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auxiliary_functions.sage
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auxiliary_functions.sage
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r"""
Auxiliary functions. Some of these functions are from:
https://github.com/bianchifrancesca/quadratic_chabauty
These functions were also used for some of the computations of
[BBBM19].
REFERENCES:
- [BBBM19] \J. S. Balakrishnan, A. Besser, F. Bianchi, J. S. Mueller,
"Explicit quadratic Chabauty over number fields". To appear in
"Isr. J. Math.".
"""
import itertools
def coleman_integrals_on_basis(H, P, Q, inverse_frob, forms, algorithm=None):
r"""
Compute the Coleman integrals `\{\int_P^Q x^i dx/2y \}_{i=0}^{2g-1}`.
The only difference with the built-in function `coleman_integrals_on_basis`
is that we input the Frobenius data `(inverse_frob, forms)`.
INPUT:
- ``H`` -- a hyperelliptic curve over a `p`-adic field `K`.
- ``P`` -- a point on `H`.
- ``Q`` -- a point on `H`.
- ``inverse_frob`` -- `(M_frob^T-1)^(-1)` where `M_frob` is the matrix of Frobenius
on Monsky-Washnitzer cohomology, with respect to the basis `(dx/2y, x dx/2y, ...x^{d-2} dx/2y)`
and with coefficients in `K`.
- ``forms`` -- list of differentials `\{f_i\}` such that
`\phi^* (x^i dx/2y) = df_i + M_frob[i]*vec(dx/2y, ..., x^{d-2} dx/2y)`;
the coefficients of the `f_i` should be in `K`.
- ``algorithm`` (optional) -- None (uses Frobenius) or teichmuller
(uses Teichmuller points).
OUTPUT:
The Coleman integrals `\{\int_P^Q x^i dx/2y \}_{i=0}^{2g-1}`.
EXAMPLES:
This example illustrates how to compute the data `(inverse_frob,forms)` for `H`::
sage: import sage.schemes.hyperelliptic_curves.monsky_washnitzer as monsky_washnitzer
sage: K = H.base_ring()
sage: M_frob, forms = monsky_washnitzer.matrix_of_frobenius_hyperelliptic(H)
sage: forms = [form.change_ring(K) for form in forms]
sage: M_sys = matrix(K, M_frob).transpose() - 1
sage: inverse_frob = M_sys**(-1)
"""
K = H.base_ring()
p = K.prime()
prec = K.precision_cap()
g = H.genus()
dim = 2*g
V = VectorSpace(K, dim)
#if P or Q is Weierstrass, use the Frobenius algorithm
if H.is_weierstrass(P):
if H.is_weierstrass(Q):
return V(0)
else:
PP = None
QQ = Q
TP = None
TQ = H.frobenius(Q)
elif H.is_weierstrass(Q):
PP = P
QQ = None
TQ = None
TP = H.frobenius(P)
elif H.is_same_disc(P, Q):
return H.tiny_integrals_on_basis(P, Q)
elif algorithm == 'teichmuller':
PP = TP = H.teichmuller(P)
QQ = TQ = H.teichmuller(Q)
evalP, evalQ = TP, TQ
else:
TP = H.frobenius(P)
TQ = H.frobenius(Q)
PP, QQ = P, Q
if TP is None:
P_to_TP = V(0)
else:
if TP is not None:
TPv = (TP[0]**g/TP[1]).valuation()
xTPv = TP[0].valuation()
else:
xTPv = TPv = +Infinity
if TQ is not None:
TQv = (TQ[0]**g/TQ[1]).valuation()
xTQv = TQ[0].valuation()
else:
xTQv = TQv = +Infinity
offset = (2*g-1)*max(TPv, TQv)
if offset == +Infinity:
offset = (2*g-1)*min(TPv,TQv)
if (offset > prec and (xTPv < 0 or xTQv < 0) and (H.residue_disc(P) == H.change_ring(GF(p))(0, 1, 0) or H.residue_disc(Q) == H.change_ring(GF(p))(0, 1, 0))):
newprec = offset + prec
K = pAdicField(p,newprec)
A = PolynomialRing(RationalField(),'x')
f = A(H.hyperelliptic_polynomials()[0])
from sage.schemes.hyperelliptic_curves.constructor import HyperellipticCurve
H = HyperellipticCurve(f).change_ring(K)
xP = P[0]
xPv = xP.valuation()
xPnew = K(sum(c * p**(xPv + i) for i, c in enumerate(xP.expansion())))
