def galois_action(self, t, N):
r"""
Suppose this cusp is `\alpha`, `G` a congruence subgroup of level `N`
and `\sigma` is the automorphism in the Galois group of
`\QQ(\zeta_N)/\QQ` that sends `\zeta_N` to `\zeta_N^t`. Then this
function computes a cusp `\beta` such that `\sigma([\alpha]) = [\beta]`,
where `[\alpha]` is the equivalence class of `\alpha` modulo `G`.
This code only needs as input the level and not the group since the
action of galois for a congruence group `G` of level `N` is compatible
with the action of the full congruence group `\Gamma(N)`.
INPUT:
- `t` -- integer that is coprime to N
- `N` -- positive integer (level)
OUTPUT:
- a cusp
.. WARNING::
In some cases `N` must fit in a long long, i.e., there
are cases where this algorithm isn't fully implemented.
.. NOTE::
Modular curves can have multiple non-isomorphic models over `\QQ`.
The action of galois depends on such a model. The model over `\QQ`
of `X(G)` used here is the model where the function field
`\QQ(X(G))` is given by the functions whose fourier expansion at
`\infty` have their coefficients in `\QQ`. For `X(N):=X(\Gamma(N))`
the corresponding moduli interpretation over `\ZZ[1/N]` is that
`X(N)` parametrizes pairs `(E,a)` where `E` is a (generalized)
elliptic curve and `a: \ZZ / N\ZZ \times \mu_N \to E` is a closed
immersion such that the weil pairing of `a(1,1)` and `a(0,\zeta_N)`
is `\zeta_N`. In this parameterisation the point `z \in H`
corresponds to the pair `(E_z,a_z)` with `E_z=\CC/(z \ZZ+\ZZ)` and
`a_z: \ZZ / N\ZZ \times \mu_N \to E` given by `a_z(1,1) = z/N` and
`a_z(0,\zeta_N) = 1/N`.
Similarly `X_1(N):=X(\Gamma_1(N))` parametrizes pairs `(E,a)` where
`a: \mu_N \to E` is a closed immersion.
EXAMPLES::
sage: Cusp(1/10).galois_action(3, 50)
1/170
sage: Cusp(oo).galois_action(3, 50)
Infinity
sage: c=Cusp(0).galois_action(3, 50); c
50/67
sage: Gamma0(50).reduce_cusp(c)
0
Here we compute the permutations of the action for t=3 on cusps for
Gamma0(50). ::
sage: N = 50; t=3; G = Gamma0(N); C = G.cusps()
sage: cl = lambda z: exists(C, lambda y:y.is_gamma0_equiv(z, N))[1]
sage: for i in range(5): print i, t^i, [cl(alpha.galois_action(t^i,N)) for alpha in C]
0 1 [0, 1/25, 1/10, 1/5, 3/10, 2/5, 1/2, 3/5, 7/10, 4/5, 9/10, Infinity]
1 3 [0, 1/25, 7/10, 2/5, 1/10, 4/5, 1/2, 1/5, 9/10, 3/5, 3/10, Infinity]
2 9 [0, 1/25, 9/10, 4/5, 7/10, 3/5, 1/2, 2/5, 3/10, 1/5, 1/10, Infinity]
3 27 [0, 1/25, 3/10, 3/5, 9/10, 1/5, 1/2, 4/5, 1/10, 2/5, 7/10, Infinity]
4 81 [0, 1/25, 1/10, 1/5, 3/10, 2/5, 1/2, 3/5, 7/10, 4/5, 9/10, Infinity]
TESTS:
Here we check that the galois action is indeed a permutation on the
cusps of Gamma1(48) and check that :trac:`13253` is fixed. ::
sage: G=Gamma1(48)
sage: C=G.cusps()
sage: for i in Integers(48).unit_gens():
... C_permuted = [G.reduce_cusp(c.galois_action(i,48)) for c in C]
... assert len(set(C_permuted))==len(C)
We test that Gamma1(19) has 9 rational cusps and check that :trac:`8998`
is fixed. ::
sage: G = Gamma1(19)
sage: [c for c in G.cusps() if c.galois_action(2,19).is_gamma1_equiv(c,19)[0]]
[2/19, 3/19, 4/19, 5/19, 6/19, 7/19, 8/19, 9/19, Infinity]
REFERENCES:
- Section 1.3 of Glenn Stevens, "Arithmetic on Modular Curves"
- There is a long comment about our algorithm in the source code for this function.
