quaternion template; fiber fixed
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@ -108,23 +108,32 @@ class as_cover:
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jumps = self.jumps
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gY = self.quotient.genus()
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n = self.height
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branch_pts = self.branch_points
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p = self.characteristic
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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)
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return p^n*(gY-1) + 1 + 1/2*sum(self.exponent_of_different(place)*len(self.fiber(place)) for place in self.branch_points)
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def exponent_of_different(self, place = 0):
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jumps = self.jumps
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n = self.height
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delta = self.nb_of_pts_at_infty
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p = self.characteristic
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return sum(p^(n-j-1)*(jumps[place][j]+1)*(p-1) for j in range(n))
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dd = [0]
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for i in range(1, n+1):
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if jumps[place][i-1] == 0:
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dd += [dd[i-1]]
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else:
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dd += [(jumps[place][i-1]+1)*(p-1) + p*dd[i-1]]
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return dd[n]
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def exponent_of_different_prim(self, place = 0):
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jumps = self.jumps
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n = self.height
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delta = self.nb_of_pts_at_infty
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p = self.characteristic
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return sum(p^(n-j-1)*(jumps[place][j])*(p-1) for j in range(n))
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dd = [0]
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for i in range(1, n+1):
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if jumps[place][i-1] == 0:
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dd += [dd[i-1]]
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else:
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dd += [jumps[place][i-1]*(p-1) + p*dd[i-1]]
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return dd[n]
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def holomorphic_differentials_basis(self, threshold = 8):
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from itertools import product
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@ -160,7 +169,8 @@ class as_cover:
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if len(forms) < self.genus():
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print("I haven't found all forms, only ", len(forms), " of ", self.genus())
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return holomorphic_differentials_basis(self, threshold = threshold + 1)
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raise ValueError("Increase threshold.")
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#return holomorphic_differentials_basis(self, threshold = threshold + 1)
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if len(forms) > self.genus():
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raise ValueError("Increase precision.")
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return forms
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@ -283,11 +293,9 @@ class as_cover:
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F = self.base_ring
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rr_space = self.at_most_poles(threshold)
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list_of_fcts = [ff for ff in rr_space if ff.valuation(place)%p != 0]
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print(len(list_of_fcts))
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list_of_fcts2 = [len(str(ff)) for ff in list_of_fcts]
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i_min = list_of_fcts2.index(min(list_of_fcts2))
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result = list_of_fcts.pop(i_min)
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print(result)
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flag = 1
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while flag == 1:
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flag = 0
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@ -298,7 +306,6 @@ class as_cover:
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list_of_fcts2 = [len(str(ff)) for ff in list_of_fcts]
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i_min = list_of_fcts2.index(min(list_of_fcts2))
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result = list_of_fcts.pop(i_min)
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print(result)
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return result
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def stabilizer(self, place = 0):
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@ -320,11 +327,12 @@ class as_cover:
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'Gives representatives for the quotient G/G_P for given place. Those are in bijection with the fiber.'
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result = [(0, 0, 0)]
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p = self.characteristic
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G = self.group
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H = self.stabilizer(place = place)
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for g in self.group.elts:
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flag = 1
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for v in result:
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if heisenberg_mult(g, heisenberg_inv(v, p), p) in H:
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if (G.elt(g)*(-G.elt(v))).as_tuple in H:
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flag = 0
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if flag:
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result += [g]
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@ -376,6 +384,15 @@ class as_cover:
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i+=1
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return result
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def upper_ramification_jumps(self, place = 0, quasiuniformizer = 0, threshold = 20):
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lj = self.ramification_jumps(place = place, quasiuniformizer = quasiuniformizer, threshold = threshold)
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result = []
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result += [lj[0]]
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for j in range(1, len(lj)):
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aux = len(self.stabilizer(place=place))//len(self.ith_ramification_gp(lj[j], place = place, quasiuniformizer = quasiuniformizer, threshold = threshold))
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result += [result[j-1] + (lj[j] - lj[j-1])//aux]
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return result
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def a_number(self):
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g = self.genus()
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return g - self.cartier_matrix().rank()
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@ -93,4 +93,29 @@ def heisenberg(p):
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inv = lambda elt : (p-elt[0], p-elt[1], (p - elt[2] - (p-elt[0])*(p-elt[1]))%p)
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gens = [(1, 0, 0), (0, 1, 0), (0, 0, 1)]
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gp = group(name, short_name, elts, one, mult, inv, gens)
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return gp
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def quaternion_mult(aa, bb):
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result = [(aa[0] + bb[0] + 2*aa[1]*bb[0])%4, (aa[1]+bb[1])%4]
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if result[1]%4 == 2 or result[1]%4 == 3:
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result[0] = (result[0] + 2)%4
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result[1] = (result[1] - 2)%4
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return tuple(result)
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def quaternion_inv(aa):
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result = [((-1)^(aa[0]*aa[1])*(-aa[0]))%4, (-aa[1])%4]
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if result[1]%4 == 2 or result[1]%4 == 3:
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result[0] = (result[0] + 2)%4