PP = P = H.lift_x(xPnew)
TP = H.frobenius(P)
xQ = Q[0]
xQv = xQ.valuation()
xQnew = K(sum(c * p**(xQv + i) for i, c in enumerate(xQ.expansion())))
QQ = Q = H.lift_x(xQnew)
TQ = H.frobenius(Q)
V = VectorSpace(K,dim)
P_to_TP = V(H.tiny_integrals_on_basis(P, TP))
if TQ is None:
TQ_to_Q = V(0)
else:
TQ_to_Q = V(H.tiny_integrals_on_basis(TQ, Q))
if PP is None:
L = [-f(QQ[0], QQ[1]) for f in forms] ##changed
elif QQ is None:
L = [f(PP[0], PP[1]) for f in forms]
else:
L = [f(PP[0], PP[1]) - f(QQ[0], QQ[1]) for f in forms]
b = V(L)
if PP is None:
b -= TQ_to_Q
elif QQ is None:
b -= P_to_TP
elif algorithm != 'teichmuller':
b -= P_to_TP + TQ_to_Q
#M_sys = matrix(K, M_frob).transpose() - 1
TP_to_TQ = inverse_frob * b
if algorithm == 'teichmuller':
return P_to_TP + TP_to_TQ + TQ_to_Q
else:
return TP_to_TQ
def Q_lift(CK, Q, p):
r"""
Compute the Teichmueller point lifting a given point over `GF(p)`.
INPUT:
- ``CK`` -- a hyperelliptic curve over `\QQ_p`.
- ``Q`` -- a point in `CK(GF(p))`.
- ``p`` -- the prime of the first two inputs.
OUTPUT: The point on `CK` lifting `Q` and fixed by Frobenius.
"""
xQ = Integers()(Q[0])
yQ = Integers()(Q[1])
if yQ == 0:
r = CK.hyperelliptic_polynomials()[0].roots()
Q_lift = CK(exists(r, lambda a : (Integers()(a[0])-xQ) % p == 0)[1][0],0)
else:
K = CK.base_ring()
xQ = K.teichmuller(K(xQ))
lifts = CK.lift_x(xQ, all=True)
for i in range(len(lifts)):
if (Integers()(lifts[i][1])-yQ) % p == 0:
Q_lift = lifts[i]
return Q_lift
def embeddings(K, p, prec):
"""r
Compute the embedding(s) of `K` in the completions of `K` at
the primes above `p`. This is taken from the code used for [BcL+16].
INPUT:
- ``K`` -- a quadratic field.
- ``p`` -- a prime.
- ``prec`` -- the `p`-adic precision.
OUTPUT:
A list of embeddings of `K` in `\QQ_p` if `p` splits in `K`;
an embedding of `K =\QQ(\sqrt(D))` in `\QQ_p(\sqrt(D))` otherwise.
REFERENCES:
- [BcL+16] \J. S. Balakrishnan, M. Ciperiani, J. Lang, B. Mirza, and R. Newton, "Shadow
lines in the arithmetic of elliptic curves". In" Directions in number theory, volume 3".
"""
Q = pAdicField(p,prec)
OK = K.maximal_order()
pOK = factor(p*OK)
if (len(pOK) == 2 and pOK[0][1] == 1):
R = Q['x']
r1, r2 = R(K.defining_polynomial()).roots()
psi1 = K.hom([r1[0]])
psi2 = K.hom([r2[0]])
return [psi1, psi2]
else:
F = Q.extension(K.defining_polynomial(),names='a')
a = F.gen()
psi = self._psis = [K.hom([a])]
return psi
def bernardi_sigma_function(E, prec=20):
r"""
Return the sigma function of Bernardi in terms of `z = log(t)`.
This is an adaptation of the code built in sage. The difference is that we input
the curve instead of the `L`-function and that we are adding `r` to `xofF`.
INPUT:
- ``E`` -- an elliptic curve over `\QQ`.
- ``prec`` -- power series precision.
OUTPUT: A power series in `z` with coefficients in \QQ approximating the sigma function
of Bernardi.
.. NOTE::
This will converge on some `p`-adic neighbourhood of the identity on `E`
for a prime `p` of good reduction.
"""
Eh = E.formal()
lo = Eh.log(prec + 5)
F = lo.reverse()
S = LaurentSeriesRing(QQ,'z',default_prec = prec)
z = S.gen()
F = F(z)
xofF = Eh.x(prec + 2)(F)
r = ( E.a1()**2 + 4*E.a2() ) / ZZ(12)
xofF = xofF + r
g = (1/z**2 - xofF ).power_series()
h = g.integral().integral()
sigma_of_z = z.power_series() * h.exp()
return sigma_of_z
def height_bernardi(P, p, prec):
"""r
Return the `p`-adic height of Bernardi at `P`.
