arbitrary p-gp covers, pt 1
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498e55d11b
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@ -24,3 +24,35 @@ def magma_module_decomposition(A, B, text = False, prefix="", sufix="", matrices
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if text:
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return result
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return(magma_free(result))
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def magma_is_isomorphic(A1, B1, A2, B2, text=0):
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q = parent(A1).base_ring().order()
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p = q.factor()[0][0]
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n = A1.dimensions()[0]
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A1 = str(list(A1))
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B1 = str(list(B1))
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A2 = str(list(A2))
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B2 = str(list(B2))
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A1 = A1.replace("(", "")
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A1 = A1.replace(")", "")
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A2 = A2.replace("(", "")
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A2 = A2.replace(")", "")
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B1 = B1.replace("(", "")
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B1 = B1.replace(")", "")
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B2 = B2.replace("(", "")
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B2 = B2.replace(")", "")
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result = ""
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if q != p:
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result += "F<a> := GF(" + str(q) + ");"
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result += "A1 := MatrixAlgebra<GF("+str(q) + "),"+ str(n) + "|"
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result += A1 + "," + B1
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result += ">;"
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result += "M1 := RModule(RSpace(GF("+str(q)+")," + str(n) + "), A1);"
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result += "A2 := MatrixAlgebra<GF("+str(q) + "),"+ str(n) + "|"
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result += A2 + "," + B2
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result += ">;"
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result += "M2 := RModule(RSpace(GF("+str(q)+")," + str(n) + "), A2);"
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result += "IsIsomorphic(M1, M2);"
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if text:
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return result
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return(magma_free(result))
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@ -47,6 +47,9 @@ class as_cech:
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f = self.f
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return as_cech(C, constant*omega, constant*f)
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def reduce(self):
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return as_cech(self.curve, self.omega0.reduce(), self.f.reduce())
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def coordinates(self, threshold=10, basis = 0):
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'''Find coordinates of self in the de Rham cohomology basis. Threshold is an argument passed to AS.de_rham_basis().'''
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AS = self.curve
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@ -94,11 +97,16 @@ class as_cech:
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return vector(coh_coordinates)+vector(self.coordinates(threshold=threshold, basis = basis))
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else:
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self.omega0 -= self.f.diffn()
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return vector(coh_coordinates) + vector(list(self.omega0.coordinates())+AS.genus()*[0])
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return vector(coh_coordinates) + vector(list(self.omega0.coordinates(basis=holo_diffs))+AS.genus()*[0])
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raise ValueError("Increase threshold.")
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def group_action(self, g):
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AS = self.curve
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omega = self.omega0
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f = self.f
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return as_cech(self.curve, omega.group_action(g), f.group_action(g))
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return as_cech(self.curve, omega.group_action(g), f.group_action(g))
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def trace(self):
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AS = self.curve
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return as_cech(AS, self.omega0.trace(), self.f.trace())
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@ -384,6 +384,50 @@ class as_cover:
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if (a*omega + fct.diffn()).is_regular_on_U0():
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return a*omega + fct.diffn()
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raise ValueError("Unknown.")
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def lift_to_de_rham_form(self, eta, threshold = 20):
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'''Given form eta regular on affine part find fct such that eta - d(fct) is regular in infty. (Works for one place at infty now)'''
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from itertools import product
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x_series = self.x_series
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y_series = self.y_series
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z_series = self.z_series
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dx_series = self.dx_series
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delta = self.nb_of_pts_at_infty
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p = self.characteristic
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n = self.height
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prec = self.prec
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C = self.quotient
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F = self.base_ring
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m = C.exponent
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r = C.polynomial.degree()
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RxyzQ, Rxyz, x, y, z = self.fct_field
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Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
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#Tworzymy zbiór S form z^i x^j y^k dx/y o waluacji >= waluacja z^(p-1)*dx/y
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S = [(eta, eta.expansion_at_infty())]
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pr = [list(GF(p)) for _ in range(n)]
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for i in range(0, threshold*r):
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for j in range(0, m):
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for k in product(*pr):
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ff = as_function(self, prod(z[i1]^(k[i1]) for i1 in range(n))/x^i*y^j)
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ff_exp = ff.diffn().expansion_at_infty()
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if ff_exp != 0*ff_exp:
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S += [(ff, ff_exp)]
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for i in range(0, threshold*r):
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for j in range(0, m):
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for k in product(*pr):
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ff = as_function(self, prod(z[i1]^(k[i1]) for i1 in range(n))*x^i*y^j)
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ff_exp = ff.diffn().expansion_at_infty()
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if ff_exp != 0*ff_exp:
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S += [(ff, ff_exp)]
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forms = holomorphic_combinations_mixed(S)
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if len(forms) <= self.genus():
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raise ValueError("Increase threshold!")
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result = []
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for ff in forms:
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if ff[0] != 0*self.dx:
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result += [as_cech(self, ff[0], -ff[1])]
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return result
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raise ValueError("Unknown.")
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def de_rham_basis(self, holo_basis = 0, cohomology_basis = 0, threshold = 30):
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if holo_basis == 0:
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@ -15,6 +15,9 @@ class as_form:
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def __repr__(self):
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return "(" + str(self.form)+") * dx"
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def __eq__(self, other):
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return self.expansion_at_infty() == other.expansion_at_infty()
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def expansion_at_infty(self, place = 0):
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C = self.curve
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delta = C.nb_of_pts_at_infty
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@ -80,6 +83,10 @@ class as_form:
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C = self.curve
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omega = self.form
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return as_form(C, constant*omega)
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def reduce(self):
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aux = as_reduction(self.curve, self.form)
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return as_form(self.curve, aux)
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def group_action(self, ZN_tuple):
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C = self.curve
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@ -121,7 +128,7 @@ class as_form:
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result = result.form
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Rxy.<x, y> = PolynomialRing(F, 2)
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Qxy = FractionField(Rxy)
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result = as_reduction(AS, result)
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result = as_reduction(C, result)
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return superelliptic_form(C_super, Qxy(result))
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def residue(self, place=0):
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@ -132,6 +132,10 @@ class as_function:
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g = self.function
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return as_function(C, g.substitute(sub_list))
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def reduce(self):
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aux = as_reduction(self.curve, self.function)
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return as_function(self.curve, aux)
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def trace(self):
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C = self.curve
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C_super = C.quotient
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@ -151,7 +155,7 @@ class as_function:
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result = result.function
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Rxy.<x, y> = PolynomialRing(F, 2)
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Qxy = FractionField(Rxy)
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result = as_reduction(AS, result)
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result = as_reduction(C, result)
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return superelliptic_function(C_super, Qxy(result))
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def coordinates(self, prec = 100, basis = 0):
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@ -11,8 +11,8 @@ class as_polyform:
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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 expansion_at_infty(self, place):
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return self.form.expansion_at_infty(place=place)*(self.curve.dx.expansion_at_infty(place=place))^(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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@ -27,6 +27,9 @@ class as_polyform:
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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 group_action(self, elt):
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return as_polyform(self.form.group_action(elt), self.mult)
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def as_holo_polydifferentials_basis(AS, mult, threshold = 8):
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'''Give the basis of H^0(Ω^⊗n) for n = mult.'''
