quaternion covers and polyforms
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@ -0,0 +1,109 @@
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class as_polyform:
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def __init__(self, form, mult):
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self.form = form
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self.curve = form.curve
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self.mult = mult
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def __add__(self, other):
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return as_polyform(self.form + other.form, self.mult)
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def __repr__(self):
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return str(self.form) + ' dx⊗' + str(self.mult)
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def expansion_at_infty(self):
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return self.form.expansion_at_infty()*(self.curve.dx.expansion_at_infty())^(self.mult)
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def coordinates(self, basis = 0):
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"""Find coordinates of the given holomorphic form self in terms of the basis forms in a list holo."""
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AS = self.curve
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if basis == 0:
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basis = AS.holomorphic_differentials_basis()
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RxyzQ, Rxyz, x, y, z = AS.fct_field
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# We need to have only polynomials to use monomial_coefficients in linear_representation_polynomials,
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# and sometimes basis elements have denominators. Thus we multiply by them.
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denom = LCM([denominator(omega.form.function) for omega in basis])
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basis = [denom*omega.form.function for omega in basis]
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self_with_no_denominator = denom*self.form.function
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return linear_representation_polynomials(Rxyz(self_with_no_denominator), [Rxyz(omega) for omega in basis])
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def as_holo_polydifferentials_basis(AS, mult, threshold = 8):
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v = AS.dx.valuation()
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result = AS.at_most_poles(mult*v, threshold=threshold)
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return [as_polyform(omega, mult) for omega in result]
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as_cover.holo_polydifferentials_basis = as_holo_polydifferentials_basis
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def as_canonical_ideal(AS, threshold=8):
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B0 = AS.holomorphic_differentials_basis(threshold=threshold)
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F = AS.base_ring
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g = AS.genus()
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B1 = [(B0[i], B0[j]) for i in range(g) for j in range(g) if i <= j]
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B2 = AS.holo_polydifferentials_basis(2, threshold = threshold)
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g = AS.genus()
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r = len(B2)
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M = matrix(F, g^2, r)
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for i in range(0, len(B1)):
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(a, b) = B1[i]
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c = as_function(AS, a.form*b.form)
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c = as_reduction(AS, c)
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c = as_function(AS, c)
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c = as_polyform(c, 2)
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#return M, c.coordinates(basis=B2)
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M[i, :] = vector(c.coordinates(basis=B2))
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K = M.kernel().basis()
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result = []
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for v in K:
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coeffs = {b : 0 for b in B1}
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for i in range(r):
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if v[i] != 0:
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coeffs[B1[i]] += v[i]
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result += [as_tensor_product_forms(B1, coeffs)]
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return result
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as_cover.canonical_ideal = as_canonical_ideal
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class as_tensor_product_forms:
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def __init__(self, pairs_of_forms, coeffs):
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self.pairs = pairs_of_forms
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self.coeffs = coeffs #dictionary
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self.curve = pairs_of_forms[0][0].curve
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def coordinates(self, basis = 0):
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AS = self.curve
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g = AS.genus()
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F = AS.base_ring
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if basis == 0:
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basis = AS.holomorphic_differentials_basis()
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result = matrix(F, g, g)
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for (omega1, omega2) in self.pairs:
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c = omega1.coordinates(basis = basis)
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d = omega2.coordinates(basis = basis)
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for i in range(g):
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for j in range(g):
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result[i, j] += self.coeffs[(omega1, omega2)]*c[i]*d[j]
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return result
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def __repr__(self):
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result = ''
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for (omega1, omega2) in self.pairs:
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if self.coeffs[(omega1, omega2)] !=0:
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result += str(self.coeffs[(omega1, omega2)]) + ' * ' + str(omega1) + '⊗' + str(omega2) + ' + '
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return result
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def polynomial(self):
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AS = self.curve
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F = AS.base_ring
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g = AS.genus()
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M = self.coordinates()
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Rg = PolynomialRing(F, 'X', g)
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X = Rg.gens()
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return sum(M[i, j] * X[i]*X[j] for i in range(g) for j in range(g))
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def group_action(self, elt):
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p = self.base_ring.characteristic()
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n = self.height
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elt1 = [p^n - a for a in elt]
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pairs_of_forms2 = [(a.group_action(elt), b.group_action(elt1)) for (a, b) in pairs_of_forms]
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coeffs2 = {(a.group_action(elt), b.group_action(elt1)) : self.coeffs[(a, b)] for (a, b) in pairs_of_forms}
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return as_tensor_product_forms(pairs_of_forms2, self.coeffs)
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@ -3,6 +3,7 @@ def as_reduction(AS, fct):
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n = AS.height
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F = AS.base_ring
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RxyzQ, Rxyz, x, y, z = AS.fct_field
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p = F.characteristic()
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ff = AS.functions
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ff = [RxyzQ(F.function) for F in ff]
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fct = RxyzQ(fct)
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@ -2,6 +2,8 @@ def teichmuller(self, n):
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AS = self.curve
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return as_witt([self] + n*[0*AS.x])
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as_function.teichmuller = teichmuller
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def Vd(self, n, m = 1):
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'''Return V^m([f] d[x]) in W_n Omega'''
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AS = self.curve
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@ -9,11 +11,45 @@ def Vd(self, n, m = 1):
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return as_witt_form([self] + n*[0*AS.x], (n+1)*[0*AS.x])
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return as_witt_form(m*[0*AS.x] + [self] + (n-m)*[0*AS.x], (n+1)*[0*AS.x])
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as_function.Vd = Vd
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def dV(self, n, m = 1):
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'''Return dV^m([f] d[x]) in W_n Omega'''
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AS = self.curve
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return as_witt_form((n+1)*[0*AS.x], m*[0*AS.x] + [self] + (n-m)*[0*AS.x])
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as_function.teichmuller = teichmuller
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as_function.Vd = Vd
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as_function.dV = dV
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as_function.dV = dV
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def witt_diffn(self, n):
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RxyzQ, Rxyz, x, y, z = AS.fct_field
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F = Rxyz.base_ring
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Rx.<x> = PolynomialRing(F)
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return 0
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as_function.witt_diffn = witt_diffn
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def auxilliary_multiplication(fct, i, form_fct, j, n):
