Skip to content
This repository has been archived by the owner on Jan 30, 2023. It is now read-only.

Commit

Permalink
trac #9320 trying to add documentation, still needs work
Browse files Browse the repository at this point in the history
  • Loading branch information
Frédéric Chapoton committed Mar 16, 2014
1 parent e63e7b2 commit 87938e0
Show file tree
Hide file tree
Showing 2 changed files with 94 additions and 64 deletions.
157 changes: 94 additions & 63 deletions src/sage/schemes/elliptic_curves/ell_number_field.py
View file
Original file line number Diff line number Diff line change
Expand Up @@ -1731,28 +1731,32 @@ def _root_number_local(E, P):
r"""
Returns the local root number of this elliptic curve at `P`.
ALGORITHM: The computation for primes not dividing 2 or 3 is based on:
ALGORITHM:
The computation for primes not dividing 2 or 3 is based on:
"Galois theory, elliptic curves, and root numbers" -- D. Rohrlich
INPUT:
- ``E`` - an elliptic curve
- ``P``- a prime ideal
OUTPUT:
The local root number of the elliptic curve at the prime ideal, either
1 or -1.
EXAMPLES::
sage: E = EllipticCurve('19a')
sage: E._root_number_local(5)
1
sage: E._root_number_local(19)
-1
sage: E = EllipticCurve('19a')
sage: E._root_number_local(5)
1
sage: E._root_number_local(19)
-1
sage: K.<a> = NumberField(x^4+2)
sage: E = EllipticCurve(K, [1, a, 0, 1+a, 0])
sage: E._root_number_local(K.ideal(a+1))
1
"""

K = E.base_field()
Expand All @@ -1763,22 +1767,21 @@ def _root_number_local(E, P):
return 1
elif d.has_split_multiplicative_reduction():
return -1
else: # additive reduction
else: # additive reduction
p = P.gen() if K is QQ else P.smallest_integer()
j = E.j_invariant()
jv = j.valuation(P) if not K is QQ else valuation(j, p)
if jv < 0:
# potential multiplicative reduction
if P.base_ring() is ZZ:
return hilbert_symbol(-1, -E.c6(), p)
else:
return generalized_hilbert_symbol(-1, -E.c6(), P)
# return hilbert_symbol_magma(-1, -E.c6(), P)
return generalized_hilbert_symbol(-1, -E.c6(), P)
# return hilbert_symbol_magma(-1, -E.c6(), P)
else:
# potential good reduction
if p==2: # prime | 2
if p == 2: # prime | 2
return E._root_number_local_2(P)
if p==3: # prime | 3
if p == 3: # prime | 3
return E._root_number_local_3(P)

if K is QQ:
Expand All @@ -1788,13 +1791,13 @@ def _root_number_local(E, P):
f = P.residue_class_degree()
e = 12 / gcd(12, E.discriminant().valuation(P))

if f%2==0 or e==1:
if f % 2 == 0 or e == 1:
eps = 1
elif e==2 or e==6:
elif e == 2 or e == 6:
eps = legendre_symbol(-1, p)
elif e==3:
elif e == 3:
eps = legendre_symbol(-3, p)
elif e==4:
elif e == 4:
eps = legendre_symbol(-2, p)
return eps

Expand All @@ -1804,35 +1807,51 @@ def _root_number_local_3(E, P):
where `P` is a prime ideal dividing 3 at which this elliptic
curve has (bad) potential good reduction.
ALGORITHM: The computation for primes dividing 3 is based on:
"The local root number of elliptic curves with wild ramification" -- S. Kobayashi
ALGORITHM:
INPUT:
The computation for primes dividing 3 is based on: "The local
root number of elliptic curves with wild ramification" --
S. Kobayashi
- ``E`` - an elliptic curve
INPUT:
- ``P`` - a prime ideal dividing 3
OUTPUT:
The local root number of the elliptic curve at the prime ideal.
EXAMPLES::
sage: E = EllipticCurve('19a')
sage: E._root_number_local_3(ZZ.ideal(3))
1
sage: E = EllipticCurve('289a')
sage: E._root_number_local_3(ZZ.ideal(3))
1
sage: K.<a> = NumberField(x^4+2)
sage: E = EllipticCurve(K, [1, a, 0, 1+a, 0])
sage: E._root_number_local_3(K.ideal(a+1))
1
"""
def _val(a, P):
if P.base_ring() is ZZ:
return valuation(a, P.gen())
return a.valuation(P)