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@ -23,18 +23,19 @@ def as_group_action_matrices_holo(AS, basis=0, threshold=10):
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as_cover.group_action_matrices_holo = as_group_action_matrices_holo
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def as_group_action_matrices_dR(AS, threshold=8):
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def as_group_action_matrices_dR(AS, basis = 0, threshold=8):
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n = AS.height
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generators = []
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F = AS.base_ring
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for i in range(n):
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ei = n*[0]
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ei[i] = 1
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generators += [ei]
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holo_basis = AS.holomorphic_differentials_basis(threshold = threshold)
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str_basis = AS.cohomology_of_structure_sheaf_basis(holo_basis = holo_basis, threshold = threshold)
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dr_basis = AS.de_rham_basis(holo_basis = holo_basis, cohomology_basis = str_basis, threshold=threshold)
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F = AS.base_ring
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basis = [holo_basis, str_basis, dr_basis]
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if basis == 0:
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holo_basis = AS.holomorphic_differentials_basis(threshold = threshold)
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str_basis = AS.cohomology_of_structure_sheaf_basis(holo_basis = holo_basis, threshold = threshold)
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dr_basis = AS.de_rham_basis(holo_basis = holo_basis, cohomology_basis = str_basis, threshold=threshold)
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basis = [holo_basis, str_basis, dr_basis]
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return as_group_action_matrices(F, basis[2], generators, basis = basis)
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as_cover.group_action_matrices_dR = as_group_action_matrices_dR
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@ -22,7 +22,7 @@ def holomorphic_combinations(S):
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# Sprawdzamy, jakim formom odpowiadają elementy V.
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forms = []
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for vec in V.basis():
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forma_holo = 0*C_AS.dx
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forma_holo = 0*S[0][0]
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forma_holo_power_series = Rt(0)
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for vec_wspolrzedna, elt_S in zip(vec, S):
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eta = elt_S[0]
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@ -32,6 +32,49 @@ def holomorphic_combinations(S):
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forms += [forma_holo]
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return forms
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def holomorphic_combinations_mixed(S):
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"""Given a list S of pairs (form, corresponding Laurent series at some pt), find their combinations holomorphic at that pt."""
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C_AS = S[0][0].curve
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p = C_AS.characteristic
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F = C_AS.base_ring
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prec = C_AS.prec
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Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
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RtQ = FractionField(Rt)
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minimal_valuation = min([g[1].valuation() for g in S])
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print(minimal_valuation)
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if minimal_valuation >= 0:
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return [s[0] for s in S]
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list_of_lists = [] #to będzie lista złożona z list współczynników część nieholomorficznych rozwinięcia form z S
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for eta, eta_exp in S:
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a = -minimal_valuation + eta_exp.valuation()
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list_coeffs = a*[0] + eta_exp.list() + (-minimal_valuation)*[0]
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list_coeffs = list_coeffs[:-minimal_valuation]
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list_of_lists += [list_coeffs]
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M = matrix(F, list_of_lists)
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V = M.kernel() #chcemy wyzerować części nieholomorficzne, biorąc kombinacje form z S
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# Sprawdzamy, jakim formom odpowiadają elementy V.
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forms = []
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for vec in V.basis():
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forma_holo = 0*S[0][0]
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forma_holo_power_series = Rt(0)
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res1 = 0*C_AS.dx
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res2 = 0*C_AS.x
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res = 0*C_AS.dx
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for vec_wspolrzedna, elt_S in zip(vec, S):
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eta = elt_S[0]
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if isinstance(eta, as_form):
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res += vec_wspolrzedna*eta
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res1 += vec_wspolrzedna*eta
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if isinstance(eta, as_function):
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res += vec_wspolrzedna*eta.diffn()
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res2 += vec_wspolrzedna*eta
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#eta_exp = elt_S[1]
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#forma_holo_power_series += vec_wspolrzedna*eta_exp
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forms += [(res1, res2)]
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return forms
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def holomorphic_combinations_fcts(S, pole_order):
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'''given a set S of (form, corresponding Laurent series at some pt), find their combinations holomorphic at that pt'''
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C_AS = S[0][0].curve
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@ -10,7 +10,7 @@ def naive_hensel(fct, F, start = 1, prec=10):
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#while fct not in RptW:
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# print(fct)
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# fct *= W
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alpha = (fct.derivative())(W = start)
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alpha = (fct.derivative())(W = RtQ(start))
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w0 = Rt(start)
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i = 1
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while(i < prec):
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58
elementary_covers/as_auxilliary.sage
Normal file
58
elementary_covers/as_auxilliary.sage
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@ -0,0 +1,58 @@
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def magma_module_decomposition(A, B, text = False, prefix="", sufix="", matrices=True):
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"""Find decomposition of Z/p^2-module given by matrices A, B into indecomposables using magma.
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If text = True, print the command for Magma. Else - return the output of Magma free."""
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q = parent(A).base_ring().order()
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p = q.factor()[0][0]
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n = A.dimensions()[0]
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A = str(list(A))
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B = str(list(B))
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A = A.replace("(", "")
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A = A.replace(")", "")
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B = B.replace("(", "")
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B = B.replace(")", "")
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result = prefix
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if q != p:
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result += "F<a> := GF(" + str(q) + ");"
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result += "A := MatrixAlgebra<GF("+str(q) + "),"+ str(n) + "|"
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result += A + "," + B
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result += ">;"
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result += "M := RModule(RSpace(GF("+str(q)+")," + str(n) + "), A);"
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result += "L := IndecomposableSummands(M); L;"
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if matrices:
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result += "for i in [1 .. #L] do print(Generators(Action(L[i]))); end for;"
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result += sufix
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if text:
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return result
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return(magma_free(result))
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def magma_is_isomorphic(A1, B1, A2, B2, text=0):
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q = parent(A1).base_ring().order()
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p = q.factor()[0][0]
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n = A1.dimensions()[0]
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A1 = str(list(A1))
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B1 = str(list(B1))
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A2 = str(list(A2))
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B2 = str(list(B2))
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A1 = A1.replace("(", "")
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A1 = A1.replace(")", "")
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A2 = A2.replace("(", "")
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A2 = A2.replace(")", "")
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B1 = B1.replace("(", "")
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B1 = B1.replace(")", "")
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B2 = B2.replace("(", "")
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B2 = B2.replace(")", "")
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result = ""
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if q != p:
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result += "F<a> := GF(" + str(q) + ");"
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result += "A1 := MatrixAlgebra<GF("+str(q) + "),"+ str(n) + "|"
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result += A1 + "," + B1
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result += ">;"
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result += "M1 := RModule(RSpace(GF("+str(q)+")," + str(n) + "), A1);"
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result += "A2 := MatrixAlgebra<GF("+str(q) + "),"+ str(n) + "|"
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result += A2 + "," + B2
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result += ">;"
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result += "M2 := RModule(RSpace(GF("+str(q)+")," + str(n) + "), A2);"
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result += "IsIsomorphic(M1, M2);"
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if text:
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return result
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return(magma_free(result))
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112
elementary_covers/as_cech_class.sage
Normal file
112
elementary_covers/as_cech_class.sage
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@ -0,0 +1,112 @@
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class as_cech:
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def __init__(self, C, omega, f):
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self.curve = C
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n = C.height
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F = C.base_ring
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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.omega0 = omega
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self.f = f
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self.omega8 = self.omega0 - self.f.diffn()
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#if self.omega0.form not in Rxyz or self.omega8.valuation() < 0:
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# raise ValueError('cech cocycle not regular')
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def __repr__(self):
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return "( " + str(self.omega0)+", " + str(self.f) + " )"
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def __add__(self, other):
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C = self.curve
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omega = self.omega0
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f = self.f
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omega1 = other.omega0
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f1 = other.f
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return as_cech(C, omega + omega1, f+f1)
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def __sub__(self, other):
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C = self.curve
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omega = self.omega0
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f = self.f
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omega1 = other.omega0
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f1 = other.f
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return as_cech(C, omega - omega1, f - f1)
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def __neg__(self):
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C = self.curve
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omega = self.omega0
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f = self.f
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return as_cech(C, -omega, -f)
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def __rmul__(self, constant):
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C = self.curve
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omega = self.omega0
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f = self.f
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return as_cech(C, constant*omega, constant*f)
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||||
|
||||
def reduce(self):
|
||||
return as_cech(self.curve, self.omega0.reduce(), self.f.reduce())
|
||||
|
||||
def coordinates(self, threshold=10, basis = 0):
|
||||
'''Find coordinates of self in the de Rham cohomology basis. Threshold is an argument passed to AS.de_rham_basis().'''
