def are_equivalent(self, x, y, trans = False):
r"""
Test whether or not cusps x and y are equivalent modulo self. If self
has a reduce_cusp() method, use that; otherwise do a slow explicit
test.
If trans = False, returns True or False. If trans = True, then return
either False or an element of self mapping x onto y.
EXAMPLE::
sage: Gamma0(7).are_equivalent(Cusp(1/3), Cusp(0), trans=True)
[ 3 -1]
[-14 5]
sage: Gamma0(7).are_equivalent(Cusp(1/3), Cusp(1/7))
False
"""
x = Cusp(x)
y = Cusp(y)
if not trans:
try:
xr = self.reduce_cusp(x)
yr = self.reduce_cusp(y)
if xr != yr:
return False
if xr == yr:
return True
except NotImplementedError:
pass
from all import SL2Z
vx = lift_to_sl2z(x.numerator(),x.denominator(), 0)
dx = SL2Z([vx[2], -vx[0], vx[3], -vx[1]])
vy = lift_to_sl2z(y.numerator(),y.denominator(), 0)
dy = SL2Z([vy[2], -vy[0], vy[3], -vy[1]])
for i in xrange(self.index()):
# Note that the width of any cusp is bounded above by the index of self.
# If self is congruence, then the level of self is a much better bound, but
# this method is written to work with non-congruence subgroups as well,
if dy * SL2Z([1,i,0,1])*(~dx) in self:
if trans:
return dy * SL2Z([1,i,0,1]) * ~dx
else:
return True
elif (self.is_odd() and dy * SL2Z([-1,-i,0,-1]) * ~dx in self):
if trans:
return dy * SL2Z([-1,-i,0,-1]) * ~dx
else:
return True
return False
def are_equivalent(self, x, y, trans=False):
r"""
Test whether or not cusps x and y are equivalent modulo self. If self
has a reduce_cusp() method, use that; otherwise do a slow explicit
test.
If trans = False, returns True or False. If trans = True, then return
either False or an element of self mapping x onto y.
EXAMPLE::
sage: Gamma0(7).are_equivalent(Cusp(1/3), Cusp(0), trans=True)
[ 3 -1]
[-14 5]
sage: Gamma0(7).are_equivalent(Cusp(1/3), Cusp(1/7))
False
"""
x = Cusp(x)
y = Cusp(y)
if not trans:
try:
xr = self.reduce_cusp(x)
yr = self.reduce_cusp(y)
if xr != yr:
return False
if xr == yr:
return True
except NotImplementedError:
pass
from all import SL2Z
vx = lift_to_sl2z(x.numerator(), x.denominator(), 0)
dx = SL2Z([vx[2], -vx[0], vx[3], -vx[1]])
vy = lift_to_sl2z(y.numerator(), y.denominator(), 0)
dy = SL2Z([vy[2], -vy[0], vy[3], -vy[1]])
for i in xrange(self.index()):
# Note that the width of any cusp is bounded above by the index of self.
# If self is congruence, then the level of self is a much better bound, but
# this method is written to work with non-congruence subgroups as well,
if dy * SL2Z([1, i, 0, 1]) * (~dx) in self:
if trans:
return dy * SL2Z([1, i, 0, 1]) * ~dx
else:
return True
elif (self.is_odd() and dy * SL2Z([-1, -i, 0, -1]) * ~dx in self):
if trans:
return dy * SL2Z([-1, -i, 0, -1]) * ~dx
else:
return True
return False
def coset_reps(self):
r"""
Return representatives for the right cosets of this congruence
subgroup in `{\rm SL}_2(\ZZ)` as a generator object.
Use ``list(self.coset_reps())`` to obtain coset reps as a
list.
EXAMPLES::
sage: list(Gamma0(5).coset_reps())
[
[1 0] [ 0 -1] [1 0] [ 0 -1] [ 0 -1] [ 0 -1]
[0 1], [ 1 0], [1 1], [ 1 2], [ 1 3], [ 1 4]
]
sage: list(Gamma0(4).coset_reps())
[
[1 0] [ 0 -1] [1 0] [ 0 -1] [ 0 -1] [1 0]
[0 1], [ 1 0], [1 1], [ 1 2], [ 1 3], [2 1]
]
sage: list(Gamma0(1).coset_reps())
[
[1 0]
[0 1]
]
"""
from .all import SL2Z
N = self.level()
if N == 1: # P1List isn't very happy working modulo 1
yield SL2Z([1, 0, 0, 1])
else:
for z in sage.modular.modsym.p1list.P1List(N):
yield SL2Z(lift_to_sl2z(z[0], z[1], N))
def random_element(self, bound=100, *args, **kwds):
r"""
Return a random element of `{\rm SL}_2(\ZZ)` with entries whose
absolute value is strictly less than bound (default 100).