def quadr_residue_symbol(a):
return Kr(a).is_square() and 1 or -1

def quadr_symbol(a):
av = _val(a, P)
if av%2==1:
if av % 2 == 1:
return -1
if P.base_ring() is ZZ:
pi = P.gen()
else:
pi = P.number_field().uniformizer(P)
b = a / pi**av
b = a / pi ** av
return quadr_residue_symbol(b)
# assume potential good reduction
Es = E.local_data(P, algorithm='generic').minimal_model(reduce=False)
Expand All @@ -1843,51 +1862,64 @@ def quadr_symbol(a):
Kr = P.residue_field()

ks = str(Es.kodaira_symbol(P))
if ks=='I0' or ks=='I0*':
assert vD%2==0, "error -- valuation of discriminant should be even"
return quadr_residue_symbol(-1)**(vD/2)
elif ks=='III' or ks=='III*':
if ks == 'I0' or ks == 'I0*':
assert vD % 2 == 0, "error -- valuation of discriminant should be even"
return quadr_residue_symbol(-1) ** (vD / 2)
elif ks == 'III' or ks == 'III*':
return quadr_residue_symbol(-2)
else:
assert ks=='II' or ks=='II*' or ks=='IV' or ks=='IV*', "error -- unexpected kodeira symbol"
else:
assert ks == 'II' or ks == 'II*' or ks == 'IV' or ks == 'IV*', "error -- unexpected kodaira symbol"
d = quadr_symbol(disc)
c = Es.a6()
cv = _val(c, P)
assert cv%3!=0, "error -- 3 should not divide the valuation of the constant term anymore"
assert cv % 3 != 0, "error -- 3 should not divide the valuation of the constant term anymore"
if P.base_ring() is ZZ:
hs = hilbert_symbol(disc, c, P.gen())
else:
hs = generalized_hilbert_symbol(disc, c, P)
qr1 = quadr_residue_symbol(cv)**vD
qr2 = quadr_residue_symbol(-1)**(vD*(vD-1)/2)
return d*hs*qr1*qr2

qr1 = quadr_residue_symbol(cv) ** vD
qr2 = quadr_residue_symbol(-1) ** (vD * (vD - 1) / 2)
return d * hs * qr1 * qr2

def _root_number_local_2(E, P):
r"""
Returns the local root number of this elliptic curve at `P`
where `P` is a prime ideal dividing 2 at which this elliptic
curve has (bad) potential good reduction.
ALGORITHM: The computation for primes dividing 2 is based on:
"Root numbers and parity ranks of elliptic curves" -- T. and V. Dokchitser
ALGORITHM:
INPUT:
The computation for primes dividing 2 is based on: "Root
numbers and parity ranks of elliptic curves" -- T. and
V. Dokchitser
- ``E`` - an elliptic curve
INPUT:
- ``P`` - a prime ideal dividing 2
OUTPUT:
The local root number of the elliptic curve at the prime ideal.
EXAMPLES::
sage: E = EllipticCurve('19a')
sage: E._root_number_local_2(ZZ.ideal(2))
1
sage: E = EllipticCurve('289a')
sage: E._root_number_local_2(ZZ.ideal(2))
-1
sage: K.<a> = NumberField(x^4+2)
sage: E = EllipticCurve(K, [1, a, 0, 1+a, 0])
sage: E._root_number_local_2(K.ideal(a))
1
"""
K = E.base_field()
rn = -1 if K is QQ else (-1)**(P.residue_class_degree() * P.ramification_index())
rn = -1 if K is QQ else (-1) ** (P.residue_class_degree() * P.ramification_index())

t = polygen(K)
f = t**3+E.a2()*t**2+E.a4()*t+E.a6() + (E.a1()*t+E.a3())**2/4
f = t ** 3 + E.a2() * t ** 2+E.a4()*t + E.a6() + (E.a1()*t+E.a3())**2/4

def _val(a, P):
if P.base_ring() is ZZ:
Expand All @@ -1897,24 +1929,26 @@ def _val(a, P):
def _change_ring(F, els):
L = F.absolute_field('l')
fromL, toL = L.structure()
PL = L.ideal([ toL(pgen) for pgen in P.gens() ])
EL = EllipticCurve(L, [ toL(a) for a in E.ainvs() ])
return (EL, PL, [ toL(e) for e in els ])
PL = L.ideal([toL(pgen) for pgen in P.gens()])
EL = EllipticCurve(L, [toL(a) for a in E.ainvs()])
return (EL, PL, [toL(e) for e in els])

