110 lines
3.6 KiB
Python
110 lines
3.6 KiB
Python
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class superelliptic_witt:
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def __init__(self, C, t, f0, f1):
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''' Define Witt function on C of the form [t] + f0([x], [y]) + V(f1). '''
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self.curve = C
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p = C.characteristic
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self.t = t #superelliptic_function
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self.f0 = reduce_rational_fct(f0, p) #polynomial/rational function over Z/p^2
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self.f1 = f1 #superelliptic_function
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def __repr__(self):
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f0 = self.f0
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f1 = self.f1
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t = self.t
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return "[" + str(t) + "] + " + str(f0).replace("x", "[x]").replace("y", "[y]") + " + V(" + str(f1) + ")"
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def __add__(self, other):
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C = self.curve
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return superelliptic_witt(C, self.t + other.t, self.f0 + other.f0, self.f1 + other.f1)
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def teichmuller_representation(self):
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'''Represents as [f] + V(g), i.e. f0 = 0.'''
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C = self.curve
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Fxy, Rxy, x, y = self.fct_field
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F = C.base_ring
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function = Rxy(self.f0)
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if self.f0 == 0:
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return self
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M = Rxy(function.monomials()[0])
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a = F(function.monomial_coefficient(M))
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fct1 = fct - superelliptic_function(C, a*M)
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function1 = fct1.function
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return teichmuller(fct1) + superelliptic_witt(C, (a.lift())^p*M.change_ring(QQ), superelliptic_function(C, function1^2*a*M + function1*(a*M)^2))
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def antiteichmuller_representation(self):
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'''Represents as f([x], [y]) + V(g), i.e. teichmuller part is zero.'''
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return 0
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def reduce_rational_fct(fct, p):
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Rxy.<x, y> = PolynomialRing(QQ)
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Fxy = FractionField(Rxy)
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fct = Fxy(fct)
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num = Rxy(fct.numerator())
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den = Rxy(fct.denominator())
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num1 = Rxy(0)
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for m in num.monomials():
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a = num.monomial_coefficient(m)
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num1 += (a%p^2)*m
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den1 = Rxy(0)
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for m in den.monomials():
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a = den.monomial_coefficient(m)
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den1 += (a%p^2)*m
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return num1/den1
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def teichmuller(fct):
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C = fct.curve
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Fxy, Rxy, x, y = C.fct_field
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F = C.base_ring
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function = Rxy(fct.function)
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if function == 0:
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return superelliptic_witt(C, 0, 0*C.x)
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M = Rxy(function.monomials()[0])
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a = F(function.monomial_coefficient(M))
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fct1 = fct - superelliptic_function(C, a*M)
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function1 = fct1.function
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return teichmuller(fct1) + superelliptic_witt(C, (a.lift())^p*M.change_ring(QQ), superelliptic_function(C, function1^2*a*M + function1*(a*M)^2))
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class superelliptic_drw_form:
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def __init__(self, C, omega_x, omega_y, omega, h):
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'''Form [omega_x] d[x] + [omega_y] d[y] + V(omega) + dV([h])'''
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self.curve = C
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self.omega_x = omega_x
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self.omega_y = omega_y
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self.omega = omega
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self.h = h
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def __eq__(self, other):
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eq1 = (self.omega1 == self.omega1)
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try:
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H = (self.h - other.h).pthroot()
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except:
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return False
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eq2 = (self.omega2 - other.omega2).cartier() - H.diffn()
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if eq1 and eq2:
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return True
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return False
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def __repr__(self):
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C = self.curve
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omega_x = self.omega_x
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omega_y = self.omega_y
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h = self.h
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return str(omega_x) + "] dx + [" + str(omega_x.form) + "] d[x] " + "+ V(" + str(omega2) + ") + dV([" + str(h) +"])"
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def mult_by_p(omega):
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C = omega.curve
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fct = omega.form
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Fxy, Rxy, x, y = C.fct_field
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omega2 = superelliptic_form(C, fct^p * x^(p-1))
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result = superelliptic_drw_form(C, 0*C.dx, omega2, 0*C.x)
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return result
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def basis_W2Omega(C):
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basis = C.holomorphic_differentials_basis()
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result = []
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for omega in basis:
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result += [mult_by_p(omega)]
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image_of_cartier = []
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return result
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