def _frobenius_matrix(self, N=None):
"""
Compute p-adic frobenius matrix to precision p^N. If N not supplied,
a default value is selected, which is the minimum needed to recover
the charpoly unambiguously.
Currently only implemented using hypellfrob, which means only works
over GF(p^1), and must have p > (2g+1)(2N-1).
TESTS::
sage: R.<t> = PolynomialRing(GF(37))
sage: H = HyperellipticCurve(t^5 + t + 2)
sage: H._frobenius_matrix()
[1258 + O(37^2) 925 + O(37^2) 132 + O(37^2) 587 + O(37^2)]
[1147 + O(37^2) 814 + O(37^2) 241 + O(37^2) 1011 + O(37^2)]
[1258 + O(37^2) 1184 + O(37^2) 1105 + O(37^2) 482 + O(37^2)]
[1073 + O(37^2) 999 + O(37^2) 772 + O(37^2) 929 + O(37^2)]
"""
p = self.base_ring().characteristic()
f, h = self.hyperelliptic_polynomials()
if h != 0:
# need y^2 = f(x)
raise NotImplementedError, "only implemented for curves y^2 = f(x)"
sign = 1
if not f.is_monic():
# at this time we need a monic f
c = f.leading_coefficient()
f = f / c
if c.is_square():
# solutions of $y^2 = c * f(x)$ correspond naturally to
# solutions of $(sqrt(c) y)^2 = f(x)$
pass
else:
# we'll count points on a twist and then correct by untwisting...
sign = -1
assert f.is_monic()
if N is None:
N = self._frobenius_coefficient_bound()
matrix_of_frobenius = hypellfrob(p, N, f)
matrix_of_frobenius = sign * matrix_of_frobenius
return matrix_of_frobenius
def _frobenius_matrix(self, N=None):
"""
Compute p-adic frobenius matrix to precision p^N. If N not supplied,
a default value is selected, which is the minimum needed to recover
the charpoly unambiguously.
Currently only implemented using hypellfrob, which means only works
over GF(p^1), and must have p > (2g+1)(2N-1).
TESTS::
sage: R.<t> = PolynomialRing(GF(37))
sage: H = HyperellipticCurve(t^5 + t + 2)
sage: H._frobenius_matrix()
[1258 + O(37^2) 925 + O(37^2) 132 + O(37^2) 587 + O(37^2)]
[1147 + O(37^2) 814 + O(37^2) 241 + O(37^2) 1011 + O(37^2)]
[1258 + O(37^2) 1184 + O(37^2) 1105 + O(37^2) 482 + O(37^2)]
[1073 + O(37^2) 999 + O(37^2) 772 + O(37^2) 929 + O(37^2)]
"""
p = self.base_ring().characteristic()
f, h = self.hyperelliptic_polynomials()
if h != 0:
# need y^2 = f(x)
raise NotImplementedError, "only implemented for curves y^2 = f(x)"
sign = 1
if not f.is_monic():
# at this time we need a monic f
c = f.leading_coefficient()
f = f / c
if c.is_square():
# solutions of $y^2 = c * f(x)$ correspond naturally to
# solutions of $(sqrt(c) y)^2 = f(x)$
pass
else:
# we'll count points on a twist and then correct by untwisting...
sign = -1
assert f.is_monic()
if N is None:
N = self._frobenius_coefficient_bound()
matrix_of_frobenius = hypellfrob(p, N, f)
matrix_of_frobenius = sign * matrix_of_frobenius
return matrix_of_frobenius
def compute_frob_matrix_and_cp_H2(f, p, prec, **kwargs):
"""
Return a p-adic matrix approximating the action of Frob on H^2 of a surface or abelian variety,
and its characteristic polynomial over ZZ
Input:
- f defining the curve or surface
- p, prime
- prec, a lower bound for the desired precision to run the computations, this increases the time exponentially
- kwargs, keyword arguments to be passed to controlledreduction
Output:
- `prec`, the minimum digits absolute precision for approximation of the Frobenius
- a matrix representing an approximation of Frob matrix with at least `prec` digits of absolute precision
- characteristic polynomial of Frob on H^2
- the number of classes omitted by working with primitive cohomology
Note: if given two or one univariate polynomial, we will try to change the model over Qpbar,
in order to work with an odd and monic model
"""
K = Qp(p, prec=prec + 10)
OK = ZpCA(p, prec=prec)
Rf = f.parent()
R = f.base_ring()
if len(Rf.gens()) == 2:
if min(f.degrees()) != 2:
