112 lines
4.2 KiB
Python
112 lines
4.2 KiB
Python
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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):
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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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C = AS.quotient
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m = C.exponent
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r = C.polynomial.degree()
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n = AS.height
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p = AS.characteristic
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RxyzQ, Rxyz, x, y, z = AS.fct_field
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if basis == 0:
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basis = [AS.holomorphic_differentials_basis(), AS.cohomology_of_structure_sheaf_basis(), AS.de_rham_basis(threshold=threshold)]
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holo_diffs = basis[0]
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coh_basis = basis[1]
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dR = basis[2]
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F = AS.base_ring
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f_products = []
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for f in coh_basis:
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f_products += [[omega.serre_duality_pairing(f) for omega in holo_diffs]]
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product_of_fct_and_omegas = []
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fct = self.f
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product_of_fct_and_omegas = [omega.serre_duality_pairing(fct) for omega in holo_diffs]
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V = (F^(AS.genus())).span_of_basis([vector(a) for a in f_products])
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coh_coordinates = V.coordinates(product_of_fct_and_omegas) #coeficients of self in the basis elts coming from cohomology of OX
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for i in range(AS.genus()):
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self -= coh_coordinates[i]*dR[i+AS.genus()]
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coh_coordinates = AS.genus()*[0] + list(coh_coordinates)
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if self.f.function not in Rxyz:
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#We remove now from f the summands which are obviously regular at infty
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pr = [list(GF(p)) for _ in range(n)]
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S = []
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from itertools import product
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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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g = (AS.x)^i*prod((AS.z[i1])^(k[i1]) for i1 in range(n))*(AS.y)^j
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S += [(g, g.expansion_at_infty())]
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S += [(self.f, self.f.expansion_at_infty())]
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fcts = holomorphic_combinations_fcts(S, 0)
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for g in fcts:
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if g.function not in Rxyz:
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for a in F:
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if (self.f.function - a*g.function in Rxyz):
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self.f.function = self.f.function - a*g.function
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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(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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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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