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'''Return V^i(fct)*V^j(form_fct dx) in W_n Omega'''
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AS = fct.curve
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p = AS.base_ring.characteristic()
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if i == 0:
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return as_witt_form(j*[0*AS.x] + [fct^(p^j) * form_fct] + (n-j)*[0*AS.x], (n+1)*[0*AS.x])
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if j == 0:
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return auxilliary_multiplication(fct, i-1, form_fct^p * AS.x^(p-1), j, n).verschiebung()
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return p*auxilliary_multiplication(fct, i-1, form_fct, j-1, n).verschiebung()
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def auxilliary_multiplication2(fct1, i, fct2, j, n):
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'''Return V^i(fct1)*dV^j(fct2) in W_n Omega'''
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AS = fct.curve
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p = AS.base_ring.characteristic()
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if i > 0 and j > 0:
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return auxilliary_multiplication2(fct1, i, fct2, j, n).verschiebung()
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if i == 0:
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aux_Vs = j*[0*AS.x] * [fct2 * fct1^(p^j - 1) * fct1.diffn()/AS.dx] + (n - j)*[0*AS.x]
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aux = as_witt_form(aux_Vs, (n+1)*[0*AS.x])
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return as_witt_form((n+1)*[0*AS.x], j*[0*AS.x] + [fct1^(p^j) * fct2] + (n-j)*[0*AS.x]) - aux
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if j == 0:
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aux_Vs = j*[0*AS.x] * [fct1 * fct2^(p^j - 1) * fct2.diffn()/AS.dx] + (n - j)*[0*AS.x]
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aux = as_witt_form(aux_Vs, (n+1)*[0*AS.x])
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return aux
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@ -38,17 +38,23 @@ class as_witt:
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result = [as_function(C, a) for a in result]
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return as_witt(result)
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def verschiebung(self):
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def verschiebung(self, i = 1):
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AS = self.curve
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return as_witt([0*AS.x] + self.f)
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return as_witt(i*[0*AS.x] + self.f[i:])
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def R(self):
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n = self.n
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return as_witt(self.f[:n])
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def frobenius(self):
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def frobenius(self, i = 1):
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n= self.n
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p = self.curve.base_ring.characteristic()
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result = self.f[:n]
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result = [a^p for a in result]
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return as_witt(result)
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result = [a^(p^i) for a in result]
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return as_witt(result)
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def diffn(self):
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n = self.n
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AS = self.curve
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f0 = self.f[0]
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return f[0].witt_diffn(n) + as_witt_form((n+1)*[0*AS.x], [0*AS.x] + self.f[1:])
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@ -25,7 +25,7 @@ class as_witt_form:
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return result
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def __add__(self, other):
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print(self, other)
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#print(self, other)
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n = self.n
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AS = self.curve
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########
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@ -33,7 +33,7 @@ class as_witt_form:
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while(i <= n and self.Vs[i].function == 0 and other.Vs[i].function == 0):
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i+=1
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if i<=n:
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print('a')
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#print('a')
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if (i <= n and self.Vs[i].function != 0 and other.Vs[i].function != 0):
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self1 = self.cutVs(i)
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other1 = other.cutVs(i)
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@ -42,7 +42,7 @@ class as_witt_form:
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other_first = other.Vs[i].teichmuller(n)
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aux = self_first + other_first
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for j in range(0, n+1 - i):
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print(i, j)
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#print(i, j)
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result += aux.f[j].Vd(n, i+j)
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return result
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if (i <= n and self.Vs[i].function != 0 and other.Vs[i].function == 0):
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@ -62,7 +62,7 @@ class as_witt_form:
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while(i <= n and self.dVs[i].function == 0 and other.dVs[i].function == 0):
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i+=1
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if i<=n:
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print('b', i)
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#print('b', i)
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if (i <= n and self.dVs[i].function != 0 and other.dVs[i].function != 0):
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self1 = self.cutdVs(i)
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other1 = other.cutdVs(i)
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@ -87,6 +87,25 @@ class as_witt_form:
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return result
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return (0*AS.x).Vd(n)
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def __rmul__(self, other):
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n = self.n
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AS = self.curve
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if parent(other) == ZZ or isinstance(other, int):
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if other == 0:
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return as_witt_form((n+1)*[0*AS.x], (n+1)*[0*AS.x])
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if other < 0:
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return -(-other)*self
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return self + (other - 1)*self
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if isinstance(other, as_witt):
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result = as_witt_form((n+1)*[0*AS.x], (n+1)*[0*AS.x])
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for i in range(0, n+1):
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for j in range(0, n+1):
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result += auxilliary_multiplication(other.f[i], i, self.Vs[j], j, n)
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result += auxilliary_multiplication(other.f[i], i, self.dVs[j], j, n)
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return result
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def verschiebung(self, i = 1):
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AS = self.curve
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p = AS.base_ring.characteristic()
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@ -13,6 +13,7 @@ load('as_covers/dual_element.sage')
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load('as_covers/ith_magical_component.sage')
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load('as_covers/combination_components.sage')
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load('as_covers/group_action_matrices.sage')
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load('as_covers/as_polyforms.sage')
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load('superelliptic_drw/second_patch.sage')
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load('superelliptic_drw/decomposition_into_g0_g8.sage')
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load('superelliptic_drw/superelliptic_witt.sage')
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@ -30,6 +31,11 @@ load('as_drw/witt_polynomials.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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##############
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##############
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def init(lista, tests = False, init=True):
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@ -0,0 +1,435 @@
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class quaternion_cover:
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def __init__(self, C, list_of_fcts, branch_points = [], prec = 10):
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self.quotient = C
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self.functions = list_of_fcts
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self.height = len(list_of_fcts)
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F = C.base_ring
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self.base_ring = F
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p = C.characteristic
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self.characteristic = p
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self.prec = prec
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#group acting
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n = self.height
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from itertools import product
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pr = [list(GF(p)) for _ in range(n)]
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self.group = QuaternionGroup()
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#########
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f = C.polynomial
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m = C.exponent
