92 lines
2.6 KiB
Python
92 lines
2.6 KiB
Python
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class superelliptic_cohomology:
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'''Cohomology of structure sheaf.'''
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def __init__(self, C, f):
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self.curve = C
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self.function = f
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def prod(self, form, prec=20):
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result = 0
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C = self.curve
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delta = C.nb_of_pts_at_infty
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fct = superelliptic_function(C, self.function)
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for i in range(delta):
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result += (fct*form).expansion_at_infty(place=i, prec=prec)[-1]
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return -result
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def __repr__(self):
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return "[" + str(self.function) + "]"
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def __add__(self, other):
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C = self.curve
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g1 = self.function
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g2 = other.function
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g = reduction(C, g1 + g2)
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return superelliptic_cohomology(C, g)
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def __sub__(self, other):
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C = self.curve
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g1 = self.function
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g2 = other.function
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g = reduction(C, g1 - g2)
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return superelliptic_cohomology(C, g)
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def __rmul__(self, constant):
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C = self.curve
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g = self.function
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return superelliptic_cohomology(C, constant*g)
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def coordinates(self, basis = 0, basis_holo = 0, prec=20):
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C = self.curve
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if basis == 0:
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basis = basis_of_cohomology(C)
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if basis_holo == 0:
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basis_holo = C.holomorphic_differentials_basis()
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g = C.genus()
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coordinates = g*[0]
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for i, omega in enumerate(basis_holo):
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coordinates[i] = self.prod(omega, prec=prec)
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return coordinates
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def frobenius(self):
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C = self.curve
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f = self.function
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p = C.base_ring.characteristic()
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return superelliptic_cohomology(C, f^p)
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def basis_of_cohomology(C):
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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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F = C.base_ring
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delta = C.nb_of_pts_at_infty
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Rx.<x> = PolynomialRing(F)
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Rxy.<x, y> = PolynomialRing(F, 2)
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Fxy = FractionField(Rxy)
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basis = []
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for j in range(1, m):
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for i in range(1, r):
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if (r*j - m*i >= delta):
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basis += [superelliptic_cohomology(C, Fxy(m*y^(j)/x^i))]
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return basis
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def frobenius_matrix(C, prec=20):
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g = C.genus()
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F = C.base_ring
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M = matrix(F, g, g)
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for i, f in enumerate(basis_of_cohomology(C)):
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M[i, :] = vector(f.frobenius().coordinates(prec=prec))
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return M
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def frobenius_kernel(C, prec=20):
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M = frobenius_matrix(C, prec=prec)
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K = M.kernel().basis()
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g = C.genus()
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result = []
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basis = basis_of_cohomology(C)
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for v in K:
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coh = superelliptic_cohomology(C, 0)
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for i in range(g):
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coh += v[i] * basis[i]
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result += [coh]
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return result
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