raise NotImplementedError("Affine curves must be hyperelliptic")
x, y = f.variables()
if f.degree(x) == 2:
f = f.substitute(x=y, y=x)
# Get Weierstrass equation
# y^2 + a*y + b == 0
b, a, _ = map(R['x'], R['x']['y'](f).monic())
# y^2 + a*y + b == 0 --> (2y + a)^2 = a^2 - 4 b
f = a**2 - 4 * b
f = find_monic_and_odd_model(f.change_ring(K), p)
cp1 = HyperellipticCurve(f.change_ring(
GF(p))).frobenius_polynomial().reverse()
F1 = hypellfrob(p, max(3, prec), f.lift())
F1 = F1.change_ring(OK)
cp, frob_matrix = from_H1_to_H2(cp1,
F1,
tensor=kwargs.get('tensor', False))
frob_matrix = frob_matrix.change_ring(K)
shift = 0
elif len(Rf.gens()) == 3 and f.total_degree() == 4 and f.is_homogeneous():
# Quartic plane curve
if p < 17:
prec = max(4, prec)
else:
prec = max(3, prec)
if 'find_better_model' in kwargs:
model = kwargs['find_better_model']
else:
# there is a speed up, but we may also lose some nice numerical stability from the original sparseness
model = binomial(2 + (prec - 1) * f.total_degree(),
2) < 2 * len(list(f**(prec - 1)))
cp1, F1 = controlledreduction(
f,
p,
min_abs_precision=prec,
frob_matrix=True,
threads=1,
find_better_model=model,
)
# change ring to OK truncates precision accordingly
F1 = F1.change_ring(OK)
cp, frob_matrix = from_H1_to_H2(cp1,
F1,
tensor=kwargs.get('tensor', False))
shift = 0
elif len(Rf.gens()) == 4 and f.total_degree() in [4, 5
] and f.is_homogeneous():
shift = 1
# we will not see the polarization
# Quartic surface
if f.total_degree() == 4:
if p == 3:
prec = max(5, prec)
elif p == 5:
prec = max(4, prec)
elif p < 43:
prec = max(3, prec)
else:
prec = max(2, prec)
elif f.total_degree() == 5:
if p in [3, 5]:
prec = max(7, prec)
elif p <= 23:
prec = max(6, prec)
else:
prec = max(5, prec)
OK = ZpCA(p, prec=prec)
# a rough estimate if it is worth to find a non degenerate mode for f
if 'find_better_model' in kwargs:
model = kwargs['find_better_model']
else:
# there is a speed up, but we may also lose some nice numerical stability from the original sparseness
model = binomial(3 + (prec - 1) * f.total_degree(),
3) < 4 * len(list(f**(prec - 1)))
threads = kwargs.get('threads', ncpus)
cp, frob_matrix = controlledreduction(f,
p,
min_abs_precision=prec,
frob_matrix=True,
find_better_model=model,
threads=threads)
frob_matrix = frob_matrix.change_ring(OK).change_ring(K)
else:
raise NotImplementedError("At the moment we only support:\n"
" - Quartic or quintic surfaces\n"
" - Jacobians of quartic curves\n"
" - Jacobians of hyperelliptic curves\n")
return prec, cp, frob_matrix, shift
def modular_symbols_from_curve(C, N, num_factors=3):
"""
Find the modular symbols spaces that shoudl correspond to the
Jacobian of the given hyperelliptic curve, up to the number of
factors we consider.
INPUT:
- C -- a hyperelliptic curve over QQ
- N -- a positive integer
- num_factors -- number of Euler factors to verify match up; this is
important, because if, e.g., there is only one factor of degree g(C), we
don't want to just immediately conclude that Jac(C) = A_f.
OUTPUT:
- list of all sign 1 simple modular symbols factor of level N
that correspond to a simple modular abelian A_f
that is isogenous to Jac(C).
EXAMPLES::
sage: from psage.modform.rational.unfiled import modular_symbols_from_curve
sage: R.<x> = ZZ[]
sage: f = x^7+4*x^6+5*x^5+x^4-3*x^3-2*x^2+1
sage: C1 = HyperellipticCurve(f)
sage: modular_symbols_from_curve(C1, 284)
[Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 39 for Gamma_0(284) of weight 2 with sign 1 over Rational Field]
sage: f = x^7-7*x^5-11*x^4+5*x^3+18*x^2+4*x-11
sage: C2 = HyperellipticCurve(f)
sage: modular_symbols_from_curve(C2, 284)
[Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 39 for Gamma_0(284) of weight 2 with sign 1 over Rational Field]
"""