Additional arguments and keywords are passed to the random_element
method of ZZ.
(Algorithm: Generate a random pair of integers at most bound. If they
are not coprime, throw them away and start again. If they are, find an
element of `{\rm SL}_2(\ZZ)` whose bottom row is that, and
left-multiply it by `\begin{pmatrix} 1 & w \\ 0 & 1\end{pmatrix}` for
an integer `w` randomly chosen from a small enough range that the
answer still has entries at most bound.)
It is, unfortunately, not true that all elements of SL2Z with entries <
bound appear with equal probability; those with larger bottom rows are
favoured, because there are fewer valid possibilities for w.
EXAMPLES::
sage: SL2Z.random_element()
[60 13]
[83 18]
sage: SL2Z.random_element(5)
[-1 3]
[ 1 -4]
Passes extra positional or keyword arguments through::
sage: SL2Z.random_element(5, distribution='1/n')
[ 1 -4]
[ 0 1]
"""
if bound <= 1: raise ValueError("bound must be greater than 1")
c = ZZ.random_element(1-bound, bound, *args, **kwds)
d = ZZ.random_element(1-bound, bound, *args, **kwds)
if gcd(c,d) != 1: # try again
return self.random_element(bound, *args, **kwds)
else:
a,b,c,d = lift_to_sl2z(c,d,0)
whi = bound
wlo = bound
if c > 0:
whi = min(whi, ((bound - a)/ZZ(c)).ceil())
wlo = min(wlo, ((bound + a)/ZZ(c)).ceil())
elif c < 0:
whi = min(whi, ((bound + a)/ZZ(-c)).ceil())
wlo = min(wlo, ((bound - a)/ZZ(-c)).ceil())
if d > 0:
whi = min(whi, ((bound - b)/ZZ(d)).ceil())
wlo = min(wlo, ((bound + b)/ZZ(d)).ceil())
elif d < 0:
whi = min(whi, ((bound + b)/ZZ(-d)).ceil())
wlo = min(wlo, ((bound - b)/ZZ(-d)).ceil())
w = ZZ.random_element(1-wlo, whi, *args, **kwds)
a += c*w
b += d*w
return self([a,b,c,d])
def coset_reps(self):
r"""
Return representatives for the right cosets of this congruence
subgroup in `{\rm SL}_2(\ZZ)` as a generator object.
Use ``list(self.coset_reps())`` to obtain coset reps as a
list.
EXAMPLES::
sage: list(Gamma0(5).coset_reps())
[
[1 0] [ 0 -1] [1 0] [ 0 -1] [ 0 -1] [ 0 -1]
[0 1], [ 1 0], [1 1], [ 1 2], [ 1 3], [ 1 4]
]
sage: list(Gamma0(4).coset_reps())
[
[1 0] [ 0 -1] [1 0] [ 0 -1] [ 0 -1] [1 0]
[0 1], [ 1 0], [1 1], [ 1 2], [ 1 3], [2 1]
]
sage: list(Gamma0(1).coset_reps())
[
[1 0]
[0 1]
]
"""
from all import SL2Z
N = self.level()
if N == 1: # P1List isn't very happy working modulo 1
yield SL2Z([1,0,0,1])
else:
for z in sage.modular.modsym.p1list.P1List(N):
yield SL2Z(lift_to_sl2z(z[0], z[1], N))
def gamma0_coset_reps(self):
r"""
Return a set of coset representatives for self \\ Gamma0(N), where N is
the level of self.
