304 lines
12 KiB
Python
304 lines
12 KiB
Python
class quaternion_form:
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def __init__(self, C, g):
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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.form = RxyzQ(g)
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def __repr__(self):
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return "(" + str(self.form)+") * dx"
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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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F = C.base_ring
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x_series = C.x_series[place]
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y_series = C.y_series[place]
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z_series = C.z_series[place]
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dx_series = C.dx_series[place]
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n = C.height
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variable_names = 'x, y'
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for j in range(n):
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variable_names += ', z' + str(j)
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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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prec = C.prec
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Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
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g = self.form
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sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}
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return g.substitute(sub_list)*dx_series
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def expansion(self, pt = 0):
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'''Same code as expansion_at_infty.'''
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C = self.curve
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F = C.base_ring
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x_series = C.x_series[pt]
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y_series = C.y_series[pt]
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z_series = C.z_series[pt]
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dx_series = C.dx_series[pt]
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n = C.height
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variable_names = 'x, y'
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for j in range(n):
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variable_names += ', z' + str(j)
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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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prec = C.prec
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Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
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g = self.form
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sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}
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return g.substitute(sub_list)*dx_series
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def __add__(self, other):
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C = self.curve
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g1 = self.form
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g2 = other.form
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return quaternion_form(C, g1 + g2)
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def __sub__(self, other):
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C = self.curve
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g1 = self.form
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g2 = other.form
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return quaternion_form(C, g1 - g2)
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def __neg__(self):
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C = self.curve
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g = self.form
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return quaternion_form(C, -g)
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def __rmul__(self, constant):
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C = self.curve
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omega = self.form
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return quaternion_form(C, constant*omega)
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def group_action(self, ZN_tuple):
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C = self.curve
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n = C.height
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RxyzQ, Rxyz, x, y, z = C.fct_field
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sub_list = {x : x, y : y} | {z[j] : z[j]+ZN_tuple[j] for j in range(n)}
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g = self.form
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return quaternion_form(C, g.substitute(sub_list))
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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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C = self.curve
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if basis == 0:
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basis = C.holomorphic_differentials_basis()
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RxyzQ, Rxyz, x, y, z = C.fct_field
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# We need to have only polynomials to use monomial_coefficients in linear_representation_polynomials,
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# and sometimes basis elements have denominators. Thus we multiply by them.
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denom = LCM([denominator(omega.form) for omega in basis])
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basis = [denom*omega for omega in basis]
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self_with_no_denominator = denom*self
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return linear_representation_polynomials(Rxyz(self_with_no_denominator.form), [Rxyz(omega.form) for omega in basis])
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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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n = C.height
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F = C.base_ring
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variable_names = 'x, y'
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for j in range(n):
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variable_names += ', z' + str(j)
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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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result = quaternion_form(C, 0)
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G = C.group
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for a in G:
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result += self.group_action(a)
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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 = quaternion_reduction(AS, 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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return self.expansion_at_infty(place = place).residue()
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def valuation(self, place=0):
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return self.expansion_at_infty(place = place).valuation()
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def serre_duality_pairing(self, fct):
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AS = self.curve
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return sum((fct*self).residue(place = _) for _ in range(AS.nb_of_pts_at_infty))
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def cartier(self):
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C = self.curve
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F = C.base_ring
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n = C.height
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ff = C.functions
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p = F.characteristic()
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C_super = C.quotient
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(RxyzQ, Rxyz, x, y, z) = C.fct_field
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fct = self.form
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Rxy.<x, y> = PolynomialRing(F, 2)
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RxyQ = FractionField(Rxy)
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x, y = Rxyz.gens()[0], Rxyz.gens()[1]
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z = Rxyz.gens()[2:]
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num = Rxyz(fct.numerator())
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den = Rxyz(fct.denominator())
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result = RxyzQ(0)
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#return (num, den, z, fct)
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if den in Rxy:
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sub_list = {x : x, y : y} | {z[j] : (z[j]^p - RxyzQ(ff[j].function)) for j in range(n)}
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num = RxyzQ(num.substitute(sub_list))
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den1 = Rxyz(num.denominator())
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num = Rxyz(num*den1^p)
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for monomial in Rxyz(num).monomials():
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degrees = [monomial.degree(z[i]) for i in range(n)]
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product_of_z = prod(z[i]^(degrees[i]) for i in range(n))
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monomial_divided_by_z = monomial/product_of_z
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product_of_z_no_p = prod(z[i]^(degrees[i]/p) for i in range(n))
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aux_form = superelliptic_form(C_super, RxyQ(monomial_divided_by_z/den))
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aux_form = aux_form.cartier()
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result += product_of_z_no_p * Rxyz(num).monomial_coefficient(monomial) * aux_form.form/den1
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return quaternion_form(C, result)
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raise ValueError("Please present first your form as sum z^i omega_i, where omega_i are forms on quotient curve.")