AUTHORS:
- William Stein, 2009-04-18
"""
if self.is_infinity(): return self
if not isinstance(t, Integer): t = Integer(t)
# Our algorithm for computing the Galois action works as
# follows (see Section 1.3 of Glenn Stevens "Arithmetic on
# Modular Curves" for a proof that the action given below is
# correct). We alternatively view the set of cusps as the
# Gamma-equivalence classes of column vectors [a;b] with
# gcd(a,b,N)=1, and the left action of Gamma by matrix
# multiplication. The action of t is induced by [a;b] |-->
# [a;t'*b], where t' is an inverse mod N of t. For [a;t'*b]
# with gcd(a,t'*b)==1, the cusp corresponding to [a;t'*b] is
# just the rational number a/(t'*b). Thus in this case, to
# compute the action of t we just do a/b <--> [a;b] |--->
# [a;t'*b] <--> a/(t'*b). IN the other case when we get
# [a;t'*b] with gcd(a,t'*b) != 1, which can and does happen,
# we have to work a bit harder. We need to find [c;d] such
# that [c;d] is congruent to [a;t'*b] modulo N, and
# gcd(c,d)=1. There is a standard lifting algorithm that is
# implemented for working with P^1(Z/NZ) [it is needed for
# modular symbols algorithms], so we just apply it to lift
# [a,t'*b] to a matrix [A,B;c,d] in SL_2(Z) with lower two
# entries congruent to [a,t'*b] modulo N. This exactly solves
# our problem, since gcd(c,d)=1.
a = self.__a
b = self.__b * t.inverse_mod(N)
if b.gcd(a) != 1:
_,_,a,b = lift_to_sl2z_llong(a,b,N)
a = Integer(a); b = Integer(b)
# Now that we've computed the Galois action, we efficiently
# construct the corresponding cusp as a Cusp object.
return Cusp(a,b,check=False)
def galois_action(self, t, N):
r"""
Suppose this cusp is `\alpha`, `G` a congruence subgroup of level `N`
and `\sigma` is the automorphism in the Galois group of
`\QQ(\zeta_N)/\QQ` that sends `\zeta_N` to `\zeta_N^t`. Then this
function computes a cusp `\beta` such that `\sigma([\alpha]) = [\beta]`,
where `[\alpha]` is the equivalence class of `\alpha` modulo `G`.
This code only needs as input the level and not the group since the
action of Galois for a congruence group `G` of level `N` is compatible
with the action of the full congruence group `\Gamma(N)`.
INPUT:
- `t` -- integer that is coprime to N
- `N` -- positive integer (level)
OUTPUT:
- a cusp
.. WARNING::
In some cases `N` must fit in a long long, i.e., there
are cases where this algorithm isn't fully implemented.
.. NOTE::
Modular curves can have multiple non-isomorphic models over `\QQ`.
The action of Galois depends on such a model. The model over `\QQ`
of `X(G)` used here is the model where the function field
`\QQ(X(G))` is given by the functions whose Fourier expansion at
`\infty` have their coefficients in `\QQ`. For `X(N):=X(\Gamma(N))`
the corresponding moduli interpretation over `\ZZ[1/N]` is that
`X(N)` parametrizes pairs `(E,a)` where `E` is a (generalized)
elliptic curve and `a: \ZZ / N\ZZ \times \mu_N \to E` is a closed
immersion such that the Weil pairing of `a(1,1)` and `a(0,\zeta_N)`
is `\zeta_N`. In this parameterisation the point `z \in H`
corresponds to the pair `(E_z,a_z)` with `E_z=\CC/(z \ZZ+\ZZ)` and
`a_z: \ZZ / N\ZZ \times \mu_N \to E` given by `a_z(1,1) = z/N` and
`a_z(0,\zeta_N) = 1/N`.
Similarly `X_1(N):=X(\Gamma_1(N))` parametrizes pairs `(E,a)` where
`a: \mu_N \to E` is a closed immersion.