EXAMPLE::
sage: GammaH(108, [1,-1]).gamma0_coset_reps()
[
[1 0] [-43 -45] [ 31 33] [-49 -54] [ 25 28] [-19 -22]
[0 1], [108 113], [108 115], [108 119], [108 121], [108 125],
<BLANKLINE>
[-17 -20] [ 47 57] [ 13 16] [ 41 52] [ 7 9] [-37 -49]
[108 127], [108 131], [108 133], [108 137], [108 139], [108 143],
<BLANKLINE>
[-35 -47] [ 29 40] [ -5 -7] [ 23 33] [-11 -16] [ 53 79]
[108 145], [108 149], [108 151], [108 155], [108 157], [108 161]
]
"""
from all import Gamma0
N = self.level()
G = Gamma0(N)
s = []
return [
G(lift_to_sl2z(0, d.lift(), N))
for d in _GammaH_coset_helper(N, self._list_of_elements_in_H())
]
def cusp_data(self, c):
r"""
Return a triple (g, w, t) where g is an element of self generating the
stabiliser of the given cusp, w is the width of the cusp, and t is 1 if
the cusp is regular and -1 if not.
EXAMPLES::
sage: Gamma1(4).cusp_data(Cusps(1/2))
(
[ 1 -1]
[ 4 -3], 1, -1
)
"""
c = Cusp(c)
# first find an element of SL2Z sending infinity to the given cusp
w = lift_to_sl2z(c.denominator(), c.numerator(), 0)
g = SL2Z([w[3], w[1], w[2], w[0]])
for d in xrange(1, 1 + self.index()):
if g * SL2Z([1, d, 0, 1]) * (~g) in self:
return (g * SL2Z([1, d, 0, 1]) * (~g), d, 1)
elif g * SL2Z([-1, -d, 0, -1]) * (~g) in self:
return (g * SL2Z([-1, -d, 0, -1]) * (~g), d, -1)
raise ArithmeticError("Can't get here!")
def random_element(self, bound=100, *args, **kwds):
r"""
Return a random element of `{\rm SL}_2(\ZZ)` with entries whose
absolute value is strictly less than bound (default 100).
Additional arguments and keywords are passed to the random_element
method of ZZ.
(Algorithm: Generate a random pair of integers at most bound. If they
are not coprime, throw them away and start again. If they are, find an
element of `{\rm SL}_2(\ZZ)` whose bottom row is that, and
left-multiply it by `\begin{pmatrix} 1 & w \\ 0 & 1\end{pmatrix}` for
an integer `w` randomly chosen from a small enough range that the
answer still has entries at most bound.)
It is, unfortunately, not true that all elements of SL2Z with entries <
bound appear with equal probability; those with larger bottom rows are
favoured, because there are fewer valid possibilities for w.
EXAMPLES::
sage: SL2Z.random_element()
[60 13]
[83 18]
sage: SL2Z.random_element(5)
[-1 3]
[ 1 -4]
Passes extra positional or keyword arguments through::
sage: SL2Z.random_element(5, distribution='1/n')
[ 1 -4]
[ 0 1]
"""
if bound <= 1: raise ValueError("bound must be greater than 1")
c = ZZ.random_element(1 - bound, bound, *args, **kwds)
d = ZZ.random_element(1 - bound, bound, *args, **kwds)
if gcd(c, d) != 1: # try again
return self.random_element(bound, *args, **kwds)
else:
a, b, c, d = lift_to_sl2z(c, d, 0)
whi = bound
wlo = bound
if c > 0:
whi = min(whi, ((bound - a) / ZZ(c)).ceil())
wlo = min(wlo, ((bound + a) / ZZ(c)).ceil())
elif c < 0:
whi = min(whi, ((bound + a) / ZZ(-c)).ceil())
wlo = min(wlo, ((bound - a) / ZZ(-c)).ceil())
if d > 0:
whi = min(whi, ((bound - b) / ZZ(d)).ceil())
wlo = min(wlo, ((bound + b) / ZZ(d)).ceil())
elif d < 0:
whi = min(whi, ((bound + b) / ZZ(-d)).ceil())
wlo = min(wlo, ((bound - b) / ZZ(-d)).ceil())
w = ZZ.random_element(1 - wlo, whi, *args, **kwds)
a += c * w
b += d * w
return self([a, b, c, d])
def cusp_data(self, c):
r"""
Return a triple (g, w, t) where g is an element of self generating the
stabiliser of the given cusp, w is the width of the cusp, and t is 1 if
the cusp is regular and -1 if not.