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r = f.degree()
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delta = GCD(m, r)
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self.nb_of_pts_at_infty = delta
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self.branch_points = list(range(delta)) + branch_points
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Rxy.<x, y> = PolynomialRing(F, 2)
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Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
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all_x_series = {}
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all_y_series = {}
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all_z_series = {}
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all_dx_series = {}
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all_jumps = {}
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for pt in self.branch_points:
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x_series = superelliptic_function(C, x).expansion(pt=pt, prec=prec)
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y_series = superelliptic_function(C, y).expansion(pt=pt, prec=prec)
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z_series = []
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jumps = []
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n = len(list_of_fcts)
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list_of_power_series = [g.expansion(pt=pt, prec=prec) for g in list_of_fcts]
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for j in range(n):
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if j == 2:
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list_of_power_series[j] += z_series[0]*list_of_power_series[0] + z_series[1]*(list_of_power_series[0] + list_of_power_series[1])
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power_series = list_of_power_series[j]
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jump, correction, t_old, z = artin_schreier_transform(power_series, prec = prec)
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x_series = x_series(t = t_old)
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y_series = y_series(t = t_old)
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z_series = [zi(t = t_old) for zi in z_series]
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z_series += [z]
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jumps += [jump]
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list_of_power_series = [g(t = t_old) for g in list_of_power_series]
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all_jumps[pt] = jumps
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all_x_series[pt] = x_series
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all_y_series[pt] = y_series
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all_z_series[pt] = z_series
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all_dx_series[pt] = x_series.derivative()
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self.jumps = all_jumps
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self.x_series = all_x_series
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self.y_series = all_y_series
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self.z_series = all_z_series
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self.dx_series = all_dx_series
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##############
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#Function field
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variable_names = 'x, y'
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for i in range(n):
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variable_names += ', z' + str(i)
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Rxyz = PolynomialRing(F, n+2, variable_names)
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x, y = Rxyz.gens()[:2]
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z = Rxyz.gens()[2:]
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RxyzQ = FractionField(Rxyz)
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self.fct_field = (RxyzQ, Rxyz, x, y, z)
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self.x = quaternion_function(self, x)
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self.y = quaternion_function(self, y)
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self.z = [quaternion_function(self, z[j]) for j in range(n)]
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self.dx = quaternion_form(self, 1)
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self.one = quaternion_function(self, 1)
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def __repr__(self):
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n = self.height
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p = self.characteristic
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if n==1:
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return "(Z/p)-cover of " + str(self.quotient)+" with the equation:\n z^" + str(p) + " - z = " + str(self.functions[0])
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result = "Q8-cover of " + str(self.quotient)+" with the equations:\n"
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for i in range(n):
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if i !=2:
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result += 'z' + str(i) + "^" + str(p) + " - z" + str(i) + " = " + str(self.functions[i]) + "\n"
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else:
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result += 'z2^2 - z2 = ' + str(self.functions[2]) + ' + ' + 'z0 * ' + str(self.functions[0]) + ' + z1 * (' + str(self.functions[0] + self.functions[1]) + ')'
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return result
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def genus(self):
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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)
|
||||
|
||||
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))
|
||||
|
||||
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))
|
||||
|
||||
def holomorphic_differentials_basis(self, threshold = 8):
|
||||
from itertools import product
|
||||
x_series = self.x_series
|
||||
y_series = self.y_series
|
||||
z_series = self.z_series
|
||||
dx_series = self.dx_series
|
||||
delta = self.nb_of_pts_at_infty
|
||||
p = self.characteristic
|
||||
n = self.height
|
||||
prec = self.prec
|
||||
C = self.quotient
|
||||
F = self.base_ring
|
||||
m = C.exponent
|
||||
r = C.polynomial.degree()
|
||||
RxyzQ, Rxyz, x, y, z = self.fct_field
|
||||
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
|
||||
#Tworzymy zbiór S form z^i x^j y^k dx/y o waluacji >= waluacja z^(p-1)*dx/y
|
||||
S = []
|
||||
pr = [list(GF(p)) for _ in range(n)]
|
||||
for i in range(0, threshold*r):
|
||||
for j in range(0, m):
|
||||
for k in product(*pr):
|
||||
eta = quaternion_form(self, x^i * prod(z[i1]^(k[i1]) for i1 in range(n))/y^j)
|
||||
eta_exp = eta.expansion(pt=self.branch_points[0])
|
||||
S += [(eta, eta_exp)]
|
||||
|
||||
forms = holomorphic_combinations(S)
|
||||
|
||||
for pt in self.branch_points[1:]:
|
||||
forms = [(omega, omega.expansion(pt=pt)) for omega in forms]
|
||||
forms = holomorphic_combinations(forms)
|
||||
|
||||
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)
|
||||
if len(forms) > self.genus():
|
||||
raise ValueError("Increase precision.")
|
||||
return forms
|
||||
|
||||
def cartier_matrix(self, prec=50):
|
||||
g = self.genus()
|
||||
F = self.base_ring
|
||||
M = matrix(F, g, g)
|
||||
for i, omega in enumerate(self.holomorphic_differentials_basis()):
|
||||
M[:, i] = vector(omega.cartier().coordinates())
|
||||
return M
|
||||
|
||||
def at_most_poles(self, pole_order, threshold = 8):
|
||||
""" Find fcts with pole order in infty's at most pole_order. Threshold gives a bound on powers of x in the function.
|
||||
If you suspect that you haven't found all the functions, you may increase it."""
|
||||
from itertools import product
|
||||
x_series = self.x_series
|
||||
y_series = self.y_series
|
||||
z_series = self.z_series
|
||||
delta = self.nb_of_pts_at_infty
|
||||
p = self.characteristic
|
||||
n = self.height
|
||||
prec = self.prec
|
||||
C = self.quotient
|
||||
F = self.base_ring
|
||||
m = C.exponent
|
||||
r = C.polynomial.degree()
|
||||
RxyzQ, Rxyz, x, y, z = self.fct_field
|
||||
F = C.base_ring
|
||||
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
|
||||
#Tworzymy zbiór S form z^i x^j y^k dx/y o waluacji >= waluacja z^(p-1)*dx/y
|
||||
S = []
|
||||
RQxyz = FractionField(Rxyz)
|
||||
pr = [list(GF(p)) for _ in range(n)]
|
||||
for i in range(0, threshold*r):
|
||||
for j in range(0, m):
|
||||
for k in product(*pr):
|
||||
eta = quaternion_function(self, x^i * prod(z[i1]^(k[i1]) for i1 in range(n))*y^j)
|
||||
eta_exp = eta.expansion_at_infty()
|
||||
S += [(eta, eta_exp)]
|
||||
|
||||
forms = holomorphic_combinations_fcts(S, pole_order)
|
||||
|
||||
for i in range(1, delta):
|
||||
forms = [(omega, omega.expansion_at_infty(place = i)) for omega in forms]
|
||||
forms = holomorphic_combinations_fcts(forms, pole_order)
|
||||
|
||||
return forms
|
||||
|
||||
def magical_element(self, threshold = 8):
|
||||
list_of_elts = self.at_most_poles(self.exponent_of_different_prim(), threshold)
|
||||
result = []
|
||||
for a in list_of_elts:
|
||||
if a.trace().function != 0:
|
||||
result += [a]
|
||||
return result
|
||||
|
||||
def pseudo_magical_element(self, threshold = 8):
|
||||
list_of_elts = self.at_most_poles(self.exponent_of_different(), threshold)
|
||||
result = []
|
||||
for a in list_of_elts:
|
||||
if a.trace().function != 0:
|
||||
result += [a]
|
||||
return result
|
||||
|
||||
def at_most_poles_forms(self, pole_order, threshold = 8):
|
||||
"""Find forms with pole order in all the points at infty equat at most to pole_order. Threshold gives a bound on powers of x in the form.
|
||||
If you suspect that you haven't found all the functions, you may increase it."""
|
||||
from itertools import product
|
||||
x_series = self.x_series
|
||||
y_series = self.y_series
|
||||
z_series = self.z_series
|
||||
delta = self.nb_of_pts_at_infty
|
||||
p = self.characteristic
|
||||
n = self.height
|
||||
prec = self.prec
|
||||
C = self.quotient
|
||||
F = self.base_ring
|
||||
m = C.exponent
|
||||
r = C.polynomial.degree()
|
||||
RxyzQ, Rxyz, x, y, z = self.fct_field
|
||||
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
|
||||
#Tworzymy zbiór S form z^i x^j y^k dx/y o waluacji >= waluacja z^(p-1)*dx/y
|
||||
S = []
|
||||
RQxyz = FractionField(Rxyz)
|
||||
pr = [list(GF(p)) for _ in range(n)]
|
||||
for i in range(0, threshold*r):
|
||||
for j in range(0, m):
|
||||
for k in product(*pr):
|
||||
eta = quaternion_form(self, x^i * prod(z[i1]^(k[i1]) for i1 in range(n))/y^j)
|
||||
eta_exp = eta.expansion_at_infty()
|
||||
S += [(eta, eta_exp)]
|
||||
|
||||
forms = holomorphic_combinations_forms(S, pole_order)
|
||||
|
||||
for pt in self.branch_points[1:]:
|
||||
forms = [(omega, omega.expansion(pt=pt)) for omega in forms]
|
||||
forms = holomorphic_combinations_forms(forms, pole_order)
|
||||
|
||||
return forms
|
||||
|
||||
def uniformizer(self, place = 0):
|
||||
'''Return uniformizer of curve self at place-th place at infinity.'''