|
||||
AS = self.curve
|
||||
C = AS.quotient
|
||||
m = C.exponent
|
||||
r = C.polynomial.degree()
|
||||
n = AS.height
|
||||
p = AS.characteristic
|
||||
RxyzQ, Rxyz, x, y, z = AS.fct_field
|
||||
if basis == 0:
|
||||
basis = [AS.holomorphic_differentials_basis(), AS.cohomology_of_structure_sheaf_basis(), AS.de_rham_basis(threshold=threshold)]
|
||||
holo_diffs = basis[0]
|
||||
coh_basis = basis[1]
|
||||
dR = basis[2]
|
||||
F = AS.base_ring
|
||||
f_products = []
|
||||
for f in coh_basis:
|
||||
f_products += [[omega.serre_duality_pairing(f) for omega in holo_diffs]]
|
||||
product_of_fct_and_omegas = []
|
||||
fct = self.f
|
||||
product_of_fct_and_omegas = [omega.serre_duality_pairing(fct) 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) #coeficients of self in the basis elts coming from cohomology of OX
|
||||
for i in range(AS.genus()):
|
||||
self -= coh_coordinates[i]*dR[i+AS.genus()]
|
||||
coh_coordinates = AS.genus()*[0] + list(coh_coordinates)
|
||||
if self.f.function not in Rxyz:
|
||||
#We remove now from f the summands which are obviously regular at infty
|
||||
pr = [list(GF(p)) for _ in range(n)]
|
||||
S = []
|
||||
from itertools import product
|
||||
for i in range(0, threshold*r):
|
||||
for j in range(0, m):
|
||||
for k in product(*pr):
|
||||
g = (AS.x)^i*prod((AS.z[i1])^(k[i1]) for i1 in range(n))*(AS.y)^j
|
||||
S += [(g, g.expansion_at_infty())]
|
||||
S += [(self.f, self.f.expansion_at_infty())]
|
||||
fcts = holomorphic_combinations_fcts(S, 0)
|
||||
for g in fcts:
|
||||
if g.function not in Rxyz:
|
||||
for a in F:
|
||||
if (self.f.function - a*g.function in Rxyz):
|
||||
self.f.function = self.f.function - a*g.function
|
||||
return vector(coh_coordinates)+vector(self.coordinates(threshold=threshold, basis = basis))
|
||||
else:
|
||||
self.omega0 -= self.f.diffn()
|
||||
return vector(coh_coordinates) + vector(list(self.omega0.coordinates(basis=holo_diffs))+AS.genus()*[0])
|
||||
|
||||
raise ValueError("Increase threshold.")
|
||||
|
||||
def group_action(self, g):
|
||||
AS = self.curve
|
||||
omega = self.omega0
|
||||
f = self.f
|
||||
return as_cech(self.curve, omega.group_action(g), f.group_action(g))
|
||||
|
||||
def trace(self):
|
||||
AS = self.curve
|
||||
return as_cech(AS, self.omega0.trace(), self.f.trace())
|
||||
444
elementary_covers/as_cover_class.sage
Normal file
444
elementary_covers/as_cover_class.sage
Normal file
@ -0,0 +1,444 @@
|
||||
class as_cover:
|
||||
def __init__(self, C, list_of_fcts, branch_points = [], prec = 10):
|
||||
self.quotient = C
|
||||
self.functions = list_of_fcts
|
||||
self.height = len(list_of_fcts)
|
||||
F = C.base_ring
|
||||
self.base_ring = F
|
||||
p = C.characteristic
|
||||
self.characteristic = p
|
||||
self.prec = prec
|
||||
#group acting
|
||||
n = self.height
|
||||
from itertools import product
|
||||
pr = [list(GF(p)) for _ in range(n)]
|
||||
group = []
|
||||
for a in product(*pr):
|
||||
group += [a]
|
||||
self.group = group
|
||||
#########
|
||||
f = C.polynomial
|
||||
m = C.exponent
|
||||
r = f.degree()
|
||||
delta = GCD(m, r)
|
||||
self.nb_of_pts_at_infty = delta
|
||||
self.branch_points = list(range(delta)) + branch_points
|
||||
Rxy.<x, y> = PolynomialRing(F, 2)
|
||||
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
|
||||
|
||||
all_x_series = {}
|
||||
all_y_series = {}
|
||||
all_z_series = {}
|
||||
all_dx_series = {}
|
||||
all_jumps = {}
|
||||
|
||||
for pt in self.branch_points:
|
||||
x_series = superelliptic_function(C, x).expansion(pt=pt, prec=prec)
|
||||
y_series = superelliptic_function(C, y).expansion(pt=pt, prec=prec)
|
||||
z_series = []
|
||||
jumps = []
|
||||
n = len(list_of_fcts)
|
||||
list_of_power_series = [g.expansion(pt=pt, prec=prec) for g in list_of_fcts]
|
||||
for j in range(n):
|
||||
power_series = list_of_power_series[j]
|
||||
jump, correction, t_old, z = artin_schreier_transform(power_series, prec = prec)
|
||||
x_series = x_series(t = t_old)
|
||||
y_series = y_series(t = t_old)
|
||||
z_series = [zi(t = t_old) for zi in z_series]
|
||||
z_series += [z]
|
||||
jumps += [jump]
|
||||
list_of_power_series = [g(t = t_old) for g in list_of_power_series]
|
||||
|
||||
all_jumps[pt] = jumps
|
||||
all_x_series[pt] = x_series
|
||||
all_y_series[pt] = y_series
|
||||
all_z_series[pt] = z_series
|
||||
all_dx_series[pt] = x_series.derivative()
|
||||
self.jumps = all_jumps
|
||||
self.x_series = all_x_series
|
||||
self.y_series = all_y_series
|
||||
self.z_series = all_z_series
|
||||
self.dx_series = all_dx_series
|
||||
##############
|
||||
#Function field
|
||||
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.fct_field = (RxyzQ, Rxyz, x, y, z)
|
||||
self.x = as_function(self, x)
|
||||
self.y = as_function(self, y)
|
||||
self.z = [as_function(self, z[j]) for j in range(n)]
|
||||
self.dx = as_form(self, 1)
|
||||
self.one = as_function(self, 1)
|
||||
|
||||
|
||||
def __repr__(self):
|
||||
n = self.height
|
||||
p = self.characteristic
|
||||
if n==1:
|
||||
return "(Z/p)-cover of " + str(self.quotient)+" with the equation:\n z^" + str(p) + " - z = " + str(self.functions[0])
|
||||
|
||||
result = "(Z/p)^"+str(self.height)+ "-cover of " + str(self.quotient)+" with the equations:\n"
|
||||
for i in range(n):
|
||||
result += 'z' + str(i) + "^" + str(p) + " - z" + str(i) + " = " + str(self.functions[i]) + "\n"
|
||||
return result
|
||||
|
||||
def genus(self):
|
||||
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 = as_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 = as_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 = as_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 = as_function(self, x)
|
||||
z = [as_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 = as_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 = as_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 lift_to_de_rham_form(self, eta, threshold = 20):
|
||||
'''Given form eta regular on affine part find fct 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 = [(eta, eta.expansion_at_infty())]
|
||||
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):
|
||||
ff = as_function(self, prod(z[i1]^(k[i1]) for i1 in range(n))/x^i*y^j)
|
||||
ff_exp = ff.diffn().expansion_at_infty()
|
||||
if ff_exp != 0*ff_exp:
|
||||
S += [(ff, ff_exp)]
|
||||
for i in range(0, threshold*r):
|
||||
for j in range(0, m):
|
||||
for k in product(*pr):
|
||||
ff = as_function(self, prod(z[i1]^(k[i1]) for i1 in range(n))*x^i*y^j)
|
||||
ff_exp = ff.diffn().expansion_at_infty()
|
||||
if ff_exp != 0*ff_exp:
|
||||
S += [(ff, ff_exp)]
|
||||
forms = holomorphic_combinations_mixed(S)
|
||||
if len(forms) <= self.genus():
|
||||
raise ValueError("Increase threshold!")