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def is_regular_on_U0(self):
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AS = self.curve
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C = AS.quotient
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m = C.exponent
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RxyzQ, Rxyz, x, y, z = AS.fct_field
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if y^(m-1)*self.form in Rxyz:
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return True
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return False
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def artin_schreier_transform(power_series, prec = 10):
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"""Given a power_series, find correction such that power_series - (correction)^p +correction has valuation
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-jump non divisible by p. Also, express t (the variable) in terms of the uniformizer at infty on the curve
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z^p - z = power_series, where z = 1/t_new^(jump) and express z in terms of the new uniformizer."""
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correction = 0
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F = power_series.parent().base()
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p = F.characteristic()
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Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
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RtQ = FractionField(Rt)
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power_series = RtQ(power_series)
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if power_series.valuation() == +Infinity:
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raise ValueError("Precision is too low.")
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if power_series.valuation() >= 0:
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# THIS IS WRONG - THERE ARE SEVERAL PLACES OVER THIS PLACE, AND IT DEPENDS
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aux = t^p - t
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z = new_reverse(aux, prec = prec)
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z = z(t = power_series)
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return(0, 0, t, z)
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while(power_series.valuation() % p == 0 and power_series.valuation() < 0):
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M = -power_series.valuation()/p
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coeff = power_series.list()[0] #wspolczynnik a_(-p) w f_AS
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correction += coeff.nth_root(p)*t^(-M)
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power_series = power_series - (coeff*t^(-p*M) - coeff.nth_root(p)*t^(-M))
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jump = max(-(power_series.valuation()), 0)
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try:
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T = nth_root2((power_series)^(-1), jump, prec=prec) #T is defined by power_series = 1/T^m
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except:
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print("no ", str(jump), "-th root; divide by", power_series.list()[0])
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return (jump, power_series.list()[0])
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T_rev = new_reverse(T, prec = prec)
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t_old = T_rev(t^p/nth_root2(1 - t^((p-1)*jump), jump, prec=prec))
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z = 1/t^(jump) + Rt(correction)(t = t_old)
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return(jump, correction, t_old, z)
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def are_forms_linearly_dependent(set_of_forms):
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from sage.rings.polynomial.toy_variety import is_linearly_dependent
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C = set_of_forms[0].curve
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F = C.base_ring
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n = C.height
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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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denominators = prod(denominator(omega.form) for omega in set_of_forms)
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return is_linearly_dependent([Rxyz(denominators*omega.form) for omega in set_of_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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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([Rt(g[1]).valuation() for g in S])
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if minimal_valuation >= -pole_order:
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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 + Rt(eta_exp).valuation()
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list_coeffs = a*[0] + Rt(eta_exp).list() + (-minimal_valuation)*[0]
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list_coeffs = list_coeffs[:-minimal_valuation - pole_order]
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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 = quaternion_function(C_AS, 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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#eta_exp = elt_S[1]
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forma_holo += vec_wspolrzedna*eta
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#forma_holo_power_series += vec_wspolrzedna*eta_exp
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forms += [forma_holo]
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return forms
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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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def holomorphic_combinations_forms(S, pole_order):
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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([Rt(g[1]).valuation() for g in S])
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if minimal_valuation >= -pole_order:
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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 + Rt(eta_exp).valuation()
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list_coeffs = a*[0] + Rt(eta_exp).list() + (-minimal_valuation)*[0]
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list_coeffs = list_coeffs[:-minimal_valuation - pole_order]
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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 = quaternion_form(C_AS, 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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#eta_exp = elt_S[1]
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forma_holo += vec_wspolrzedna*eta
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#forma_holo_power_series += vec_wspolrzedna*eta_exp
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forms += [forma_holo]
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return forms
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#print only forms that are log at the branch pts, but not holomorphic
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def only_log_forms(C_AS):
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list1 = AS.at_most_poles_forms(0)
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list2 = AS.at_most_poles_forms(1)
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result = []
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for a in list2:
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if not(are_forms_linearly_dependent(list1 + result + [a])):
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result += [a]
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return result
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