|
||||
p = self.characteristic
|
||||
n = self.height
|
||||
F = self.base_ring
|
||||
RxyzQ, Rxyz, x, y, z = self.fct_field
|
||||
fx = quaternion_function(self, x)
|
||||
z = [quaternion_function(self, zi) for zi in z]
|
||||
# We create a list of functions. We add there all variables...
|
||||
list_of_fcts = [fx]+z
|
||||
vfx = fx.valuation(place)
|
||||
vz = [zi.valuation(place) for zi in z]
|
||||
|
||||
# Then we subtract powers of variables with the same valuation (so that 1/t^(kp) cancels) and add to this list.
|
||||
for j1 in range(n):
|
||||
for j2 in range(n):
|
||||
if j1>j2:
|
||||
a = gcd(vz[j1] , vz[j2])
|
||||
vz1 = vz[j1]/a
|
||||
vz2 = vz[j2]/a
|
||||
for b in F:
|
||||
if (z[j1]^(vz2) - b*z[j2]^(vz1)).valuation(place) > (z[j2]^(vz1)).valuation(place):
|
||||
list_of_fcts += [z[j1]^(vz2) - b*z[j2]^(vz1)]
|
||||
for j1 in range(n):
|
||||
a = gcd(vz[j1], vfx)
|
||||
vzj = vz[j1] /a
|
||||
vfx = vfx/a
|
||||
for b in F:
|
||||
if (fx^(vzj) - b*z[j1]^(vfx)).valuation(place) > (z[j1]^(vfx)).valuation(place):
|
||||
list_of_fcts += [fx^(vzj) - b*z[j1]^(vfx)]
|
||||
#Finally, we check if on the list there are two elements with the same valuation.
|
||||
for f1 in list_of_fcts:
|
||||
for f2 in list_of_fcts:
|
||||
d, a, b = xgcd(f1.valuation(place), f2.valuation(place))
|
||||
if d == 1:
|
||||
return f1^a*f2^b
|
||||
raise ValueError("My method of generating fcts with relatively prime valuation failed.")
|
||||
|
||||
|
||||
def ith_ramification_gp(self, i, place = 0):
|
||||
'''Find ith ramification group at place at infty of nb place.'''
|
||||
G = self.group
|
||||
t = self.uniformizer(place)
|
||||
Gi = [G[0]]
|
||||
for g in G:
|
||||
if g != G[0]:
|
||||
tg = t.group_action(g)
|
||||
v = (tg - t).valuation(place)
|
||||
if v >= i+1:
|
||||
Gi += [g]
|
||||
return Gi
|
||||
|
||||
def ramification_jumps(self, place = 0):
|
||||
'''Return list of lower ramification jumps at at place at infty of nb place.'''
|
||||
G = self.group
|
||||
ramification_jps = []
|
||||
i = 0
|
||||
while len(G) > 1:
|
||||
Gi = self.ith_ramification_gp(i+1, place)
|
||||
if len(Gi) < len(G):
|
||||
ramification_jps += [i]
|
||||
G = Gi
|
||||
i+=1
|
||||
return ramification_jps
|
||||
|
||||
def a_number(self):
|
||||
g = self.genus()
|
||||
return g - self.cartier_matrix().rank()
|
||||
|
||||
def cohomology_of_structure_sheaf_basis(self, holo_basis = 0, threshold = 8):
|
||||
if holo_basis == 0:
|
||||
holo_basis = self.holomorphic_differentials_basis(threshold = threshold)
|
||||
from itertools import product
|
||||
x_series = self.x_series
|
||||
y_series = self.y_series
|
||||
z_series = self.z_series
|
||||
delta = self.nb_of_pts_at_infty
|
||||
p = self.characteristic
|
||||
n = self.height
|
||||
prec = self.prec
|
||||
C = self.quotient
|
||||
F = self.base_ring
|
||||
m = C.exponent
|
||||
r = C.polynomial.degree()
|
||||
RxyzQ, Rxyz, x, y, z = self.fct_field
|
||||
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
|
||||
#Tworzymy zbiór S form z^i x^j y^k dx/y o waluacji >= waluacja z^(p-1)*dx/y
|
||||
result_fcts = []
|
||||
V = VectorSpace(F,self.genus())
|
||||
S = V.subspace([])
|
||||
RQxyz = FractionField(Rxyz)
|
||||
pr = [list(GF(p)) for _ in range(n)]
|
||||
i = 0
|
||||
while len(result_fcts) < self.genus():
|
||||
for j in range(0, m):
|
||||
for k in product(*pr):
|
||||
f = quaternion_function(self, prod(z[i1]^(k[i1]) for i1 in range(n))/x^i*y^j)
|
||||
f_products = [omega.serre_duality_pairing(f) for omega in holo_basis]
|
||||
if vector(f_products) not in S:
|
||||
S = S+V.subspace([V(f_products)])
|
||||
result_fcts += [f]
|
||||
i += 1
|
||||
return result_fcts
|
||||
|
||||
def lift_to_de_rham(self, fct, threshold = 30):
|
||||
'''Given function fct, find form eta regular on affine part such that eta - d(fct) is regular in infty. (Works for one place at infty now)'''
|
||||
from itertools import product
|
||||
x_series = self.x_series
|
||||
y_series = self.y_series
|
||||
z_series = self.z_series
|
||||
dx_series = self.dx_series
|
||||
delta = self.nb_of_pts_at_infty
|
||||
p = self.characteristic
|
||||
n = self.height
|
||||
prec = self.prec
|
||||
C = self.quotient
|
||||
F = self.base_ring
|
||||
m = C.exponent
|
||||
r = C.polynomial.degree()
|
||||
RxyzQ, Rxyz, x, y, z = self.fct_field
|
||||
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
|
||||
#Tworzymy zbiór S form z^i x^j y^k dx/y o waluacji >= waluacja z^(p-1)*dx/y
|
||||
S = [(fct.diffn(), fct.diffn().expansion_at_infty())]
|
||||
pr = [list(GF(p)) for _ in range(n)]
|
||||
holo = self.holomorphic_differentials_basis(threshold = threshold)
|
||||
for i in range(0, threshold*r):
|
||||
for j in range(0, m):
|
||||
for k in product(*pr):
|
||||
eta = quaternion_form(self, x^i*prod(z[i1]^(k[i1]) for i1 in range(n))/y^j)
|
||||
eta_exp = eta.expansion_at_infty()
|
||||
S += [(eta, eta_exp)]
|
||||
forms = holomorphic_combinations(S)
|
||||
if len(forms) <= self.genus():
|
||||
raise ValueError("Increase threshold!")
|
||||
for omega in forms:
|
||||
for a in F:
|
||||
if (a*omega + fct.diffn()).is_regular_on_U0():
|
||||
return a*omega + fct.diffn()
|
||||
raise ValueError("Unknown.")
|
||||
|
||||
def de_rham_basis(self, holo_basis = 0, cohomology_basis = 0, threshold = 30):
|
||||
if holo_basis == 0:
|
||||
holo_basis = self.holomorphic_differentials_basis(threshold = threshold)
|
||||
if cohomology_basis == 0:
|
||||
cohomology_basis = self.cohomology_of_structure_sheaf_basis(holo_basis = holo_basis, threshold = threshold)
|
||||
result = []
|
||||
for omega in holo_basis:
|
||||
result += [quaternion_cech(self, omega, quaternion_function(self, 0))]
|
||||
for f in cohomology_basis:
|
||||
omega = self.lift_to_de_rham(f, threshold = threshold)
|
||||
result += [quaternion_cech(self, omega, f)]
|
||||
return result
|
||||
|
||||
def holomorphic_combinations(S):
|
||||
"""Given a list S of pairs (form, corresponding Laurent series at some pt), find their combinations holomorphic at that pt."""