|
||||
result = []
|
||||
for ff in forms:
|
||||
if ff[0] != 0*self.dx:
|
||||
result += [as_cech(self, ff[0], -ff[1])]
|
||||
return result
|
||||
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 += [as_cech(self, omega, as_function(self, 0))]
|
||||
for f in cohomology_basis:
|
||||
omega = self.lift_to_de_rham(f, threshold = threshold)
|
||||
result += [as_cech(self, omega, f)]
|
||||
return result
|
||||
|
||||
205
elementary_covers/as_form_class.sage
Normal file
205
elementary_covers/as_form_class.sage
Normal file
@ -0,0 +1,205 @@
|
||||
class as_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 __eq__(self, other):
|
||||
return self.expansion_at_infty() == other.expansion_at_infty()
|
||||
|
||||
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 as_form(C, g1 + g2)
|
||||
|
||||
def __sub__(self, other):
|
||||
C = self.curve
|
||||
g1 = self.form
|
||||
g2 = other.form
|
||||
return as_form(C, g1 - g2)
|
||||
|
||||
def __neg__(self):
|
||||
C = self.curve
|
||||
g = self.form
|
||||
return as_form(C, -g)
|
||||
|
||||
def __rmul__(self, constant):
|
||||
C = self.curve
|
||||
omega = self.form
|
||||
return as_form(C, constant*omega)
|
||||
|
||||
def reduce(self):
|
||||
aux = as_reduction(self.curve, self.form)
|
||||
return as_form(self.curve, aux)
|
||||
|
||||
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 as_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 = as_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 = as_reduction(C, 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 as_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 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 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
|
||||
202
elementary_covers/as_function_class.sage
Normal file
202
elementary_covers/as_function_class.sage
Normal file
@ -0,0 +1,202 @@
|
||||
class as_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 = as_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 = as_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 as_function(C, g1 + g2)
|
||||
|
||||
def __sub__(self, other):
|
||||
C = self.curve
|
||||
g1 = self.function
|
||||
g2 = other.function
|
||||
return as_function(C, g1 - g2)
|
||||
|
||||
def __rmul__(self, constant):
|
||||
C = self.curve
|
||||
g = self.function
|
||||
return as_function(C, constant*g)
|
||||
|
||||
def __neg__(self):
|
||||
C = self.curve
|
||||
g = self.function
|
||||
return as_function(C, -g)
|
||||
|
||||
def __mul__(self, other):
|
||||
if isinstance(other, as_function):
|
||||
C = self.curve
|
||||
g1 = self.function
|
||||
g2 = other.function
|
||||
return as_function(C, g1*g2)
|
||||
if isinstance(other, as_form):
|
||||
C = self.curve
|
||||
g1 = self.function
|
||||
g2 = other.form
|
||||
return as_form(C, g1*g2)
|
||||
|
||||
def __truediv__(self, other):
|
||||
C = self.curve
|
||||
g1 = self.function
|
||||
g2 = other.function
|
||||
return as_function(C, g1/g2)
|
||||
|
||||
def __pow__(self, exponent):
|
||||
C = self.curve
|
||||
g1 = self.function
|
||||
return as_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, ZN_tuple):
|
||||
C = self.curve
|
||||
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)
|
||||
sub_list = {x : x, y : y} | {z[j] : z[j]+ZN_tuple[j] for j in range(n)}
|
||||
g = self.function
|
||||
return as_function(C, g.substitute(sub_list))
|
||||
|
||||
def reduce(self):
|
||||
aux = as_reduction(self.curve, self.function)
|
||||
return as_function(self.curve, aux)
|
||||
|
||||
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 = as_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 = as_reduction(C, 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 as_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()
|
||||
221
elementary_covers/as_polyforms.sage
Normal file
221
elementary_covers/as_polyforms.sage
Normal file
@ -0,0 +1,221 @@
|
||||
class as_polyform:
|
||||
def __init__(self, form, mult):
|
||||
'''Elements of H^0(Ω^⊗n). Usage: mult is n, form should be as_function.'''
|
||||
self.form = form
|
||||
self.curve = form.curve
|
||||
self.mult = mult
|
||||
|
||||
def __add__(self, other):
|
||||
return as_polyform(self.form + other.form, self.mult)
|
||||
|
||||
def __repr__(self):
|
||||
return '(' + str(self.form) + ') dx⊗' + str(self.mult)
|
||||
|
||||
def expansion_at_infty(self, place):
|
||||
return self.form.expansion_at_infty(place=place)*(self.curve.dx.expansion_at_infty(place=place))^(self.mult)
|
||||
|
||||
def coordinates(self, basis = 0):
|
||||
"""Find coordinates of the given holomorphic form self in terms of the basis forms in a list holo."""
|
||||
AS = self.curve
|
||||
if basis == 0:
|
||||
basis = AS.holo_polydifferentials_basis(mult = self.mult)
|
||||
RxyzQ, Rxyz, x, y, z = AS.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.function) for omega in basis])
|
||||
basis = [denom*omega.form.function for omega in basis]
|
||||
self_with_no_denominator = denom*self.form.function
|
||||
return linear_representation_polynomials(Rxyz(self_with_no_denominator), [Rxyz(omega) for omega in basis])
|
||||
|
||||
def group_action(self, elt):
|
||||
return as_polyform(self.form.group_action(elt), self.mult)
|
||||
|
||||
|
||||
def as_holo_polydifferentials_basis(AS, mult, threshold = 8):
|
||||
'''Give the basis of H^0(Ω^⊗n) for n = mult.'''
|
||||
v = AS.dx.valuation()
|
||||
result = AS.at_most_poles(mult*v, threshold=threshold)
|
||||
result = [as_polyform(omega, mult) for omega in result]
|
||||
if mult == 1 and len(result) < AS.genus():
|
||||
raise ValueError('Increase threshold, not all forms found.')
|
||||
if mult > 1 and len(result) < (2*mult - 1)*(AS.genus() - 1):
|
||||
raise ValueError('Increase threshold, not all forms found.')
|
||||
return result
|
||||
|
||||
as_cover.holo_polydifferentials_basis = as_holo_polydifferentials_basis
|
||||
|
||||
def as_symmetric_power_basis(AS, n, threshold = 8):
|
||||
'''Give the basis of H^0(Ω)^⊙n for n = mult.'''
|
||||
g = AS.genus()
|
||||
B0 = AS.holomorphic_differentials_basis(threshold=threshold)
|
||||
from itertools import product
|
||||
indices = [list(range(g)) for i in range(n)]
|
||||
indices_nonrepeating = []
|
||||
for i in product(*indices):
|
||||
if non_decreasing(i):
|
||||
indices_nonrepeating += [i]
|
||||
result = []
|
||||
for i in indices_nonrepeating:
|
||||
tensor_form = [B0[i[j]] for j in range(n)]
|
||||
tensor_form = [1] + tensor_form
|
||||
result += [tensor_form]
|
||||
result = [as_symmetric_product_forms([a]) for a in result]
|
||||
return result
|
||||
|
||||
def as_canonical_ideal(AS, n, threshold=8):
|
||||
'''Return the n-th homogeneous part of the canonical ideal.'''
