as transform for unramified points fixed
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@ -176,49 +176,6 @@ class as_form:
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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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if jump != 0:
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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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if jump != 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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if jump == 0:
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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, correction, t, 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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45
as_covers/as_transform.sage
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45
as_covers/as_transform.sage
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@ -0,0 +1,45 @@
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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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print('as transform for ', power_series)
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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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print("a")
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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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if jump != 0:
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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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if jump != 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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if jump == 0:
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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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z = z + correction
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print('corr', correction, 'ps', power_series, 'z', z)
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return(0, correction, t, z)
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@ -1,12 +1,13 @@
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p = 3
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F = GF(3^2, 'a')
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F = GF(3)
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#F.<a> = GF(3^2)
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Rx.<x> = PolynomialRing(F)
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P1 = superelliptic(x^2 + 1, 2)
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fct1 = (P1.x)^2
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fct2 = fct1 + (P1.x)/(P1.y - P1.x)
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fct3 = (P1.x)^4
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C = heisenberg_cover(P1, [1/2*fct1, fct2, fct3], prec=500)
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print(C)
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B = C.holomorphic_differentials_basis()
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fct2 = (fct1 + P1.one/(P1.y - P1.x))
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fct3 = 0*P1.x
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C = heisenberg_cover(P1, [fct1, fct2, fct3], prec=200)
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print(C, '\n', C.genus(), '\n', C.jumps)
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#B = C.holomorphic_differentials_basis()
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#print("Computed basis")
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#a1, b1, c1 = heisenberg_group_action_matrices_holo(C, basis = B)
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@ -194,9 +194,11 @@ class heisenberg_cover:
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forms = holomorphic_combinations_fcts(S, pole_order)
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for i in range(1, delta):
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forms = [(omega, omega.expansion_at_infty(place = i)) for omega in forms]
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forms = holomorphic_combinations_fcts(forms, pole_order)
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for i in range(delta):
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for g in [(0, i, 0) for i in range(p)]:
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if i!=0 or g != (0, 0, 0):
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forms = [(omega, omega.expansion_at_infty(place = i)) for omega in forms]
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forms = holomorphic_combinations_fcts(forms, pole_order)
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return forms
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@ -217,7 +219,7 @@ class heisenberg_cover:
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return result
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def at_most_poles_forms(self, pole_order, threshold = 8):
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"""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.
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"""Find forms with pole order in all the points at infty equal at most to pole_order. Threshold gives a bound on powers of x in the form.
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If you suspect that you haven't found all the functions, you may increase it."""
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from itertools import product
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x_series = self.x_series
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@ -405,6 +407,75 @@ class heisenberg_cover:
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result += [heisenberg_cech(self, omega, f)]
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return result
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def stabilizer(self, place = 0):
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result = []
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for g in self.group:
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flag = 1
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for i in range(self.height):
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if self.z[i].valuation(place = place) > 0:
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fct = self.z[i]
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elif self.z[i].valuation(place = place) < 0:
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fct = self.one/self.z[i]
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if fct.group_action(g).valuation(place = place) <= 0:
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flag = 0
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if flag:
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result += [g]
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return result
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def stabilizer_coset_reps(self, place = 0):
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result = [(0, 0, 0)]
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p = self.characteristic
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H = self.stabilizer(place = place)
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for g in self.group:
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flag = 1
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for v in result:
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if heisenberg_mult(g, heisenberg_inv(v, p), p) in H:
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flag = 0
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if flag:
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result += [g]
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return result
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##############
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def at_most_poles2(self, pole_orders, threshold = 8):
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""" 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.
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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."""
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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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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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F = C.base_ring
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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 = []
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RQxyz = FractionField(Rxyz)
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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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eta = heisenberg_function(self, x^i * prod(z[i1]^(k[i1]) for i1 in range(n))*y^j)
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eta_exp = eta.expansion_at_infty()
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S += [(eta, eta_exp)]
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forms = holomorphic_combinations_fcts(S, pole_orders[(0, (0, 0, 0))])
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for i in range(delta):
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for g in self.stabilizer_coset_reps(place = i):
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if i!=0 or g != (0, 0, 0):
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forms = [(omega, omega.expansion_at_infty(place = i)) for omega in forms]
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forms = holomorphic_combinations_fcts(forms, pole_orders[(i, g)])
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return forms
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##############
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def heisenberg_holomorphic_combinations(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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@ -201,48 +201,6 @@ class heisenberg_form:
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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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if jump != 0:
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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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if jump != 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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if jump == 0:
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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, correction, t, 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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@ -3,6 +3,7 @@ load('superelliptic/superelliptic_function_class.sage')
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load('superelliptic/superelliptic_form_class.sage')
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load('superelliptic/superelliptic_cech_class.sage')
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load('superelliptic/frobenius_kernel.sage')
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load('as_covers/as_transform.sage')
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load('as_covers/as_cover_class.sage')
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load('as_covers/as_function_class.sage')
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load('as_covers/as_form_class.sage')
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@ -43,6 +44,9 @@ load('heisenberg_covers/heisenberg_form_class.sage')
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load('heisenberg_covers/heisenberg_polyforms.sage')
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load('heisenberg_covers/heisenberg_reduction.sage')
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load('heisenberg_covers/heisenberg_group_action_matrices.sage')
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load('heisenberg_covers/dual_element.sage')
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load('heisenberg_covers/ith_magical_component.sage')
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load('heisenberg_covers/heisenberg_group.sage')
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##############
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##############
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def init(lista, tests = False, init=True):
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@ -195,49 +195,6 @@ class quaternion_form:
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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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if jump != 0:
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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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if jump != 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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if jump == 0:
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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, correction, t, 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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