def _H(E, PO, ell=2):
H = 1
L = [(ZZ.ideal(2),1)] if PO.base_ring() is ZZ else PO.factor()
for P,_ in L:
L = [(ZZ.ideal(2), 1)] if PO.base_ring() is ZZ else PO.factor()
for P, _ in L:
t = E.tamagawa_number(P)
cv = valuation(t,ell) + (0 if ell==3 else _u_of_E(E, P))
if cv%2==1: H *= -1
cv = valuation(t, ell) + (0 if ell == 3 else _u_of_E(E, P))
if cv % 2 == 1:
H *= -1
return H

def _Hp(E, r, PO, ell=2):
# translate with two isogeny
E = E.change_weierstrass_model(1,r,0,0)
E = E.change_weierstrass_model(1, r, 0, 0)
assert E.a1() == 0 and E.a3() == 0
E = EllipticCurve(E.base_field(), [0,-2*E.a2(),0,E.a2()**2-4*E.a4(),0])
E = EllipticCurve(E.base_field(), [0, -2 * E.a2(), 0,
E.a2() ** 2 - 4 * E.a4(), 0])
return _H(E, PO, ell)

def _u_of_E(E, P):
Expand All @@ -1923,36 +1957,33 @@ def _u_of_E(E, P):
discmin = Emin.discriminant()
if E.base_field() is QQ:
return Integer(_val(disc/discmin, P) / 12)
else:
return Integer(P.residue_class_degree() * _val(disc/discmin, P) / 12)

return Integer(P.residue_class_degree() * _val(disc/discmin, P) / 12)

disc = E.discriminant()
roots = f.roots(multiplicities=False)
if len(roots) == 3: # d=1
if len(roots) == 3: # d=1
return rn * _H (E, P) * _Hp(E, roots[0], P) * _Hp(E, roots[1], P) * _Hp(E, roots[2], P)

elif len(roots) == 1: # d=2
E = E.change_weierstrass_model(1,roots[0],0,0) # E: y^2 = ... x
elif len(roots) == 1: # d=2
E = E.change_weierstrass_model(1, roots[0], 0, 0) # E: y^2 = ... x
K2 = K.extension(t**2+E.a2()*t+E.a4(), 'a2')
EL, PL, [rL] = _change_ring(K2, [K2('a2')])
return rn * _H (E, P) * _Hp(E, 0, P) * _Hp(EL, 0, PL) * _Hp(EL, rL, PL)

elif disc.is_square(): # d=3
elif disc.is_square(): # d=3
K3 = K.extension(f, 'a3')
EL, PL, [rL] = _change_ring(K3, [K3('a3')])
return rn * _H(EL, PL) * _Hp(EL, rL, PL)

else: # d=6
K2 = K.extension(t**2-disc, 'a2')
else: # d=6
K2 = K.extension(t ** 2 - disc, 'a2')
K3 = K.extension(f, 'a3')
K6 = K3.extension(t**2-disc, 'a6')
EM, PM, [] = _change_ring(K2, [])
K6 = K3.extension(t ** 2 - disc, 'a6')
EM, PM, [] = _change_ring(K2, [])
EL, PL, [rL] = _change_ring(K3, [K3('a3')])
EF, PF, [rF] = _change_ring(K6, [K3('a3')])
return rn * _H(EL, PL) * _Hp(EL, rL, PL) * _H(EF, PF) * _H(EF, PF, 3) * _Hp(EF, rF, PF) * _H(EM, PM, 3)


def _torsion_bound(self,number_of_places = 20):
r"""
An upper bound on the order of the torsion subgroup.
Expand Down
1 change: 0 additions & 1 deletion src/sage/schemes/elliptic_curves/ell_rational_field.py
View file
Original file line number Diff line number Diff line change
Expand Up @@ -91,7 +91,6 @@
from sage.functions.other import gamma_inc
from math import sqrt
from sage.interfaces.all import gp
from sage.misc.cachefunc import cached_method
from copy import copy

Q = RationalField()
Expand Down

0 comments on commit 87938e0

Please sign in to comment.