|
||||
B0 = AS.holomorphic_differentials_basis(threshold=threshold)
|
||||
F = AS.base_ring
|
||||
g = AS.genus()
|
||||
RxyzQ, Rxyz, x, y, z = AS.fct_field
|
||||
from itertools import product
|
||||
B1 = as_symmetric_power_basis(AS, n, threshold = 8)
|
||||
B2 = AS.holo_polydifferentials_basis(n, threshold = threshold)
|
||||
g = AS.genus()
|
||||
r = len(B2)
|
||||
M = matrix(F, len(B1), r)
|
||||
for i in range(0, len(B1)):
|
||||
c = B1[i].multiply()
|
||||
#return M, c.coordinates(basis=B2)
|
||||
M[i, :] = vector(c.coordinates(basis=B2))
|
||||
K = M.kernel().basis()
|
||||
result = []
|
||||
for v in K:
|
||||
kernel_vector = 0*B1[0]
|
||||
for i in range(len(v)):
|
||||
kernel_vector += v[i]*B1[i]
|
||||
result += [kernel_vector]
|
||||
return result
|
||||
|
||||
as_cover.canonical_ideal = as_canonical_ideal
|
||||
|
||||
def as_canonical_ideal_polynomials(AS, n, threshold=8):
|
||||
'''Return the polynomials defining n-th homogeneous part of the canonical ideal.'''
|
||||
return [a.polynomial() for a in AS.canonical_ideal(n, threshold=threshold)]
|
||||
|
||||
as_cover.canonical_ideal_polynomials = as_canonical_ideal_polynomials
|
||||
|
||||
class as_symmetric_product_forms:
|
||||
def __init__(self, forms_and_coeffs):
|
||||
'''Elements of forms_and_coeffs are of the form [coeff, form_1, ..., form_n]'''
|
||||
self.n = len(forms_and_coeffs[0]) - 1
|
||||
forms_and_coeffs1 = []
|
||||
for atuple in forms_and_coeffs:
|
||||
if atuple[0] != 0 and atuple[1:] not in [a[1:] for a in forms_and_coeffs1]:
|
||||
forms_and_coeffs1 += [atuple]
|
||||
elif atuple[1:] in [a[1:] for a in forms_and_coeffs1]:
|
||||
i = [a[1:] for a in forms_and_coeffs1].index(atuple[1:])
|
||||
forms_and_coeffs1[i][0] += atuple[0]
|
||||
if len(forms_and_coeffs1) == 0:
|
||||
forms_and_coeffs1 = [[0] + forms_and_coeffs[0][1:]]
|
||||
self.tuples = forms_and_coeffs1
|
||||
self.curve = forms_and_coeffs1[0][1].curve
|
||||
|
||||
def coordinates(self, basis = 0):
|
||||
AS = self.curve
|
||||
g = AS.genus()
|
||||
n = self.n
|
||||
F = AS.base_ring
|
||||
if basis == 0:
|
||||
basis = AS.holomorphic_differentials_basis()
|
||||
from itertools import product
|
||||
indices = [list(range(g)) for i in range(self.n)]
|
||||
result = {i : 0 for i in product(*indices)}
|
||||
for atuple in self.tuples:
|
||||
coors = [omega.coordinates(basis = basis) for omega in atuple[1:]]
|
||||
for i in product(*indices):
|
||||
aux_product = 1
|
||||
for j in range(n):
|
||||
aux_product *= coors[j][i[j]]
|
||||
result[i] += atuple[0]*aux_product
|
||||
return result
|
||||
|
||||
def __repr__(self):
|
||||
result = ''
|
||||
for atuple in self.tuples:
|
||||
if atuple[0] !=0:
|
||||
result += str(atuple[0]) + ' * '
|
||||
for j in range(1, self.n+1):
|
||||
result += "(" + str(atuple[j]) + ")"
|
||||
if j != self.n:
|
||||
result + '⊗'
|
||||
result += ' + '
|
||||
return result
|
||||
|
||||
def __add__(self, other):
|
||||
return as_symmetric_product_forms(self.tuples + other.tuples)
|
||||
|
||||
def __rmul__(self, other):
|
||||
AS = self.curve
|
||||
F = AS.base_ring
|
||||
if isinstance(other, int) or other in ZZ or other in F:
|
||||
aux_tuples = []
|
||||
for atuple in self.tuples:
|
||||
atuple1 = [other * atuple[0]] + atuple[1:]
|
||||
aux_tuples += [atuple1]
|
||||
return as_symmetric_product_forms(aux_tuples)
|
||||
|
||||
def multiply(self):
|
||||
'''The map H^0(Ω)^⊙n ---> H^0(Ω^⊗n)'''
|
||||
n = self.n
|
||||
AS = self.curve
|
||||
RxyzQ, Rxyz, x, y, z = AS.fct_field
|
||||
result = as_polyform(0*AS.x, n)
|
||||
for atuple in self.tuples:
|
||||
aux_product = Rxyz(1)
|
||||
for fct in atuple[1:]:
|
||||
aux_product = aux_product * fct.form
|
||||
aux_product = as_function(AS, aux_product)
|
||||
aux_product = as_reduction(AS, aux_product)
|
||||
aux_product = as_function(AS, aux_product)
|
||||
result += as_polyform(atuple[0]*aux_product, n)
|
||||
return result
|
||||
|
||||
def polynomial(self):
|
||||
'''Return the associated polynomial.'''
|
||||
AS = self.curve
|
||||
F = AS.base_ring
|
||||
g = AS.genus()
|
||||
M = self.coordinates()
|
||||
n = self.n
|
||||
Rg = PolynomialRing(F, 'X', g)
|
||||
X = Rg.gens()
|
||||
from itertools import product
|
||||
indices = [list(range(g)) for i in range(self.n)]
|
||||
result = Rg(0)
|
||||
for i in product(*indices):
|
||||
aux_product = Rg(1)
|
||||
for j in range(n):
|
||||
aux_product *= X[i[j]]
|
||||
result += M[i] * aux_product
|
||||
return result
|
||||
|
||||
def group_action(self, elt):
|
||||
p = self.curve.base_ring.characteristic()
|
||||
n = self.curve.height
|
||||
aux_tuples = []
|
||||
for atuple in self.tuples:
|
||||
aux_tuple = [atuple[0]] + [a.group_action(elt) for a in atuple[1:]]
|
||||
aux_tuples += [aux_tuple]
|
||||
return as_symmetric_product_forms(aux_tuples)
|
||||
|
||||
def non_decreasing(L):
|
||||
return all(x<=y for x, y in zip(L, L[1:]))
|
||||
|
||||
def as_matrices_group_action_canonical_ideal(AS, mult, threshold = 8):
|
||||
'''Return the group action matrices for the n-th homogeneous part of the canonical ideal.'''
|
||||
K = as_canonical_ideal(AS, mult, threshold = threshold)
|
||||
n = AS.height
|
||||
F = AS.base_ring
|
||||
K_polynomials = [a.polynomial() for a in K]
|
||||
r = len(K)
|
||||
matrices = []
|
||||
for i in range(n):
|
||||
M = matrix(F, r, r)
|
||||
K_group_action_polynomials = [a.group_action([j == i for j in range(n)]).polynomial() for a in K]
|
||||
for i in range(r):
|
||||
M[i, :] = vector(linear_representation_polynomials(K_group_action_polynomials[i], K_polynomials))
|
||||
matrices += [M]
|
||||
return matrices
|
||||
|
||||
as_cover.group_action_canonical_ideal = as_matrices_group_action_canonical_ideal
|
||||
33
elementary_covers/as_reduction.sage
Normal file
33
elementary_covers/as_reduction.sage
Normal file
@ -0,0 +1,33 @@
|
||||
def as_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 = as_function(AS, fct2)
|
||||
denom_norm = prod(as_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 as_reduction(AS, fct1)/as_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 as_reduction(AS, RxyzQ(result))
|
||||
42
elementary_covers/as_transform.sage
Normal file
42
elementary_covers/as_transform.sage
Normal file
@ -0,0 +1,42 @@
|
||||
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:
|
||||
if jump != 0:
|
||||
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])
|
||||
if jump != 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)
|
||||
if jump == 0:
|
||||
aux = t^p - t
|
||||
z = new_reverse(aux, prec = prec)
|
||||
z = z(t = power_series)
|
||||
z = z + correction
|
||||
return(0, correction, t, z)
|
||||
14
elementary_covers/combination_components.sage
Normal file
14
elementary_covers/combination_components.sage
Normal file
@ -0,0 +1,14 @@
|
||||
def combination_components(omega, zmag, w):
|
||||
'''Given a form omega on AS cover and normal basis element zmag, find the decomposition
|
||||
sum_g g(zmag) omega_g and return sum_g g(w) omega_g.'''