|
||||
C_AS = S[0][0].curve
|
||||
p = C_AS.characteristic
|
||||
F = C_AS.base_ring
|
||||
prec = C_AS.prec
|
||||
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
|
||||
RtQ = FractionField(Rt)
|
||||
minimal_valuation = min([g[1].valuation() for g in S])
|
||||
if minimal_valuation >= 0:
|
||||
return [s[0] for s in S]
|
||||
list_of_lists = [] #to będzie lista złożona z list współczynników część nieholomorficznych rozwinięcia form z S
|
||||
for eta, eta_exp in S:
|
||||
a = -minimal_valuation + eta_exp.valuation()
|
||||
list_coeffs = a*[0] + eta_exp.list() + (-minimal_valuation)*[0]
|
||||
list_coeffs = list_coeffs[:-minimal_valuation]
|
||||
list_of_lists += [list_coeffs]
|
||||
M = matrix(F, list_of_lists)
|
||||
V = M.kernel() #chcemy wyzerować części nieholomorficzne, biorąc kombinacje form z S
|
||||
|
||||
|
||||
# Sprawdzamy, jakim formom odpowiadają elementy V.
|
||||
forms = []
|
||||
for vec in V.basis():
|
||||
forma_holo = quaternion_form(C_AS, 0)
|
||||
forma_holo_power_series = Rt(0)
|
||||
for vec_wspolrzedna, elt_S in zip(vec, S):
|
||||
eta = elt_S[0]
|
||||
#eta_exp = elt_S[1]
|
||||
forma_holo += vec_wspolrzedna*eta
|
||||
#forma_holo_power_series += vec_wspolrzedna*eta_exp
|
||||
forms += [forma_holo]
|
||||
return forms
|
||||
|
|
@ -0,0 +1,303 @@
|
|||
class quaternion_form:
|
||||
def __init__(self, C, g):
|
||||
self.curve = C
|
||||
n = C.height
|
||||
F = C.base_ring
|
||||
variable_names = 'x, y'
|
||||
for i in range(n):
|
||||
variable_names += ', z' + str(i)
|
||||
Rxyz = PolynomialRing(F, n+2, variable_names)
|
||||
x, y = Rxyz.gens()[:2]
|
||||
z = Rxyz.gens()[2:]
|
||||
RxyzQ = FractionField(Rxyz)
|
||||
self.form = RxyzQ(g)
|
||||
|
||||
def __repr__(self):
|
||||
return "(" + str(self.form)+") * dx"
|
||||
|
||||
def expansion_at_infty(self, place = 0):
|
||||
C = self.curve
|
||||
delta = C.nb_of_pts_at_infty
|
||||
F = C.base_ring
|
||||
x_series = C.x_series[place]
|
||||
y_series = C.y_series[place]
|
||||
z_series = C.z_series[place]
|
||||
dx_series = C.dx_series[place]
|
||||
n = C.height
|
||||
variable_names = 'x, y'
|
||||
for j in range(n):
|
||||
variable_names += ', z' + str(j)
|
||||
Rxyz = PolynomialRing(F, n+2, variable_names)
|
||||
x, y = Rxyz.gens()[:2]
|
||||
z = Rxyz.gens()[2:]
|
||||
RxyzQ = FractionField(Rxyz)
|
||||
prec = C.prec
|
||||
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
|
||||
g = self.form
|
||||
sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}
|
||||
return g.substitute(sub_list)*dx_series
|
||||
|
||||
def expansion(self, pt = 0):
|
||||
'''Same code as expansion_at_infty.'''
|
||||
C = self.curve
|
||||
F = C.base_ring
|
||||
x_series = C.x_series[pt]
|
||||
y_series = C.y_series[pt]
|
||||
z_series = C.z_series[pt]
|
||||
dx_series = C.dx_series[pt]
|
||||
n = C.height
|
||||
variable_names = 'x, y'
|
||||
for j in range(n):
|
||||
variable_names += ', z' + str(j)
|
||||
Rxyz = PolynomialRing(F, n+2, variable_names)
|
||||
x, y = Rxyz.gens()[:2]
|
||||
z = Rxyz.gens()[2:]
|
||||
RxyzQ = FractionField(Rxyz)
|
||||
prec = C.prec
|
||||
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
|
||||
g = self.form
|
||||
sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}
|
||||
return g.substitute(sub_list)*dx_series
|
||||
|
||||
def __add__(self, other):
|
||||
C = self.curve
|
||||
g1 = self.form
|
||||
g2 = other.form
|
||||
return quaternion_form(C, g1 + g2)
|
||||
|
||||
def __sub__(self, other):
|
||||
C = self.curve
|
||||
g1 = self.form
|
||||
g2 = other.form
|
||||
return quaternion_form(C, g1 - g2)
|
||||
|
||||
def __neg__(self):
|
||||
C = self.curve
|
||||
g = self.form
|
||||
return quaternion_form(C, -g)
|
||||
|
||||
def __rmul__(self, constant):
|
||||
C = self.curve
|
||||
omega = self.form
|
||||
return quaternion_form(C, constant*omega)
|
||||
|
||||
def group_action(self, ZN_tuple):
|
||||
C = self.curve
|
||||
n = C.height
|
||||
RxyzQ, Rxyz, x, y, z = C.fct_field
|
||||
sub_list = {x : x, y : y} | {z[j] : z[j]+ZN_tuple[j] for j in range(n)}
|
||||
g = self.form
|
||||
return quaternion_form(C, g.substitute(sub_list))
|
||||
|
||||
def coordinates(self, basis = 0):
|
||||
"""Find coordinates of the given holomorphic form self in terms of the basis forms in a list holo."""
|
||||
C = self.curve
|
||||
if basis == 0:
|
||||
basis = C.holomorphic_differentials_basis()
