130 lines
3.9 KiB
Python
130 lines
3.9 KiB
Python
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class superelliptic_form:
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def __init__(self, C, g):
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F = C.base_ring
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Rxy.<x, y> = PolynomialRing(F, 2)
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Fxy = FractionField(Rxy)
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g = Fxy(reduction_form(C, g))
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self.form = g
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self.curve = C
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def __add__(self, other):
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C = self.curve
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g1 = self.form
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g2 = other.form
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g = reduction(C, g1 + g2)
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return superelliptic_form(C, g)
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def __sub__(self, other):
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C = self.curve
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g1 = self.form
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g2 = other.form
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g = reduction(C, g1 - g2)
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return superelliptic_form(C, g)
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def __repr__(self):
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g = self.form
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if len(str(g)) == 1:
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return str(g) + ' dx'
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return '('+str(g) + ') dx'
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def __rmul__(self, constant):
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C = self.curve
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omega = self.form
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return superelliptic_form(C, constant*omega)
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def cartier(self):
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C = self.curve
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m = C.exponent
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p = C.characteristic
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f = C.polynomial
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F = C.base_ring
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Rx.<x> = PolynomialRing(F)
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Fx = FractionField(Rx)
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FxRy.<y> = PolynomialRing(Fx)
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Fxy = FractionField(FxRy)
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result = superelliptic_form(C, FxRy(0))
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mult_order = Integers(m)(p).multiplicative_order()
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M = Integer((p^(mult_order)-1)/m)
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for j in range(1, m):
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fct_j = self.jth_component(j)
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h = Rx(fct_j*f^(M*j))
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j1 = (p^(mult_order-1)*j)%m
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B = floor(p^(mult_order-1)*j/m)
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result += superelliptic_form(C, polynomial_part(p, h)/(f^B*y^(j1)))
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return result
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def coordinates(self):
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C = self.curve
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F = C.base_ring
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m = C.exponent
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Rx.<x> = PolynomialRing(F)
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Fx = FractionField(Rx)
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FxRy.<y> = PolynomialRing(Fx)
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g = C.genus()
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degrees_holo = C.degrees_holomorphic_differentials()
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degrees_holo_inv = {b:a for a, b in degrees_holo.items()}
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basis = C.holomorphic_differentials_basis()
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for j in range(1, m):
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omega_j = Fx(self.jth_component(j))
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if omega_j != Fx(0):
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d = degree_of_rational_fctn(omega_j, F)
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index = degrees_holo_inv[(d, j)]
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a = coeff_of_rational_fctn(omega_j, F)
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a1 = coeff_of_rational_fctn(basis[index].jth_component(j), F)
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elt = self - (a/a1)*basis[index]
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return elt.coordinates() + a/a1*vector([F(i == index) for i in range(0, g)])
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return vector(g*[0])
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def jth_component(self, j):
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g = self.form
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C = self.curve
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F = C.base_ring
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Rx.<x> = PolynomialRing(F)
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Fx = FractionField(Rx)
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FxRy.<y> = PolynomialRing(Fx)
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Fxy = FractionField(FxRy)
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Ryinv.<y_inv> = PolynomialRing(Fx)
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g = Fxy(g)
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g = g(y = 1/y_inv)
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g = Ryinv(g)
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return coff(g, j)
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def is_regular_on_U0(self):
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C = self.curve
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F = C.base_ring
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m = C.exponent
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Rx.<x> = PolynomialRing(F)
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for j in range(1, m):
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if self.jth_component(j) not in Rx:
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return 0
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return 1
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def is_regular_on_Uinfty(self):
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C = self.curve
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F = C.base_ring
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m = C.exponent
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f = C.polynomial
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r = f.degree()
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delta = GCD(m, r)
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M = m/delta
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R = r/delta
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for j in range(1, m):
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A = self.jth_component(j)
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d = degree_of_rational_fctn(A, F)
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if(-d*M + j*R -(M+1)<0):
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return 0
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return 1
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def expansion_at_infty(self, i = 0, prec=10):
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g = self.form
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C = self.curve
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g = superelliptic_function(C, g)
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g = g.expansion_at_infty(i = i, prec=prec)
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x_series = superelliptic_function(C, x).expansion_at_infty(i = i, prec=prec)
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dx_series = x_series.derivative()
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return g*dx_series
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