|
||||
AS = omega.curve
|
||||
p = AS.characteristic
|
||||
group_elts = [(j1, j2) for j1 in range(p) for j2 in range(p)]
|
||||
zvee = dual_elt(AS, zmag)
|
||||
result = as_form(AS, 0)
|
||||
for g in AS.group:
|
||||
omegag = ith_magical_component(omega, zvee, g)
|
||||
aux_fct1 = w.group_action(g).function
|
||||
aux_fct2 = omegag.form
|
||||
result += as_form(AS, aux_fct1*aux_fct2)
|
||||
return result
|
||||
26
elementary_covers/dual_element.sage
Normal file
26
elementary_covers/dual_element.sage
Normal file
@ -0,0 +1,26 @@
|
||||
def dual_elt(AS, zmag):
|
||||
'''Find the trace dual of a given elt zmag in the function field of an Artin-Schreier cover AS.'''
|
||||
p = AS.characteristic
|
||||
n = AS.height
|
||||
G = AS.group
|
||||
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)
|
||||
M = matrix(RxyzQ, p^n, p^n)
|
||||
for i in range(p^n):
|
||||
for j in range(p^n):
|
||||
elt = (zmag.group_action(G[i])*zmag.group_action(G[j])).trace().function
|
||||
elt = Rxyz(elt.numerator())/Rxyz(elt.denominator())
|
||||
M[i, j] = RxyzQ(elt)
|
||||
main_det = M.determinant()
|
||||
zvee = as_function(AS, 0)
|
||||
for i in range(p^n):
|
||||
Mprim = matrix(RxyzQ, M)
|
||||
Mprim[:, i] = vector([(j == 0) for j in range(p^n)])
|
||||
fi = Mprim.determinant()/main_det
|
||||
zvee += as_function(AS, fi*(zmag.group_action(G[i]).function))
|
||||
return zvee
|
||||
53
elementary_covers/group_action_matrices.sage
Normal file
53
elementary_covers/group_action_matrices.sage
Normal file
@ -0,0 +1,53 @@
|
||||
def as_group_action_matrices(F, space, list_of_group_elements, basis):
|
||||
n = len(list_of_group_elements)
|
||||
d = len(space)
|
||||
A = [matrix(F, d, d) for i in range(n)]
|
||||
for i, g in enumerate(list_of_group_elements):
|
||||
for j, omega in enumerate(space):
|
||||
omega1 = omega.group_action(g)
|
||||
v1 = omega1.coordinates(basis = basis)
|
||||
A[i][:, j] = vector(v1)
|
||||
return A
|
||||
|
||||
def as_group_action_matrices_holo(AS, basis=0, threshold=10):
|
||||
n = AS.height
|
||||
generators = []
|
||||
for i in range(n):
|
||||
ei = n*[0]
|
||||
ei[i] = 1
|
||||
generators += [ei]
|
||||
if basis == 0:
|
||||
basis = AS.holomorphic_differentials_basis(threshold=threshold)
|
||||
F = AS.base_ring
|
||||
return as_group_action_matrices(F, basis, generators, basis = basis)
|
||||
|
||||
as_cover.group_action_matrices_holo = as_group_action_matrices_holo
|
||||
|
||||
def as_group_action_matrices_dR(AS, basis = 0, threshold=8):
|
||||
n = AS.height
|
||||
generators = []
|
||||
F = AS.base_ring
|
||||
for i in range(n):
|
||||
ei = n*[0]
|
||||
ei[i] = 1
|
||||
generators += [ei]
|
||||
if basis == 0:
|
||||
holo_basis = AS.holomorphic_differentials_basis(threshold = threshold)
|
||||
str_basis = AS.cohomology_of_structure_sheaf_basis(holo_basis = holo_basis, threshold = threshold)
|
||||
dr_basis = AS.de_rham_basis(holo_basis = holo_basis, cohomology_basis = str_basis, threshold=threshold)
|
||||
basis = [holo_basis, str_basis, dr_basis]
|
||||
return as_group_action_matrices(F, basis[2], generators, basis = basis)
|
||||
|
||||
as_cover.group_action_matrices_dR = as_group_action_matrices_dR
|
||||
|
||||
def as_group_action_matrices_log_holo(AS):
|
||||
n = AS.height
|
||||
generators = []
|
||||
for i in range(n):
|
||||
ei = n*[0]
|
||||
ei[i] = 1
|
||||
generators += [ei]
|
||||
F = AS.base_ring
|
||||
return as_group_action_matrices(F, AS.at_most_poles_forms(1), generators, basis = AS.at_most_poles_forms(1))
|
||||
|
||||
as_cover.group_action_matrices_log_holo = as_group_action_matrices_log_holo
|
||||
149
elementary_covers/holomorphic_combinations.sage
Normal file
149
elementary_covers/holomorphic_combinations.sage
Normal file
@ -0,0 +1,149 @@
|
||||
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 = 0*S[0][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
|
||||
|
||||
def holomorphic_combinations_mixed(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])
|
||||
print(minimal_valuation)
|
||||
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 = 0*S[0][0]
|
||||
forma_holo_power_series = Rt(0)
|
||||
res1 = 0*C_AS.dx
|
||||
res2 = 0*C_AS.x
|
||||
res = 0*C_AS.dx
|
||||
for vec_wspolrzedna, elt_S in zip(vec, S):
|
||||
eta = elt_S[0]
|
||||
if isinstance(eta, as_form):
|
||||
res += vec_wspolrzedna*eta
|
||||
res1 += vec_wspolrzedna*eta
|
||||
if isinstance(eta, as_function):
|
||||
res += vec_wspolrzedna*eta.diffn()
|
||||
res2 += vec_wspolrzedna*eta
|
||||
#eta_exp = elt_S[1]
|
||||
#forma_holo_power_series += vec_wspolrzedna*eta_exp
|
||||
forms += [(res1, res2)]
|
||||
return 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()
|
||||
if eta_exp !=0:
|
||||
list_coeffs = a*[0] + Rt(eta_exp).list() + (-minimal_valuation)*[0]
|
||||
list_coeffs = list_coeffs[:-minimal_valuation - pole_order]
|
||||
else:
|
||||
list_coeffs = (-minimal_valuation - pole_order)*[0]
|
||||
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 = 0*C_AS.x
|
||||
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
|
||||
|
||||
def holomorphic_combinations_forms(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 = 0*C_AS.dx
|
||||
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
|
||||
6
elementary_covers/ith_magical_component.sage
Normal file
6
elementary_covers/ith_magical_component.sage
Normal file
@ -0,0 +1,6 @@
|
||||
def ith_magical_component(omega, zvee, g):
|
||||
'''Given a form omega on AS cover, element g of group AS.group and normal basis element zmag, find the decomposition
|
||||
sum_g g(zmag) omega_g and return omega_g.'''
