices.sage')
-load('heisenberg_covers/dual_element.sage')
+#load('heisenberg_covers/dual_element.sage')
-load('heisenberg_covers/ith_magical_component.sage')
+#load('heisenberg_covers/ith_magical_component.sage')
-load('heisenberg_covers/heisenberg_group.sage')
+#load('heisenberg_covers/heisenberg_group.sage')
 ##############
 ##############
 def init(lista, tests = False, init=True):