# We will use the Eichler-Shimura relation and David Harvey's
# p-adic point counting hypellfrob. Harvey's code has the
# constraint: p > (2*g + 1)*(2*prec - 1).
# So, find the smallest p not dividing N that satisfies the
# above constraint, for our given choice of prec.
f, f2 = C.hyperelliptic_polynomials()
if f2 != 0:
raise NotImplementedError, "curve must be of the form y^2 = f(x)"
if f.degree() % 2 == 0:
raise NotImplementedError, "curve must be of the form y^2 = f(x) with f(x) odd"
prec = 1
g = C.genus()
B = (2 * g + 1) * (2 * prec - 1)
from sage.rings.all import next_prime
p = B
# We use that if F(X) is the characteristic polynomial of the
# Hecke operator T_p, then X^g*F(X+p/X) is the characteristic
# polynomial of Frob_p, because T_p = Frob_p + p/Frob_p, according
# to Eichler-Shimura. Use this to narrow down the factors.
from sage.all import ModularSymbols, Integers, get_verbose
D = ModularSymbols(
N, sign=1).cuspidal_subspace().new_subspace().decomposition()
D = [A for A in D if A.dimension() == g]
from sage.schemes.hyperelliptic_curves.hypellfrob import hypellfrob
while num_factors > 0:
p = next_prime(p)
while N % p == 0:
p = next_prime(p)
R = Integers(p**prec)['X']
X = R.gen()
D2 = []
# Compute the charpoly of Frobenius using hypellfrob
M = hypellfrob(p, 1, f)
H = R(M.charpoly())
for A in D:
F = R(A.hecke_polynomial(p))
# Compute charpoly of Frobenius from F(X)
G = R(F.parent()(X**g * F(X + p / X)))
if get_verbose(): print(p, G, H)
if G == H:
D2.append(A)
D = D2
num_factors -= 1
return D
def modular_symbols_from_curve(C, N, num_factors=3):
"""
Find the modular symbols spaces that shoudl correspond to the
Jacobian of the given hyperelliptic curve, up to the number of
factors we consider.
INPUT:
- C -- a hyperelliptic curve over QQ
- N -- a positive integer
- num_factors -- number of Euler factors to verify match up; this is
important, because if, e.g., there is only one factor of degree g(C), we
don't want to just immediately conclude that Jac(C) = A_f.
OUTPUT:
- list of all sign 1 simple modular symbols factor of level N
that correspond to a simple modular abelian A_f
that is isogenous to Jac(C).
EXAMPLES::
sage: from psage.modform.rational.unfiled import modular_symbols_from_curve
sage: R.<x> = ZZ[]
sage: f = x^7+4*x^6+5*x^5+x^4-3*x^3-2*x^2+1
sage: C1 = HyperellipticCurve(f)
sage: modular_symbols_from_curve(C1, 284)
[Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 39 for Gamma_0(284) of weight 2 with sign 1 over Rational Field]
sage: f = x^7-7*x^5-11*x^4+5*x^3+18*x^2+4*x-11
sage: C2 = HyperellipticCurve(f)
sage: modular_symbols_from_curve(C2, 284)
[Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 39 for Gamma_0(284) of weight 2 with sign 1 over Rational Field]
"""
# We will use the Eichler-Shimura relation and David Harvey's
# p-adic point counting hypellfrob. Harvey's code has the
# constraint: p > (2*g + 1)*(2*prec - 1).
# So, find the smallest p not dividing N that satisfies the
# above constraint, for our given choice of prec.
f, f2 = C.hyperelliptic_polynomials()
if f2 != 0:
raise NotImplementedError, "curve must be of the form y^2 = f(x)"
if f.degree() % 2 == 0:
raise NotImplementedError, "curve must be of the form y^2 = f(x) with f(x) odd"
prec = 1
g = C.genus()
B = (2*g + 1)*(2*prec - 1)
from sage.rings.all import next_prime
p = B
# We use that if F(X) is the characteristic polynomial of the
# Hecke operator T_p, then X^g*F(X+p/X) is the characteristic
# polynomial of Frob_p, because T_p = Frob_p + p/Frob_p, according
# to Eichler-Shimura. Use this to narrow down the factors.
from sage.all import ModularSymbols, Integers, get_verbose
D = ModularSymbols(N,sign=1).cuspidal_subspace().new_subspace().decomposition()
D = [A for A in D if A.dimension() == g]
from sage.schemes.hyperelliptic_curves.hypellfrob import hypellfrob
while num_factors > 0:
p = next_prime(p)
while N % p == 0: p = next_prime(p)
R = Integers(p**prec)['X']
X = R.gen()
D2 = []
# Compute the charpoly of Frobenius using hypellfrob
M = hypellfrob(p, 1, f)
H = R(M.charpoly())
for A in D:
F = R(A.hecke_polynomial(p))
# Compute charpoly of Frobenius from F(X)
G = R(F.parent()(X**g * F(X + p/X)))
if get_verbose(): print (p, G, H)
if G == H:
D2.append(A)
D = D2
num_factors -= 1
return D