|
||||
z_vee_g = zvee.group_action(g)
|
||||
new_form = z_vee_g*omega
|
||||
return new_form.trace()
|
||||
26
elementary_covers/tests/as_cover_test.sage
Normal file
26
elementary_covers/tests/as_cover_test.sage
Normal file
@ -0,0 +1,26 @@
|
||||
p = 5
|
||||
m = 2
|
||||
F = GF(p^2, 'a')
|
||||
a = F.gens()[0]
|
||||
Rx.<x> = PolynomialRing(F)
|
||||
f = x^3 + x^2 + 1
|
||||
C_super = superelliptic(f, m)
|
||||
Rxy.<x, y> = PolynomialRing(GF(p), 2)
|
||||
fArS1 = superelliptic_function(C_super, y*x)
|
||||
fArS2 = superelliptic_function(C_super, y*x^2)
|
||||
fArS3 = superelliptic_function(C_super, y + x)
|
||||
AS1 = as_cover(C_super, [fArS1, fArS2, fArS3], prec=150)
|
||||
AS2 = as_cover(C_super, [fArS2, fArS3, fArS1], prec=150)
|
||||
print(AS1.genus() == AS2.genus())
|
||||
##################
|
||||
p = 5
|
||||
m = 2
|
||||
Rx.<x> = PolynomialRing(GF(p))
|
||||
f = x^3 + x^2 + 1
|
||||
C_super = superelliptic(f, m)
|
||||
Rxy.<x, y> = PolynomialRing(GF(p), 2)
|
||||
fArS1 = superelliptic_function(C_super, y*x)
|
||||
fArS2 = superelliptic_function(C_super, y*x^2)
|
||||
AS1 = as_cover(C_super, [fArS1, fArS2], prec=1000)
|
||||
omega = as_form(AS1, 1/y)
|
||||
print(omega.expansion_at_infty().valuation()==AS1.exponent_of_different())
|
||||
27
elementary_covers/tests/as_polyforms_test.sage
Normal file
27
elementary_covers/tests/as_polyforms_test.sage
Normal file
@ -0,0 +1,27 @@
|
||||
p = 2
|
||||
F = GF(p)
|
||||
Rx.<x> = PolynomialRing(F)
|
||||
C = superelliptic(x, 1)
|
||||
AS = as_cover(C, [C.x^3, C.x^5], prec = 200)
|
||||
B = AS.holo_polydifferentials_basis(3)
|
||||
print(len(B) == (2 * 3 - 1) * (AS.genus() - 1)) #is the dimension as predicted by Riemann--Roch?
|
||||
I2 = AS.canonical_ideal_polynomials(2)
|
||||
R = I2[0].parent() #ring, to which the polynomials belong
|
||||
J2 = R.ideal(I2) #ideal defined by the set I
|
||||
print(J2.is_prime(), len(R.gens()) - J2.dimension() - 1)
|
||||
I3 = AS.canonical_ideal_polynomials(3)
|
||||
J3 = R.ideal(I2 + I3)
|
||||
print(J3.is_prime(), len(R.gens()) - J3.dimension() - 1)
|
||||
|
||||
p = 5
|
||||
F = GF(p)
|
||||
Rx.<x> = PolynomialRing(F)
|
||||
C = superelliptic(x, 1)
|
||||
AS = as_cover(C, [C.x^8], prec = 200)
|
||||
I2 = AS.canonical_ideal_polynomials(2, threshold=15)
|
||||
R = I2[0].parent() #ring, to which the polynomials belong
|
||||
J2 = R.ideal(I2) #ideal defined by the set I
|
||||
print(J2.is_prime(), len(R.gens()) - J2.dimension() - 1)
|
||||
I3 = AS.canonical_ideal_polynomials(3, threshold=20)
|
||||
J3 = R.ideal(I2 + I3)
|
||||
print(J3.is_prime(), len(R.gens()) - J3.dimension() - 1)
|
||||
11
elementary_covers/tests/cartier_test.sage
Normal file
11
elementary_covers/tests/cartier_test.sage
Normal file
@ -0,0 +1,11 @@
|
||||
p = 5
|
||||
m = 2
|
||||
F = GF(p)
|
||||
Rx.<x> = PolynomialRing(F)
|
||||
f = x^3 + x + 1
|
||||
C = superelliptic(f, m)
|
||||
g = f(x^p - x)
|
||||
C1 = superelliptic(g, m)
|
||||
ff = superelliptic_function(C, x)
|
||||
AS = as_cover(C, [ff])
|
||||
print(C1.cartier_matrix().rank() == AS.cartier_matrix().rank())
|
||||
15
elementary_covers/tests/diffn_test.sage
Normal file
15
elementary_covers/tests/diffn_test.sage
Normal file
@ -0,0 +1,15 @@
|
||||
p = 3
|
||||
m = 1
|
||||
F = GF(p)
|
||||
Rx.<x> = PolynomialRing(F)
|
||||
f = x^2 + 1
|
||||
C_super = superelliptic(f, m)
|
||||
|
||||
Rxy.<x, y> = PolynomialRing(F, 2)
|
||||
f1 = superelliptic_function(C_super, x^2)
|
||||
f2 = superelliptic_function(C_super, x^4)
|
||||
AS = as_cover(C_super, [f1, f2], prec=1000)
|
||||
RxyzQ, Rxyz, x, y, z = AS.fct_field
|
||||
f_z = as_function(AS, z[0]*z[1]*y)
|
||||
df_z = f_z.diffn()
|
||||
print(df_z.form == -z[0]*z[1]*x + y*z[1]*x - y*z[0]*x^3)
|
||||
20
elementary_covers/tests/dual_element_test.sage
Normal file
20
elementary_covers/tests/dual_element_test.sage
Normal file
@ -0,0 +1,20 @@
|
||||
p = 5
|
||||
m = 1
|
||||
F = GF(p)
|
||||
Rx.<x> = PolynomialRing(F)
|
||||
f = x
|
||||
C_super = superelliptic(f, m)
|
||||
|
||||
Rxy.<x, y> = PolynomialRing(F, 2)
|
||||
f1 = superelliptic_function(C_super, x^2)
|
||||
f2 = superelliptic_function(C_super, x^3)
|
||||
AS = as_cover(C_super, [f1, f2], prec=500)
|
||||
zmag = (AS.magical_element())[0]
|
||||
zdual = dual_elt(AS, zmag)
|
||||
|
||||
for i in range(p):
|
||||
for j in range(p):
|
||||
if (i, j) == (0, 0):
|
||||
print((zmag*(zdual.group_action([i, j]))).trace().function == 1)
|
||||
else:
|
||||
print((zmag*(zdual.group_action([i, j]))).trace().function == 0)
|
||||
18
elementary_covers/tests/group_action_matrices_test.sage
Normal file
18
elementary_covers/tests/group_action_matrices_test.sage
Normal file
@ -0,0 +1,18 @@
|
||||
p = 3
|
||||
m = 2
|
||||
F = GF(p)
|
||||
Rx.<x> = PolynomialRing(F)
|
||||
f = x^3 + x
|
||||
C_super = superelliptic(f, m)
|
||||
|
||||
f1 = C_super.x^2*C_super.y
|
||||
f2 = C_super.x^3
|
||||
AS = as_cover(C_super, [f1, f2], prec=1000)
|
||||
|
||||
A, B = AS.group_action_matrices_holo()
|
||||
n = A.dimensions()[0]
|
||||
print(A*B == B*A)
|
||||
print(A^p == identity_matrix(n))
|
||||
print(B^p == identity_matrix(n))
|
||||
|
||||
print(magma_module_decomposition(A, B))
|
||||
15
elementary_covers/tests/ith_component_test.sage
Normal file
15
elementary_covers/tests/ith_component_test.sage
Normal file
@ -0,0 +1,15 @@
|
||||
p = 5
|
||||
m = 1
|
||||
F = GF(p)
|
||||
Rx.<x> = PolynomialRing(F)
|
||||
f = x
|
||||
C_super = superelliptic(f, m)
|
||||
|
||||
Rxy.<x, y> = PolynomialRing(F, 2)
|
||||
f1 = superelliptic_function(C_super, x^2)
|
||||
f2 = superelliptic_function(C_super, x^3)
|
||||
AS = as_cover(C_super, [f1, f2], prec=500)
|
||||
zmag = (AS.magical_element())[0]
|
||||
|
||||