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result[1] = (result[1] - 2)%4
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return tuple(result)
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def quaternion_gp():
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name = "Q8"
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short_name = name
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elts = [(i, j) for i in range(4) for j in range(2)]
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mult = quaternion_mult
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inv = quaternion_inv
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gens = [(1, 0), (0, 1)]
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one = (0, 0)
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gp = group(name, short_name, elts, one, mult, inv, gens)
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return gp
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@ -37,6 +37,11 @@ def elementary_template(p, n):
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gp_action = [[z[j] + (i == j) for j in range(n)] for i in range(n)]
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return template(height, field, group, fcts, gp_action)
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def elementary_cover(list_of_fcts, prec=10):
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n = len(list_of_fcts)
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C = list_of_fcts[0].curve
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return as_cover(C, elementary_template(p, n), list_of_fcts, branch_points = [], prec = prec)
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def heisenberg_template(p):
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group = heisenberg(p)
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field = GF(p)
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@ -57,6 +62,10 @@ def heisenberg_template(p):
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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]]
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return template(height, field, group, fcts, gp_action)
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def heisenberg_cover(list_of_fcts, prec=10):
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n = len(list_of_fcts)
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C = list_of_fcts[0].curve
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return as_cover(C, heisenberg_template(p), list_of_fcts, branch_points = [], prec = prec)
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def witt_pol(X, p, n):
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n = len(X)
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@ -133,4 +142,33 @@ def witt_template(p, n):
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########
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aux
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gp_action = [gp_action]
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return template(height, field, group, fcts, gp_action)
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return template(height, field, group, fcts, gp_action)
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def witt_cover(list_of_fcts, prec=10):
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n = len(list_of_fcts)
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C = list_of_fcts[0].curve
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return as_cover(C, witt_template(p, n), list_of_fcts, branch_points = [], prec = prec)
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def quaternion_template():
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field = GF(2)
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height = 3
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n = 3
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variable_names = ''
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for i in range(n):
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variable_names += 'z'+str(i)+','
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for i in range(n):
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variable_names += 'f'+str(i)
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if i!=n-1:
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variable_names += ','
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R = PolynomialRing(field, 2*n, variable_names)
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z = R.gens()[:n]
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f = R.gens()[n:]
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group = quaternion_gp()
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fcts = [f[0], f[1], f[2] + z[0]*f[0]+z[1]*(f[0] + f[1])]
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gp_action = [[z[0]+1, z[1], z[2] + z[0]], [z[0], z[1] + 1, z[2] + z[1] + z[0]]]
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return template(height, field, group, fcts, gp_action)
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def quaternion_cover(list_of_fcts, prec=10):
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n = len(list_of_fcts)
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C = list_of_fcts[0].curve
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return as_cover(C, quaternion_template(), list_of_fcts, branch_points = [], prec = prec)
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30
init.sage
30
init.sage
@ -35,21 +35,21 @@ load('auxilliaries/laurent_analytic_part.sage')
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#load('as_drw/as_witt.sage')
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#load('as_drw/as_witt_form.sage')
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#load('as_drw/as_compability.sage')
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load('quaternion_covers/quaternion_covers.sage')
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load('quaternion_covers/quaternion_function_class.sage')
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load('quaternion_covers/quaternion_form_class.sage')
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load('quaternion_covers/quaternion_polyforms.sage')
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load('quaternion_covers/quaternion_reduction.sage')
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load('quaternion_covers/quaternion_group_action_matrices.sage')
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load('heisenberg_covers/heisenberg_covers.sage')
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load('heisenberg_covers/heisenberg_function_class.sage')
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load('heisenberg_covers/heisenberg_form_class.sage')
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load('heisenberg_covers/heisenberg_polyforms.sage')
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load('heisenberg_covers/heisenberg_reduction.sage')
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load('heisenberg_covers/heisenberg_group_action_matrices.sage')
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load('heisenberg_covers/dual_element.sage')
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load('heisenberg_covers/ith_magical_component.sage')
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load('heisenberg_covers/heisenberg_group.sage')
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#load('quaternion_covers/quaternion_covers.sage')
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#load('quaternion_covers/quaternion_function_class.sage')
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#load('quaternion_covers/quaternion_form_class.sage')
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#load('quaternion_covers/quaternion_polyforms.sage')
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#load('quaternion_covers/quaternion_reduction.sage')
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#load('quaternion_covers/quaternion_group_action_matrices.sage')
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#load('heisenberg_covers/heisenberg_covers.sage')
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#load('heisenberg_covers/heisenberg_function_class.sage')
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#load('heisenberg_covers/heisenberg_form_class.sage')
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#load('heisenberg_covers/heisenberg_polyforms.sage')
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#load('heisenberg_covers/heisenberg_reduction.sage')
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#load('heisenberg_covers/heisenberg_group_action_matrices.sage')
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#load('heisenberg_covers/dual_element.sage')
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#load('heisenberg_covers/ith_magical_component.sage')
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#load('heisenberg_covers/heisenberg_group.sage')
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##############
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##############
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def init(lista, tests = False, init=True):
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