EXAMPLES::
sage: Cusp(1/10).galois_action(3, 50)
1/170
sage: Cusp(oo).galois_action(3, 50)
Infinity
sage: c=Cusp(0).galois_action(3, 50); c
50/67
sage: Gamma0(50).reduce_cusp(c)
0
Here we compute the permutations of the action for t=3 on cusps for
Gamma0(50). ::
sage: N = 50; t=3; G = Gamma0(N); C = G.cusps()
sage: cl = lambda z: exists(C, lambda y:y.is_gamma0_equiv(z, N))[1]
sage: for i in range(5):
....: print((i, t^i))
....: print([cl(alpha.galois_action(t^i,N)) for alpha in C])
(0, 1)
[0, 1/25, 1/10, 1/5, 3/10, 2/5, 1/2, 3/5, 7/10, 4/5, 9/10, Infinity]
(1, 3)
[0, 1/25, 7/10, 2/5, 1/10, 4/5, 1/2, 1/5, 9/10, 3/5, 3/10, Infinity]
(2, 9)
[0, 1/25, 9/10, 4/5, 7/10, 3/5, 1/2, 2/5, 3/10, 1/5, 1/10, Infinity]
(3, 27)
[0, 1/25, 3/10, 3/5, 9/10, 1/5, 1/2, 4/5, 1/10, 2/5, 7/10, Infinity]
(4, 81)
[0, 1/25, 1/10, 1/5, 3/10, 2/5, 1/2, 3/5, 7/10, 4/5, 9/10, Infinity]
TESTS:
Here we check that the Galois action is indeed a permutation on the
cusps of Gamma1(48) and check that :trac:`13253` is fixed. ::
sage: G = Gamma1(48)
sage: C = G.cusps()
sage: for i in Integers(48).unit_gens():
....: C_permuted = [G.reduce_cusp(c.galois_action(i,48)) for c in C]
....: assert len(set(C_permuted))==len(C)
We test that Gamma1(19) has 9 rational cusps and check that :trac:`8998`
is fixed. ::
sage: G = Gamma1(19)
sage: [c for c in G.cusps() if c.galois_action(2,19).is_gamma1_equiv(c,19)[0]]
[2/19, 3/19, 4/19, 5/19, 6/19, 7/19, 8/19, 9/19, Infinity]
REFERENCES:
- Section 1.3 of Glenn Stevens, "Arithmetic on Modular Curves"
- There is a long comment about our algorithm in the source code for this function.
AUTHORS:
- William Stein, 2009-04-18
"""
if self.is_infinity():
return self
if not isinstance(t, Integer):
t = Integer(t)
# Our algorithm for computing the Galois action works as
# follows (see Section 1.3 of Glenn Stevens "Arithmetic on
# Modular Curves" for a proof that the action given below is
# correct). We alternatively view the set of cusps as the
# Gamma-equivalence classes of column vectors [a;b] with
# gcd(a,b,N)=1, and the left action of Gamma by matrix
# multiplication. The action of t is induced by [a;b] |-->
# [a;t'*b], where t' is an inverse mod N of t. For [a;t'*b]
# with gcd(a,t'*b)==1, the cusp corresponding to [a;t'*b] is
# just the rational number a/(t'*b). Thus in this case, to
# compute the action of t we just do a/b <--> [a;b] |--->
# [a;t'*b] <--> a/(t'*b). IN the other case when we get
# [a;t'*b] with gcd(a,t'*b) != 1, which can and does happen,
# we have to work a bit harder. We need to find [c;d] such
# that [c;d] is congruent to [a;t'*b] modulo N, and
# gcd(c,d)=1. There is a standard lifting algorithm that is
# implemented for working with P^1(Z/NZ) [it is needed for
# modular symbols algorithms], so we just apply it to lift
# [a,t'*b] to a matrix [A,B;c,d] in SL_2(Z) with lower two
# entries congruent to [a,t'*b] modulo N. This exactly solves
# our problem, since gcd(c,d)=1.
a = self.__a
b = self.__b * t.inverse_mod(N)
if b.gcd(a) != 1:
_, _, a, b = lift_to_sl2z_llong(a, b, N)
a = Integer(a)
b = Integer(b)
# Now that we've computed the Galois action, we efficiently
# construct the corresponding cusp as a Cusp object.
return Cusp(a, b, check=False)
def galois_action(self, t, N):
r"""
Suppose this cusp is `\alpha`, `G` is the congruence subgroup
`\Gamma_0(N)` of level `N`, and `\sigma` is the automorphism in
the Galois group of `\QQ(\zeta_N)/\QQ` that sends `\zeta_N` to
`\zeta_N^t`. Then this function computes a cusp `\beta` such
that `\sigma([\alpha]) = [\beta]`, where `[\alpha]` is the
equivalence class of `\alpha` modulo `G`.