|
||||
RxyzQ, Rxyz, x, y, z = C.fct_field
|
||||
# We need to have only polynomials to use monomial_coefficients in linear_representation_polynomials,
|
||||
# and sometimes basis elements have denominators. Thus we multiply by them.
|
||||
denom = LCM([denominator(omega.form) for omega in basis])
|
||||
basis = [denom*omega for omega in basis]
|
||||
self_with_no_denominator = denom*self
|
||||
return linear_representation_polynomials(Rxyz(self_with_no_denominator.form), [Rxyz(omega.form) for omega in basis])
|
||||
|
||||
def trace(self):
|
||||
C = self.curve
|
||||
C_super = C.quotient
|
||||
n = C.height
|
||||
F = C.base_ring
|
||||
variable_names = 'x, y'
|
||||
for j in range(n):
|
||||
variable_names += ', z' + str(j)
|
||||
Rxyz = PolynomialRing(F, n+2, variable_names)
|
||||
x, y = Rxyz.gens()[:2]
|
||||
z = Rxyz.gens()[2:]
|
||||
RxyzQ = FractionField(Rxyz)
|
||||
result = quaternion_form(C, 0)
|
||||
G = C.group
|
||||
for a in G:
|
||||
result += self.group_action(a)
|
||||
result = result.form
|
||||
Rxy.<x, y> = PolynomialRing(F, 2)
|
||||
Qxy = FractionField(Rxy)
|
||||
result = quaternion_reduction(AS, result)
|
||||
return superelliptic_form(C_super, Qxy(result))
|
||||
|
||||
def residue(self, place=0):
|
||||
return self.expansion_at_infty(place = place).residue()
|
||||
|
||||
def valuation(self, place=0):
|
||||
return self.expansion_at_infty(place = place).valuation()
|
||||
|
||||
def serre_duality_pairing(self, fct):
|
||||
AS = self.curve
|
||||
return sum((fct*self).residue(place = _) for _ in range(AS.nb_of_pts_at_infty))
|
||||
|
||||
def cartier(self):
|
||||
C = self.curve
|
||||
F = C.base_ring
|
||||
n = C.height
|
||||
ff = C.functions
|
||||
p = F.characteristic()
|
||||
C_super = C.quotient
|
||||
(RxyzQ, Rxyz, x, y, z) = C.fct_field
|
||||
fct = self.form
|
||||
Rxy.<x, y> = PolynomialRing(F, 2)
|
||||
RxyQ = FractionField(Rxy)
|
||||
x, y = Rxyz.gens()[0], Rxyz.gens()[1]
|
||||
z = Rxyz.gens()[2:]
|
||||
num = Rxyz(fct.numerator())
|
||||
den = Rxyz(fct.denominator())
|
||||
result = RxyzQ(0)
|
||||
#return (num, den, z, fct)
|
||||
if den in Rxy:
|
||||
sub_list = {x : x, y : y} | {z[j] : (z[j]^p - RxyzQ(ff[j].function)) for j in range(n)}
|
||||
num = RxyzQ(num.substitute(sub_list))
|
||||
den1 = Rxyz(num.denominator())
|
||||
num = Rxyz(num*den1^p)
|
||||
for monomial in Rxyz(num).monomials():
|
||||
degrees = [monomial.degree(z[i]) for i in range(n)]
|
||||
product_of_z = prod(z[i]^(degrees[i]) for i in range(n))
|
||||
monomial_divided_by_z = monomial/product_of_z
|
||||
product_of_z_no_p = prod(z[i]^(degrees[i]/p) for i in range(n))
|
||||
aux_form = superelliptic_form(C_super, RxyQ(monomial_divided_by_z/den))
|
||||
aux_form = aux_form.cartier()
|
||||
result += product_of_z_no_p * Rxyz(num).monomial_coefficient(monomial) * aux_form.form/den1
|
||||
return quaternion_form(C, result)
|
||||
raise ValueError("Please present first your form as sum z^i omega_i, where omega_i are forms on quotient curve.")
|
||||
|
||||
def is_regular_on_U0(self):
|
||||
AS = self.curve
|
||||
C = AS.quotient
|
||||
m = C.exponent
|
||||
RxyzQ, Rxyz, x, y, z = AS.fct_field
|
||||
if y^(m-1)*self.form in Rxyz:
|
||||
return True
|
||||
return False
|
||||
|
||||
def artin_schreier_transform(power_series, prec = 10):
|
||||
"""Given a power_series, find correction such that power_series - (correction)^p +correction has valuation
|
||||
-jump non divisible by p. Also, express t (the variable) in terms of the uniformizer at infty on the curve
|
||||
z^p - z = power_series, where z = 1/t_new^(jump) and express z in terms of the new uniformizer."""
|
||||
correction = 0
|
||||
F = power_series.parent().base()
|
||||
p = F.characteristic()
|
||||
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
|
||||
RtQ = FractionField(Rt)
|
||||
power_series = RtQ(power_series)
|
||||
if power_series.valuation() == +Infinity:
|
||||
raise ValueError("Precision is too low.")
|
||||
if power_series.valuation() >= 0:
|
||||
# THIS IS WRONG - THERE ARE SEVERAL PLACES OVER THIS PLACE, AND IT DEPENDS
|
||||
aux = t^p - t
|
||||
z = new_reverse(aux, prec = prec)
|
||||
z = z(t = power_series)
|
||||
return(0, 0, t, z)
|
||||
|
||||
while(power_series.valuation() % p == 0 and power_series.valuation() < 0):
|
||||
M = -power_series.valuation()/p
|
||||
coeff = power_series.list()[0] #wspolczynnik a_(-p) w f_AS
|
||||
correction += coeff.nth_root(p)*t^(-M)
|
||||
power_series = power_series - (coeff*t^(-p*M) - coeff.nth_root(p)*t^(-M))
|
||||
jump = max(-(power_series.valuation()), 0)
|
||||
try:
|
||||
T = nth_root2((power_series)^(-1), jump, prec=prec) #T is defined by power_series = 1/T^m
|
||||
except:
|
||||
print("no ", str(jump), "-th root; divide by", power_series.list()[0])
|
||||
return (jump, power_series.list()[0])
|
||||
T_rev = new_reverse(T, prec = prec)
|
||||
t_old = T_rev(t^p/nth_root2(1 - t^((p-1)*jump), jump, prec=prec))
|
||||
z = 1/t^(jump) + Rt(correction)(t = t_old)
|
||||
return(jump, correction, t_old, z)
|
||||
|
||||
|
||||
def are_forms_linearly_dependent(set_of_forms):
|
||||
from sage.rings.polynomial.toy_variety import is_linearly_dependent
|
||||
C = set_of_forms[0].curve
|
||||
F = C.base_ring
|
||||
n = C.height
|
||||
variable_names = 'x, y'
|
||||
for i in range(n):
|
||||
variable_names += ', z' + str(i)
|
||||
Rxyz = PolynomialRing(F, n+2, variable_names)
|
||||
denominators = prod(denominator(omega.form) for omega in set_of_forms)
|
||||
return is_linearly_dependent([Rxyz(denominators*omega.form) for omega in set_of_forms])
|
||||
|
||||
def holomorphic_combinations_fcts(S, pole_order):
|
||||
'''given a set S of (form, corresponding Laurent series at some pt), find their combinations holomorphic at that pt'''
|
||||
C_AS = S[0][0].curve
|
||||
p = C_AS.characteristic
|
||||
F = C_AS.base_ring
|
||||
prec = C_AS.prec
|
||||
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
|
||||
RtQ = FractionField(Rt)
|
||||
minimal_valuation = min([Rt(g[1]).valuation() for g in S])
|
||||
if minimal_valuation >= -pole_order:
|
||||
return [s[0] for s in S]
|
||||
list_of_lists = [] #to będzie lista złożona z list współczynników część nieholomorficznych rozwinięcia form z S
|
||||
for eta, eta_exp in S:
|
||||
a = -minimal_valuation + Rt(eta_exp).valuation()
|
||||
list_coeffs = a*[0] + Rt(eta_exp).list() + (-minimal_valuation)*[0]
|
||||
list_coeffs = list_coeffs[:-minimal_valuation - pole_order]
|
||||
list_of_lists += [list_coeffs]
|
||||
M = matrix(F, list_of_lists)
|
||||
V = M.kernel() #chcemy wyzerować części nieholomorficzne, biorąc kombinacje form z S