om = AS.holomorphic_differentials_basis()[4]
|
||||
print(combination_components(om, zmag, zmag).form == om.form)
|
||||
20
elementary_covers/tests/ith_ramification_gp_test.sage
Normal file
20
elementary_covers/tests/ith_ramification_gp_test.sage
Normal file
@ -0,0 +1,20 @@
|
||||
p = 3
|
||||
m = 1
|
||||
F = GF(p)
|
||||
Rx.<x> = PolynomialRing(F)
|
||||
f = x
|
||||
C_super = superelliptic(f, m)
|
||||
|
||||
Rxy.<x, y> = PolynomialRing(F, 2)
|
||||
f1 = superelliptic_function(C_super, x^7)
|
||||
f2 = superelliptic_function(C_super, x^4)
|
||||
AS = as_cover(C_super, [f1, f2], prec=1000)
|
||||
n = AS.height
|
||||
d_test = (p^n - 1)
|
||||
Gi = AS.group
|
||||
i = 1
|
||||
while(len(Gi) > 1):
|
||||
Gi = AS.ith_ramification_gp(i)
|
||||
d_test += len(Gi) - 1
|
||||
i+=1
|
||||
print(d_test == AS.exponent_of_different())
|
||||
25
elementary_covers/tests/ramification_jumps_test.sage
Normal file
25
elementary_covers/tests/ramification_jumps_test.sage
Normal file
@ -0,0 +1,25 @@
|
||||
p = 5
|
||||
m = 1
|
||||
F = GF(p)
|
||||
Rx.<x> = PolynomialRing(F)
|
||||
f = x
|
||||
C_super = superelliptic(f, m)
|
||||
|
||||
Rxy.<x, y> = PolynomialRing(F, 2)
|
||||
f1 = superelliptic_function(C_super, x^2)
|
||||
f2 = superelliptic_function(C_super, x^3)
|
||||
AS = as_cover(C_super, [f1, f2], prec=500)
|
||||
m = AS.jumps[0]
|
||||
m1 = m[0]
|
||||
m2 = m[1]
|
||||
#We compute jumps from jumps in the tower of two Z/p-covers.
|
||||
if m1 >= m2:
|
||||
M1 = m2
|
||||
M2 = m2 + p*(m1 - m2)
|
||||
else:
|
||||
M1 = m1
|
||||
M2 = m2
|
||||
|
||||
theoretical_jumps = [M1, M2]
|
||||
theoretical_jumps.sort()
|
||||
print(theoretical_jumps == AS.ramification_jumps())
|
||||
12
elementary_covers/tests/uniformizer_test.sage
Normal file
12
elementary_covers/tests/uniformizer_test.sage
Normal file
@ -0,0 +1,12 @@
|
||||
p = 3
|
||||
m = 1
|
||||
F = GF(p)
|
||||
Rx.<x> = PolynomialRing(F)
|
||||
f = x
|
||||
C_super = superelliptic(f, m)
|
||||
|
||||
Rxy.<x, y> = PolynomialRing(F, 2)
|
||||
f1 = superelliptic_function(C_super, x^7)
|
||||
f2 = superelliptic_function(C_super, x^4)
|
||||
AS = as_cover(C_super, [f1, f2], prec=1000)
|
||||
print(AS.uniformizer().valuation() == 1)
|
||||
@ -474,4 +474,43 @@ def at_most_poles2(self, pole_orders, threshold = 8):
|
||||
print('iteration')
|
||||
forms = [(omega, omega.group_action(g).expansion_at_infty(place = i)) for omega in forms]
|
||||
forms = holomorphic_combinations_fcts(forms, pole_orders[(i, g)])
|
||||
return forms
|
||||
|
||||
def at_most_poles_forms2(self, pole_orders, threshold = 8):
|
||||
""" Find fcts with pole order in infty's at most pole_order from the given dictionary. The keys of the dictionary are pairs (place_at_infty, group element). The items are the poles orders at those places.
|
||||
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 = heisenberg_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_orders[(0, (0, 0, 0))])
|
||||
print('iteration')
|
||||
for i in range(delta):
|
||||
for g in self.fiber(place = i):
|
||||
if i!=0 or g != (0, 0, 0):
|
||||
print('iteration')
|
||||
forms = [(omega, omega.group_action(g).expansion_at_infty(place = i)) for omega in forms]
|
||||
forms = holomorphic_combinations_forms(forms, pole_orders[(i, g)])
|
||||
return forms
|
||||
@ -15,6 +15,14 @@ class heisenberg_form:
|
||||
def __repr__(self):
|
||||
return "(" + str(self.form)+") * dx"
|
||||
|
||||
def __eq__(self, other):
|
||||
AS = self.curve
|
||||
for i in len(AS.nb_of_pts_at_infty):
|
||||
for g in AS.fiber(place = i):
|
||||
if (self - other).group_action(g).expansion_at_infty(place = i) != 0:
|
||||
return False
|
||||
return True
|
||||
|
||||
def expansion_at_infty(self, place = 0):
|
||||
C = self.curve
|
||||
delta = C.nb_of_pts_at_infty
|
||||
@ -146,7 +154,8 @@ class heisenberg_form:
|
||||
result = result.form
|
||||
Rxy.<x, y> = PolynomialRing(F, 2)
|
||||
Qxy = FractionField(Rxy)
|
||||
result = heisenberg_reduction(AS, result)
|
||||
result = heisenberg_reduction(C, result)
|
||||
#return heisenberg_form(C, result)
|
||||
return superelliptic_form(C_super, Qxy(result))
|
||||
|
||||
def residue(self, place=0):
|
||||
|
||||
@ -14,7 +14,13 @@ class heisenberg_function:
|
||||
#self.function = heisenberg_reduction(AS, RxyzQ(g))
|
||||
|
||||
def __repr__(self):
|
||||
return str(self.function)
|
||||
result = str(self.function)
|
||||
result = result.replace('x', 'H.x')
|
||||
result = result.replace('y', 'H.y')
|
||||
result = result.replace('z0', 'H.z[0]')
|
||||
result = result.replace('z1', 'H.z[1]')
|
||||
result = result.replace('z2', 'H.z[2]')
|
||||
return result
|
||||
|
||||
def __eq__(self, other):
|
||||
AS = self.curve
|
||||
@ -148,7 +154,6 @@ class heisenberg_function:
|
||||
return self.group_action(elt1).group_action((0, 0, 1))
|
||||
|
||||
def trace(self):
|
||||
print('trace ')
|
||||
C = self.curve
|
||||
C_super = C.quotient
|
||||
n = C.height
|
||||
|
||||
@ -140,7 +140,7 @@ class quaternion_form:
|
||||
result = result.form
|
||||
Rxy.<x, y> = PolynomialRing(F, 2)
|
||||
Qxy = FractionField(Rxy)
|
||||
result = quaternion_reduction(AS, result)
|
||||
result = quaternion_reduction(C, result)
|
||||
return superelliptic_form(C_super, Qxy(result))
|
||||
|
||||
def residue(self, place=0):
|
||||
|
||||
@ -160,7 +160,7 @@ class quaternion_function:
|
||||
result = result.function
|
||||
Rxy.<x, y> = PolynomialRing(F, 2)
|
||||
Qxy = FractionField(Rxy)
|
||||
result = quaternion_reduction(AS, result)
|
||||
result = quaternion_reduction(C, result)
|
||||
return superelliptic_function(C_super, Qxy(result))
|
||||
|
||||
def coordinates(self, prec = 100, basis = 0):
|
||||
|
||||
Loading…
Reference in New Issue
Block a user