BIG WARNING: Despite its general name, this function only
computes something that defines an action on `\Gamma_0(N)`
equivalence classes; it does not compute the action on
`\Gamma_1(N)` classes (see trac 8998).
INPUT:
- `t` -- integer that is coprime to N
- `N` -- positive integer (level)
OUTPUT:
- a cusp
EXAMPLES::
sage: Cusp(1/10).galois_action(3, 50)
1/170
sage: Cusp(oo).galois_action(3, 50)
Infinity
sage: Cusp(0).galois_action(3, 50)
0
Here we compute explicitly the permutations of the action for
t=3 on cusps for Gamma0(50)::
sage: N = 50; t=3; G = Gamma0(N); C = G.cusps()
sage: cl = lambda z: exists(C, lambda y:y.is_gamma0_equiv(z, N))[1]
sage: for i in range(5): print i, t^i, [cl(alpha.galois_action(t^i,N)) for alpha in C]
0 1 [0, 1/25, 1/10, 1/5, 3/10, 2/5, 1/2, 3/5, 7/10, 4/5, 9/10, Infinity]
1 3 [0, 1/25, 7/10, 2/5, 1/10, 4/5, 1/2, 1/5, 9/10, 3/5, 3/10, Infinity]
2 9 [0, 1/25, 9/10, 4/5, 7/10, 3/5, 1/2, 2/5, 3/10, 1/5, 1/10, Infinity]
3 27 [0, 1/25, 3/10, 3/5, 9/10, 1/5, 1/2, 4/5, 1/10, 2/5, 7/10, Infinity]
4 81 [0, 1/25, 1/10, 1/5, 3/10, 2/5, 1/2, 3/5, 7/10, 4/5, 9/10, Infinity]
REFERENCES:
- Section 1.3 of Glenn Stevens, "Arithmetic on Modular Curves"
- There is a long comment about our algorithm in the source code for this function.
WARNING: In some cases `N` must fit in a long long, i.e., there
are cases where this algorithm isn't fully implemented.
AUTHORS:
- William Stein, 2009-04-18
"""
if self.is_infinity() or not self.__a: return self
if not isinstance(t, Integer): t = Integer(t)
# Our algorithm for computing the Galois action works as
# follows (see Section 1.3 of Glenn Stevens "Arithmetic on
# Modular Curves" for a proof that the action given below is
# correct). We alternatively view the set of cusps as the
# Gamma-equivalence classes of column vectors [a;b] with
# gcd(a,b,N)=1, and the left action of Gamma by matrix
# multiplication. The action of t is induced by [a;b] |-->
# [a;t'*b], where t' is an inverse mod N of t. For [a;t'*b]
# with gcd(a,t'*b)==1, the cusp corresponding to [a;t'*b] is
# just the rational number a/(t'*b). Thus in this case, to
# compute the action of t we just do a/b <--> [a;b] |--->
# [a;t'*b] <--> a/(t'*b). IN the other case when we get
# [a;t'*b] with gcd(a,t'*b) != 1, which can and does happen,
# we have to work a bit harder. We need to find [c;d] such
# that [c;d] is congruent to [a;t'*b] modulo N, and
# gcd(c,d)=1. There is a standard lifting algorithm that is
# implemented for working with P^1(Z/NZ) [it is needed for
# modular symbols algorithms], so we just apply it to lift
# [a,t'*b] to a matrix [A,B;c,d] in SL_2(Z) with lower two
# entries congruent to [a,t'*b] modulo N. This exactly solves
# our problem, since gcd(c,d)=1.
a = self.__a
b = self.__b * t.inverse_mod(N)
if b.gcd(a) != 1:
_,_,a,b = lift_to_sl2z_llong(a,b,N)
a = Integer(a); b = Integer(b)
# Now that we've computed the Galois action, we efficiently
# construct the corresponding cusp as a Cusp object.
return Cusp(a,b,check=False)