INPUT:
- ``P`` -- a point on an elliptic curve `E` over `\QQ`.
- ``p`` -- an odd prime of good reduction for `E`.
- ``prec`` -- integer. The precision of the computation.
OUTPUT:
A `p`-adic number; the Bernardi `p`-adic height of `P`.
"""
E = P.scheme()
tam = E.tamagawa_numbers()
tam.append(E.Np(p))
m = lcm(tam)
Q = m*P
dQ = ZZ(Q[0].denominator().sqrt())
sigma = bernardi_sigma_function(E,prec=prec+5)
sigma = sigma(E.formal().log(prec+5))
sigmaQ = Qp(p, prec)(sigma(-Qp(p,prec+5)(Q[0]/Q[1])))
return (-2*log(sigmaQ/dQ))/m^2
############## SOLVERS OF SYSTEMS OF MULTIVARIABLE p-ADIC POLYNOMIALS ##############
def hensel_lift_n(flist, p, prec):
r"""
Multivariable Hensel lifter for roots that are simple
modulo `p`.
This is essentially the code from [S15] with some minor
modifications.
INPUT:
- ``flist`` -- A list of `n` polynomials in `n` variables
over a `p`-adic field. Each polynomial should have coefficients in
`\ZZ_p` and be normalised so that the minimal valaution of its
coefficients is `0`.
- ``p`` -- the prime number of the first input item.
- ``prec`` -- `p`-adic precision. In order for the output to be given
modulo `p^{prec}`, the coefficients of the polynomials in `flist`
should be known at least modulo `p^{prec}`.
OUTPUT:
A tuple consisting of:
- A list `L` of common roots in `Qp(p, prec)` of the polynomials in `flist`
(each root is returned as an `(n x 1)`-matrix, where the `i`-th row
corresponds to the `i`-th variable).
Each root in this list lift uniquely to a root in `\QQ_p`.
- An integer: the number of roots modulo `p` of `flist` for which the
determinant of the Jacobian matrix vanishes modulo `p`. If this is zero,
then the `L` contains all the roots of `flist`.
EXAMPLES:
An example with no double roots modulo `p`::
sage: _.<s, t> = PolynomialRing(Qp(5, 10))
sage: f1 = s + t - 2*s*t
sage: f2 = s - t
sage: a, b = hensel_lift_n([f1, f2], 5, 10)
sage: print a
[[0]
[0], [1 + O(5^10)]
[1 + O(5^10)]]
sage: print b
0
An example with only double roots modulo `p`::
sage: _.<s,t> = PolynomialRing(Qp(5,10))
sage: f1 = s - 11* t + 5*s*t
sage: f2 = s - t
sage: a, b = hensel_lift_n([f1, f2], 5, 10)
sage: print a
[]
sage: print b
5
REFERENCES:
- [S15]: \B. Schmidt, "Solutions to Systems of Multivariate p-adic Power Series".
Oxford MSc Thesis, 2015.