|
||||
|
||||
|
||||
# Sprawdzamy, jakim formom odpowiadają elementy V.
|
||||
forms = []
|
||||
for vec in V.basis():
|
||||
forma_holo = quaternion_function(C_AS, 0)
|
||||
forma_holo_power_series = Rt(0)
|
||||
for vec_wspolrzedna, elt_S in zip(vec, S):
|
||||
eta = elt_S[0]
|
||||
#eta_exp = elt_S[1]
|
||||
forma_holo += vec_wspolrzedna*eta
|
||||
#forma_holo_power_series += vec_wspolrzedna*eta_exp
|
||||
forms += [forma_holo]
|
||||
return forms
|
||||
|
||||
#given a set S of (form, corresponding Laurent series at some pt), find their combinations holomorphic at that pt
|
||||
def holomorphic_combinations_forms(S, pole_order):
|
||||
C_AS = S[0][0].curve
|
||||
p = C_AS.characteristic
|
||||
F = C_AS.base_ring
|
||||
prec = C_AS.prec
|
||||
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
|
||||
RtQ = FractionField(Rt)
|
||||
minimal_valuation = min([Rt(g[1]).valuation() for g in S])
|
||||
if minimal_valuation >= -pole_order:
|
||||
return [s[0] for s in S]
|
||||
list_of_lists = [] #to będzie lista złożona z list współczynników część nieholomorficznych rozwinięcia form z S
|
||||
for eta, eta_exp in S:
|
||||
a = -minimal_valuation + Rt(eta_exp).valuation()
|
||||
list_coeffs = a*[0] + Rt(eta_exp).list() + (-minimal_valuation)*[0]
|
||||
list_coeffs = list_coeffs[:-minimal_valuation - pole_order]
|
||||
list_of_lists += [list_coeffs]
|
||||
M = matrix(F, list_of_lists)
|
||||
V = M.kernel() #chcemy wyzerować części nieholomorficzne, biorąc kombinacje form z S
|
||||
|
||||
|
||||
# Sprawdzamy, jakim formom odpowiadają elementy V.
|
||||
forms = []
|
||||
for vec in V.basis():
|
||||
forma_holo = quaternion_form(C_AS, 0)
|
||||
forma_holo_power_series = Rt(0)
|
||||
for vec_wspolrzedna, elt_S in zip(vec, S):
|
||||
eta = elt_S[0]
|
||||
#eta_exp = elt_S[1]
|
||||
forma_holo += vec_wspolrzedna*eta
|
||||
#forma_holo_power_series += vec_wspolrzedna*eta_exp
|
||||
forms += [forma_holo]
|
||||
return forms
|
||||
|
||||
#print only forms that are log at the branch pts, but not holomorphic
|
||||
def only_log_forms(C_AS):
|
||||
list1 = AS.at_most_poles_forms(0)
|
||||
list2 = AS.at_most_poles_forms(1)
|
||||
result = []
|
||||
for a in list2:
|
||||
if not(are_forms_linearly_dependent(list1 + result + [a])):
|
||||
result += [a]
|
||||
return result
|
||||
|
|
@ -0,0 +1,207 @@
|
|||
class quaternion_function:
|
||||
def __init__(self, C, g):
|
||||
self.curve = C
|
||||
F = C.base_ring
|
||||
n = C.height
|
||||
variable_names = 'x, y'
|
||||
for i in range(n):
|
||||
variable_names += ', z' + str(i)
|
||||
Rxyz = PolynomialRing(F, n+2, variable_names)
|
||||
x, y = Rxyz.gens()[:2]
|
||||
z = Rxyz.gens()[2:]
|
||||
RxyzQ = FractionField(Rxyz)
|
||||
self.function = RxyzQ(g)
|
||||
#self.function = quaternion_reduction(AS, RxyzQ(g))
|
||||
|
||||
def __repr__(self):
|
||||
return str(self.function)
|
||||
|
||||
def __eq__(self, other):
|
||||
AS = self.curve
|
||||
RxyzQ, Rxyz, x, y, z = AS.fct_field
|
||||
aux = self - other
|
||||
aux = RxyzQ(aux.function)
|
||||
aux = aux.numerator()
|
||||
aux = quaternion_function(AS, aux)
|
||||
aux = aux.expansion_at_infty()
|
||||
if aux.valuation() >= 1:
|
||||
return True
|
||||
return False
|
||||
|
||||
def __add__(self, other):
|
||||
C = self.curve
|
||||
g1 = self.function
|
||||
g2 = other.function
|
||||
return quaternion_function(C, g1 + g2)
|
||||
|
||||
def __sub__(self, other):
|
||||
C = self.curve
|
||||
g1 = self.function
|
||||
g2 = other.function
|
||||
return quaternion_function(C, g1 - g2)
|
||||
|
||||
def __rmul__(self, constant):
|
||||
C = self.curve
|
||||
g = self.function
|
||||
return quaternion_function(C, constant*g)
|
||||
|
||||
def __neg__(self):
|
||||
C = self.curve
|
||||
g = self.function
|
||||
return quaternion_function(C, -g)
|
||||
|
||||
def __mul__(self, other):
|
||||
if isinstance(other, quaternion_function):
|
||||
C = self.curve
|
||||
g1 = self.function
|
||||
g2 = other.function
|
||||
return quaternion_function(C, g1*g2)
|
||||
if isinstance(other, quaternion_form):
|
||||
C = self.curve
|
||||
g1 = self.function
|
||||
g2 = other.form
|
||||
return quaternion_form(C, g1*g2)
|
||||
|
||||
def __truediv__(self, other):
|
||||
C = self.curve
|
||||
g1 = self.function
|
||||
g2 = other.function
|
||||
return quaternion_function(C, g1/g2)
|
||||
|
||||
def __pow__(self, exponent):
|
||||
C = self.curve
|
||||
g1 = self.function
|
||||
return quaternion_function(C, g1^(exponent))
|
||||
|
||||
def expansion_at_infty(self, place = 0):
|
||||
C = self.curve
|
||||
delta = C.nb_of_pts_at_infty
|
||||
F = C.base_ring
|
||||
x_series = C.x_series[place]
|
||||
y_series = C.y_series[place]
|
||||
z_series = C.z_series[place]
|
||||
n = C.height
|
||||
variable_names = 'x, y'
|
||||
for j in range(n):
|
||||
variable_names += ', z' + str(j)
|
||||
Rxyz = PolynomialRing(F, n+2, variable_names)
|
||||
x, y = Rxyz.gens()[:2]
|
||||
z = Rxyz.gens()[2:]
|
||||
RxyzQ = FractionField(Rxyz)
|
||||
prec = C.prec
|
||||
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
|
||||
g = self.function
|
||||
g = RxyzQ(g)
|
||||
sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}
|
||||
return g.substitute(sub_list)
|
||||
|
||||
def expansion(self, pt = 0):
|
||||
C = self.curve
|
||||
delta = C.nb_of_pts_at_infty
|
||||
F = C.base_ring
|
||||
x_series = C.x_series[pt]
|
||||
y_series = C.y_series[pt]
|
