def to_polredabs(K):
"""
INPUT:
* "K" - a number field
OUTPUT:
* "phi" - an isomorphism K -> L, where L = QQ['x']/f and f a polynomial such that f = polredabs(f)
"""
R = PolynomialRing(QQ, 'x')
x = R.gen(0)
if K == QQ:
L = QQ.extension(x, 'w')
return QQ.hom(L)
L = K.absolute_field('a')
m1 = L.structure()[1]
f = L.absolute_polynomial()
g = pari(f).polredabs(1)
g, h = g[0].sage(locals={'x': x}), g[1].lift().sage(locals={'x': x})
if debug:
print('f', f)
print('g', g)
print('h', h)
M = QQ.extension(g, 'w')
m2 = L.hom([h(M.gen(0))])
return m2 * m1
def to_polredabs(K):
"""
INPUT:
* "K" - a number field
OUTPUT:
* "phi" - an isomorphism K -> L, where L = QQ['x']/f and f a polynomial such that f = polredabs(f)
"""
R = PolynomialRing(QQ,'x')
x = R.gen(0)
if K == QQ:
L = QQ.extension(x,'w')
return QQ.hom(L)
L = K.absolute_field('a')
m1 = L.structure()[1]
f = L.absolute_polynomial()
g = pari(f).polredabs(1)
g,h = g[0].sage(locals={'x':x}),g[1].lift().sage(locals={'x':x})
if debug:
print 'f',f
print 'g',g
print 'h',h
M = QQ.extension(g,'w')
m2 = L.hom([h(M.gen(0))])
return m2*m1
def compute_alpha_point(g,
iaj,
alpha,
P0,
digits,
power,
verbose,
aggressive=True,
append=""):
# input:
# * g, defining polynomial of the hyperelliptic curve
# * iaj = InvertAJglobal class
# * alpha = analytic representation of the endomorphism acting on the tangent space
# * digits = how many digits to work with
# * power, we will divide by 2**power before inverting the abel jacobi map
# * P0, a point in the curve for which we will compute alpha*2(P - \infty) = P1 + P2 - \infty
# * verbose
# * agressive, raise ZeroDivisionError if we get a P1[0] = P2[0] or one of the points is infty
# * append, a string to append to the numberfield generators
#
# output: a dictionary with the following keys
# TODO add the keys
output = {}
K = alpha.base_ring()
CCap = iaj.C
prec = CCap.precision()
output['prec'] = prec
x0, y0 = P0
Kz = PolynomialRing(K, "z")
z = Kz.gen()
if verbose:
print "compute_alpha_point()"
print "P0 = %s" % (P0, )
#deal with the field of definition
if y0 in K:
if K is QQ:
L = K
from_K_to_L = QQ.hom(1, QQ)
else:
L = K.change_names("b" + append)
from_K_to_L = L.structure()[1]
else:
L, from_K_to_L = (z**2 - g(x0)).splitting_field("b" + append,
simplify_all=True,
map=True)
output['L'] = L
# output['L_str'] = sage_str_numberfield(L, 'x','b'+append);
output['from_K_to_L'] = from_K_to_L
# output['L_gen'] = toCCap(L.gen(), 53);
# figure out y0 in L
y0_ap = toCCap(y0, prec)
y0s = [elt for elt, _ in (z**2 - g(x0)).roots(L)]
assert len(y0s) == 2
y0s_ap = [toCCap(elt, prec) for elt in y0s]
if norm(y0_ap - y0s_ap[0]) < norm(y0_ap - y0s_ap[1]):
y0 = y0s[0]
else:
y0 = y0s[1]
P0 = vector(L, [x0, y0])
alpha_L = Matrix(L, [[from_K_to_L(elt) for elt in row]
for row in alpha.rows()])
alpha_ap = Matrix(CCap, [toCCap_list(row, prec) for row in alpha_L.rows()])
output['P'] = P0
output['alpha'] = alpha_L
if verbose:
print "L = %s" % L
print "%s = %s" % (L.gen(), toCCap(L.gen(), 53))
print "P0 = %s" % (P0, )
print "alpha = %s" % ([[elt for elt in row]
for row in alpha_L.rows()], )
P0_ap = vector(CCap, toCCap_list(P0, prec + 192))
if verbose:
print "P0_ap = %s" % (vector(CC, P0_ap), )
if verbose:
print "Computing AJ of P0..."
c, w = cputime(), walltime()
ajP0 = vector(CCap, AJ1_digits(g, P0_ap, digits))
print "Time: CPU %.2f s, Wall: %.2f s" % (
cputime(c),
walltime(w),
)
print
else:
ajP0 = vector(CCap, AJ1_digits(g, P0_ap, digits))
output['ajP'] = ajP0
aj2P0 = 2 * ajP0
alpha2P0_an = alpha_ap * aj2P0
if verbose:
print "Working over %s" % CCap
invertAJ_tries = 3
for i in range(invertAJ_tries):
#try invertAJ_tries times to invertAJ
try:
if verbose:
print "\n\ninverting AJ..."
c, w = cputime(), walltime()
alpha2P0_div = iaj.invertAJ(alpha2P0_an, power)
print "Time: CPU %.2f s, Wall: %.2f s" % (
cputime(c),
walltime(w),
)
print "\n\n"
else:
alpha2P0_div = iaj.invertAJ(alpha2P0_an, power)
break
except AssertionError as error:
print error
if verbose:
print "an assertion failed while inverting AJ.."
if i == invertAJ_tries - 1:
if verbose:
print "retrying with a new P0"
print
raise ZeroDivisionError
else:
iaj.iajlocal.set_basepoints()
if verbose:
print "retrying again with a new set of base points"
if verbose:
print "Computing the Mumford coordinates of alpha(2*P0 - 2*W)"
c, w = cputime(), walltime()
R0, R1 = alpha2P0_div.coordinates()
print "Time: CPU %.2f s, Wall: %.2f s" % (
cputime(c),
walltime(w),
)
else:
R0, R1 = alpha2P0_div.coordinates()
points = [R0, R1]
x_poly = alpha2P0_div.x_coordinates()
output['R'] = points
output['x_poly'] = x_poly
if (R0 in [+Infinity, -Infinity]) or (R1 in [+Infinity, -Infinity]):
if aggressive:
# we want to avoid these situations
if verbose:
print "One of the coordinates is at infinity"
if R0 in [+Infinity, -Infinity]:
print "R0 = %s" % (R0, )
else:
print "R1 = %s" % (R1, )
print "retrying with a new P0"
print
raise ZeroDivisionError
else:
return output
buffer = "# R0 = %s\n" % (vector(CC, R0), )
buffer += "# R1 = %s\n" % (vector(CC, R1), )
buffer += "# and \n# x_poly = %s\n" % (vector(CC, x_poly), )
if verbose:
print buffer
assert len(x_poly) == 3
algx_poly = [NF_embedding(coeff, L) for coeff in x_poly]
buffer = "algx_poly = %s;\n" % (algx_poly, )
if L != QQ:
buffer += "#where %s ~ %s\n" % (L.gen(), L.gen().complex_embedding())
buffer += "\n"
if verbose:
print buffer
sys.stdout.flush()
sys.stderr.flush()
output['algx_poly'] = algx_poly
if None in algx_poly:
if aggressive:
if verbose:
print "No algebraic expression for the polynomial"
print "retrying with a new P0"
sys.stdout.flush()
sys.stderr.flush()
raise ZeroDivisionError
else:
return output
#c, b, a = algx_poly
#if aggressive and b**2 - 4*a*c == 0:
# raise ZeroDivisionError
if verbose:
print "Done compute_alpha_point()"
return output