"""
precvec = []
k = 0
for F in flist:
R1 = flist[k].parent()
F1 = R1.base_ring()
if F1.is_exact():
precision1 = prec
else:
precision1 = F1.precision_cap()
if prec > F1.precision_cap():
print 'Cannot get %s digits of precision due to precision of inputs of f1; raise precision of inputs' %prec
if prec < F1.precision_cap():
precision1 = prec
precvec.append(precision1)
k = k+1
precision = min(precvec);
#print 'Precision is', precision
slist = list(var('s%d' % i) for i in (1..len(flist)))
s = ''
for i in range(len(flist)):
s += str(slist[i])+','
a = s[0:len(s)-1]
R = Qp(p, precision,'capped-rel')[a]
flistnew = []
for F in flist:
F = R(F)
flistnew.append(F)
Jlist=[]
for F in flistnew:
for cars in list(flistnew[0].parent().gens()):
Jlist.append(F.derivative(cars))
J = matrix(len(flistnew), len(flistnew), Jlist)
M = J.determinant()
from itertools import product
lists = []
for i in range(len(flistnew)):
lists.append([j for j in range(p)])
coords = []
for items in product(*lists):
coords.append(items)
roots = []
roots_info = []
nonroots = 0
for i in range(len(coords)):
valuesval = [(F(*coords[i]).valuation()) for F in flistnew]
min_valuesval = min(valuesval)
ord_det_J = (M(*coords[i])).valuation()
#if min_valuesval > 2*ord_det_J: #FB: changed this
if min(valuesval) > 0 and M(*coords[i]).valuation() == 0:
roots.append(coords[i])
roots_info.append([min_valuesval - 2*ord_det_J, ord_det_J])
elif min_valuesval > 0 :
nonroots += 1
#print 'Roots =', roots
actual_roots = []
for r in roots:
ind_roots = roots.index(r)
rt_info = roots_info[ind_roots]
if rt_info[0] == infinity:
actual_roots.append(matrix(len(flist), 1, r))
else:
variables = []
k = 0
i_l = matrix(len(flist), 1, r)
Jeval = matrix(len(flistnew), len(flistnew), [f(*r) for f in Jlist])
B = (Jeval.transpose() * Jeval)
while k < ceil(log(RR((prec-rt_info[1])/rt_info[0]))/log(2.)) + 1 and (Jeval.transpose()*Jeval).determinant() != 0: #FB:changed this
A = matrix(len(flistnew), 1, [-f(*r) for f in flistnew]);
i_l = i_l + ~B*Jeval.transpose()*A #NB: ~B*Jeval.transpose() == Jeval.inverse()
for i in (0..len(flist)-1):
variables.append(i_l[i, 0])
r = variables
variables = []
k = k+1;
Jeval = matrix(len(flistnew), len(flistnew), [f(*r) for f in Jlist])
#FB: added the following line
B = (Jeval.transpose() * Jeval)
actual_roots.append(i_l)
return actual_roots, nonroots
def two_variable_padic_system_solver(G, H, p, prec1, prec2):
r"""
Solve systems of two `p`-adic polynomials in two variables
by combining naive lifting of roots with the multivariable
Hensel's lemma. See Appendix A, Algorithm 1 (4) of [BBBM19].
INPUT:
- ``G``, ``H`` -- polynomials over `\ZZ_p`, normalised so
that the minimal valuation of the coefficients is `0`.
- ``p`` -- the prime of the first input.
- ``prec1`` -- precision for initial naive lifting.
- ``prec2`` -- the `p`-adic precision at which we would like to
compute the roots of `G` and `H`. `prec2` should be at most
equal to the precision of the coefficients of `G` and `H`.
OUTPUT:
A tuple consisting of:
- A list `L` of common roots in `Qp(p, prec2')^2` of `G` and `H`
(each root is returned as a `2`-tuple, where the `i`-th entry
corresponds to the `i`-th variable; prec2' is an integer `\le prec2`).
- an integer `n`: the number of roots in `L` which might not lift to
a root in `\QQ_p^2` or might not lift uniquely. In particular, if `n`
is zero, then there is a bijection between `L` and the common roots of
`G` and `H`.
If `n` is positive, then the roots in `L` which force `n` to be
positive are returned modulo `p^{prec1-3}`.
EXAMPLES:
An example with no double roots modulo `p` (the same example as `hensel_lift_n`)::
sage: _.<s, t> = PolynomialRing(Qp(5, 10))
sage: f1 = s + t - 2*s*t
sage: f2 = s - t
sage: a, b = two_variable_padic_system_solver(f1,f2, 5, 4, 10)
sage: print a
[(1 + O(5^10), 1 + O(5^10)), (0, 0)]
sage: print b
0
An example with double roots modulo `p` (the same example as `hensel_lift_n`)::
sage: _.<s, t> = PolynomialRing(Qp(5, 10))