||||
z_series = C.z_series[pt]
|
||||
n = C.height
|
||||
variable_names = 'x, y'
|
||||
for j in range(n):
|
||||
variable_names += ', z' + str(j)
|
||||
Rxyz = PolynomialRing(F, n+2, variable_names)
|
||||
x, y = Rxyz.gens()[:2]
|
||||
z = Rxyz.gens()[2:]
|
||||
RxyzQ = FractionField(Rxyz)
|
||||
prec = C.prec
|
||||
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
|
||||
g = self.function
|
||||
g = RxyzQ(g)
|
||||
sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}
|
||||
return g.substitute(sub_list)
|
||||
|
||||
def group_action(self, elt):
|
||||
Q8 = QuaternionGroup()
|
||||
Qi, Qj = Q8.gens()
|
||||
AS = self.curve
|
||||
RxyzQ, Rxyz, x, y, z = AS.fct_field
|
||||
if elt == Qi^4:
|
||||
sub_list = {x : x, y : y} | {z[0] : z[0], z[1] : z[1], z[2]: z[2]}
|
||||
if elt == Qi:
|
||||
sub_list = {x : x, y : y} | {z[0] : z[0]+1, z[1] : z[1], z[2]: z[2] + z[0]}
|
||||
if elt == Qj:
|
||||
sub_list = {x : x, y : y} | {z[0] : z[0], z[1] : z[1] + 1, z[2]: z[2] + z[1] + z[0]}
|
||||
if elt == Qi^2:
|
||||
sub_list = {x : x, y : y} | {z[0] : z[0], z[1] : z[1], z[2]: z[2] + 1}
|
||||
if elt == Qi * Qj:
|
||||
sub_list = {x : x, y : y} | {z[0] : z[0] + 1, z[1] : z[1] + 1, z[2]: z[2] + z[1] + 1}
|
||||
if elt == Qj * Qi:
|
||||
sub_list = {x : x, y : y} | {z[0] : z[0] + 1, z[1] : z[1] + 1, z[2]: z[2] + z[1]}
|
||||
if elt == Qi^3:
|
||||
sub_list = {x : x, y : y} | {z[0] : z[0]+1, z[1] : z[1], z[2]: z[2] + z[0] + 1}
|
||||
if elt == Qj^3:
|
||||
sub_list = {x : x, y : y} | {z[0] : z[0], z[1] : z[1] + 1, z[2]: z[2] + z[1] + z[0] + 1}
|
||||
g = self.function
|
||||
return quaternion_function(AS, g.substitute(sub_list))
|
||||
|
||||
def trace(self):
|
||||
C = self.curve
|
||||
C_super = C.quotient
|
||||
n = C.height
|
||||
F = C.base_ring
|
||||
variable_names = 'x, y'
|
||||
for j in range(n):
|
||||
variable_names += ', z' + str(j)
|
||||
Rxyz = PolynomialRing(F, n+2, variable_names)
|
||||
x, y = Rxyz.gens()[:2]
|
||||
z = Rxyz.gens()[2:]
|
||||
RxyzQ = FractionField(Rxyz)
|
||||
result = quaternion_function(C, 0)
|
||||
G = C.group
|
||||
for a in G:
|
||||
result += self.group_action(a)
|
||||
result = result.function
|
||||
Rxy.<x, y> = PolynomialRing(F, 2)
|
||||
Qxy = FractionField(Rxy)
|
||||
result = quaternion_reduction(AS, result)
|
||||
return superelliptic_function(C_super, Qxy(result))
|
||||
|
||||
def coordinates(self, prec = 100, basis = 0):
|
||||
"Return coordinates in H^1(X, OX)."
|
||||
AS = self.curve
|
||||
if basis == 0:
|
||||
basis = [AS.holomorphic_differentials_basis(), AS.cohomology_of_structure_sheaf_basis()]
|
||||
holo_diffs = basis[0]
|
||||
coh_basis = basis[1]
|
||||
f_products = []
|
||||
for f in coh_basis:
|
||||
f_products += [[omega.serre_duality_pairing(f) for omega in holo_diffs]]
|
||||
product_of_fct_and_omegas = []
|
||||
product_of_fct_and_omegas = [omega.serre_duality_pairing(self) for omega in holo_diffs]
|
||||
|
||||
V = (F^(AS.genus())).span_of_basis([vector(a) for a in f_products])
|
||||
coh_coordinates = V.coordinates(product_of_fct_and_omegas)
|
||||
return coh_coordinates
|
||||
|
||||
def diffn(self):
|
||||
C = self.curve
|
||||
C_super = C.quotient
|
||||
n = C.height
|
||||
RxyzQ, Rxyz, x, y, z = C.fct_field
|
||||
fcts = C.functions
|
||||
f = self.function
|
||||
y_super = superelliptic_function(C_super, y)
|
||||
dy_super = y_super.diffn().form
|
||||
dz = []
|
||||
for i in range(n):
|
||||
dfct = fcts[i].diffn().form
|
||||
dz += [-dfct]
|
||||
result = f.derivative(x)
|
||||
result += f.derivative(y)*dy_super
|
||||
for i in range(n):
|
||||
result += f.derivative(z[i])*dz[i]
|
||||
return quaternion_form(C, result)
|
||||
|
||||
def valuation(self, place = 0):
|
||||
'''Return valuation at i-th place at infinity.'''
|
||||
C = self.curve
|
||||
F = C.base_ring
|
||||
Rt.<t> = LaurentSeriesRing(F)
|
||||
return Rt(self.expansion_at_infty(place = place)).valuation()
|
||||
|
|
@ -0,0 +1,33 @@
|
|||
def quaternion_reduction(AS, fct):
|
||||
'''Simplify rational function fct as a function in the function field of AS, so that z[i] appear in powers <p and only in numerator'''
|
||||
n = AS.height
|
||||
F = AS.base_ring
|
||||
RxyzQ, Rxyz, x, y, z = AS.fct_field
|
||||
p = F.characteristic()
|
||||
ff = AS.functions
|
||||
ff = [RxyzQ(F.function) for F in ff]
|
||||
fct = RxyzQ(fct)
|
||||
fct1 = numerator(fct)
|
||||
fct2 = denominator(fct)
|
||||
denom = quaternion_function(AS, fct2)
|
||||
denom_norm = prod(quaternion_function(AS, fct2).group_action(g) for g in AS.group if list(g) != n*[0])
|
||||
fct1 = Rxyz(fct1*denom_norm.function)
|
||||
fct2 = Rxyz(fct2*denom_norm.function)
|
||||
if fct2 != 1:
|
||||
return quaternion_reduction(AS, fct1)/quaternion_reduction(AS, fct2)
|
||||
|
||||
result = RxyzQ(0)
|
||||
change = 0
|
||||
for a in fct1.monomials():
|
||||
degrees_zi = [a.degree(z[i]) for i in range(n)]
|
||||
d_div = [a.degree(z[i])//p for i in range(n)]
|
||||
if d_div != n*[0]:
|
||||
change = 1
|
||||
d_rem = [a.degree(z[i])%p for i in range(n)]
|
||||
monomial = fct1.monomial_coefficient(a)*x^(a.degree(x))*y^(a.degree(y))*prod(z[i]^(d_rem[i]) for i in range(n))*prod((z[i] + ff[i])^(d_div[i]) for i in range(n))
|
||||
result += RxyzQ(monomial)
|
||||
|
||||
if change == 0:
|
||||
return RxyzQ(result)
|
||||
else:
|
||||
return quaternion_reduction(AS, RxyzQ(result))
|
||||
Loading…
Reference in New Issue