sage: f1 = s - 11* t + 5*s*t
sage: f2 = s - t
sage: a, b = two_variable_padic_system_solver(f1, f2, 5, 6, 10)
sage: print a
[(2 + O(5^5), 2 + O(5^5)), (0, 0)]
sage: print b
0
An example of an actual double root::
sage: _.<s, t> = PolynomialRing(Qp(5, 10))
sage: f1 = s*t
sage: f2 = s - t
sage: a, b = two_variable_padic_system_solver(f1, f2, 5, 6, 10)
sage: print a
[(O(5^3), O(5^3))]
sage: print b
1
"""
K = Qp(p, prec2)
sols = []
x,y = G.variables()
Zxy.<x,y> = PolynomialRing(ZZ)
gprec = Zxy(G)
hprec = Zxy(H)
#Find roots modulo p^prec1 by naive lifting
for i in range(1, prec1 + 1):
modulus_one_less = p^(i-1)
tempsols = []
temp_new_list = []
temp_fct_list = []
if i == 1:
for k in range(p):
x1 = GF(p)(k)
for j in range(p):
y1 = GF(p)(j)
if gprec(x1, y1) % p == 0:
if hprec(x1, y1) % p == 0:
tempsols.append(vector([ZZ(x1), ZZ(y1)]))
temp_fct_list.append([gprec, hprec])
temp_new_list.append(vector([ZZ(x1), ZZ(y1)]))
sols = tempsols
fct_list = temp_fct_list
new_list = temp_new_list
else:
for ind in range(len(sols)):
gnew = Zxy(fct_list[ind][0](sols[ind][0] + p*x, sols[ind][1] + p*y)/p)
hnew = Zxy(fct_list[ind][1](sols[ind][0] + p*x, sols[ind][1] + p*y)/p)
for k in range(p):
x1 = GF(p)(k)
for j in range(p):
y1 = GF(p)(j)
one = gnew(x1, y1)
if one % p == 0:
two = hnew(x1, y1)
if two % p == 0:
xnew = new_list[ind][0] + k*modulus_one_less
ynew = new_list[ind][1] + j*modulus_one_less
tempsols.append(vector([ZZ(x1), ZZ(y1)]))
temp_fct_list.append([gnew, hnew])
temp_new_list.append([xnew, ynew])
sols = tempsols
fct_list = temp_fct_list
new_list = temp_new_list
#Reduce the roots modulo prec1-3 to avoid spurious sols
sols = [(K(x) + O(p^(prec1-3)), K(y) + O(p^(prec1-3))) for (x,y) in new_list]
sols = sorted(set(sols))
#Now apply multivariable Hensel on the roots that are
#simple modulo prec1-3
flist = [G,H]
precvec = []
k = 0
for F in flist:
R1 = flist[k].parent()
F1 = R1.base_ring()
if F1.is_exact():
precision1 = prec2
else:
precision1 = F1.precision_cap()
if prec2 > F1.precision_cap():
print 'Cannot get %s digits of precision due to precision of inputs of f1; raise precision of inputs' %prec
if prec2 < F1.precision_cap():
precision1 = prec2
precvec.append(precision1)
k = k+1
precision = min(precvec);
#print 'Precision is', precision
slist = list(var('s%d' % i) for i in (1..len(flist)))
s = ''
for i in range(len(flist)):
s += str(slist[i])+','
a = s[0:len(s)-1]
R = Qp(p,precision,'capped-rel')[a]
flistnew=[]
for F in flist:
F = R(F)
flistnew.append(F)
Jlist=[]
for F in flistnew:
for cars in list(flistnew[0].parent().gens()):
Jlist.append(F.derivative(cars))
J = matrix(len(flistnew), len(flistnew), Jlist)
M = J.determinant()
from itertools import product
roots = []
roots_info = []
roots2 = []
for i in range(len(sols)):
valuesval = [(F(*sols[i]).valuation()) for F in flistnew]
min_valuesval = min(valuesval)
ord_det_J = (M(*sols[i])).valuation()
if min_valuesval > 2*ord_det_J:
roots.append(sols[i])
roots_info.append([min_valuesval - 2*ord_det_J,ord_det_J])
else:
roots2.append(sols[i])
actual_roots = list(roots2)
for r in roots:
ind_roots = roots.index(r)
rt_info = roots_info[ind_roots]
if rt_info[0] == infinity:
actual_roots.append((K(ZZ(r[0])),K(ZZ(r[1]))))
else:
ind_roots = roots.index(r)
rt_info = roots_info[ind_roots]
variables = []
k = 0
r = [ZZ(r[0]),ZZ(r[1])]
i_l = matrix(len(flist),1,r)
Jeval = matrix(len(flistnew),len(flistnew),[f(*r) for f in Jlist])
B=(Jeval.transpose() * Jeval)
while k < ceil(log(RR((prec2-rt_info[1])/rt_info[0]))/log(2.))+1 and (Jeval.transpose()*Jeval).determinant() != 0:
A = matrix(len(flistnew),1,[-f(*r) for f in flistnew]);
i_l = i_l + ~B*Jeval.transpose()*A
for i in (0..len(flist)-1):
variables.append(i_l[i,0])
r = variables
variables = []
k = k+1;
Jeval = matrix(len(flistnew), len(flistnew), [f(*r) for f in Jlist])
#FB: added the following line
B = (Jeval.transpose() * Jeval)
actual_roots.append((K(i_l[0][0]), K(i_l[1][0])))
return actual_roots, len(roots2)