EXAMPLES::
sage: Gamma1(4).cusp_data(Cusps(1/2))
(
[ 1 -1]
[ 4 -3], 1, -1
)
"""
c = Cusp(c)
# first find an element of SL2Z sending infinity to the given cusp
w = lift_to_sl2z(c.denominator(), c.numerator(), 0)
g = SL2Z([w[3], w[1], w[2],w[0]])
for d in xrange(1,1+self.index()):
if g * SL2Z([1,d,0,1]) * (~g) in self:
return (g * SL2Z([1,d,0,1]) * (~g), d, 1)
elif g * SL2Z([-1,-d,0,-1]) * (~g) in self:
return (g * SL2Z([-1,-d,0,-1]) * (~g), d, -1)
raise ArithmeticError("Can't get here!")
def gamma0_coset_reps(self):
r"""
Return a set of coset representatives for self \\ Gamma0(N), where N is
the level of self.
EXAMPLES::
sage: GammaH(108, [1,-1]).gamma0_coset_reps()
[
[1 0] [-43 -2] [ 31 2] [-49 -5] [ 25 3] [-19 -3]
[0 1], [108 5], [108 7], [108 11], [108 13], [108 17],
<BLANKLINE>
[-17 -3] [ 47 10] [ 13 3] [ 41 11] [ 7 2] [-37 -12]
[108 19], [108 23], [108 25], [108 29], [108 31], [108 35],
<BLANKLINE>
[-35 -12] [ 29 11] [ -5 -2] [ 23 10] [-11 -5] [ 53 26]
[108 37], [108 41], [108 43], [108 47], [108 49], [108 53]
]
"""
from .all import SL2Z
N = self.level()
return [
SL2Z(lift_to_sl2z(0, d.lift(), N))
for d in _GammaH_coset_helper(N, self._list_of_elements_in_H())
]
def lift_to_gamma1(g, m, n):
r"""
If ``g = [a,b,c,d]`` is a list of integers defining a `2 \times 2` matrix
whose determinant is `1 \pmod m`, return a list of integers giving the
entries of a matrix which is congruent to `g \pmod m` and to
`\begin{pmatrix} 1 & * \\ 0 & 1 \end{pmatrix} \pmod n`. Here `m` and `n`
must be coprime.
Here `m` and `n` should be coprime positive integers. Either of `m` and `n`
can be `1`. If `n = 1`, this still makes perfect sense; this is what is
called by the function :func:`~lift_matrix_to_sl2z`. If `m = 1` this is a
rather silly question, so we adopt the convention of always returning the
identity matrix.
The result is always a list of Sage integers (unlike ``lift_to_sl2z``,
which tends to return Python ints).
EXAMPLE::
sage: from sage.modular.local_comp.liftings import lift_to_gamma1
sage: A = matrix(ZZ, 2, lift_to_gamma1([10, 11, 3, 11], 19, 5)); A
[371 68]
[ 60 11]
sage: A.det() == 1
True
sage: A.change_ring(Zmod(19))
[10 11]
[ 3 11]
sage: A.change_ring(Zmod(5))
[1 3]
[0 1]
sage: m = list(SL2Z.random_element())
sage: n = lift_to_gamma1(m, 11, 17)
sage: assert matrix(Zmod(11), 2, n) == matrix(Zmod(11),2,m)
sage: assert matrix(Zmod(17), 2, [n[0], 0, n[2], n[3]]) == 1
sage: type(lift_to_gamma1([10,11,3,11],19,5)[0])
<type 'sage.rings.integer.Integer'>
Tests with `m = 1` and with `n = 1`::
sage: lift_to_gamma1([1,1,0,1], 5, 1)
[1, 1, 0, 1]
sage: lift_to_gamma1([2,3,11,22], 1, 5)
[1, 0, 0, 1]
"""
if m == 1:
return [ZZ(1), ZZ(0), ZZ(0), ZZ(1)]
a, b, c, d = [ZZ(x) for x in g]
if not (a * d - b * c) % m == 1:
raise ValueError("Determinant is {0} mod {1}, should be 1".format(
(a * d - b * c) % m, m))
c2 = crt(c, 0, m, n)
d2 = crt(d, 1, m, n)
a3, b3, c3, d3 = [ZZ(_) for _ in lift_to_sl2z(c2, d2, m * n)]
r = (a3 * b - b3 * a) % m
return [a3 + r * c3, b3 + r * d3, c3, d3]
def lift_to_gamma1(g, m, n):
r"""
If ``g = [a,b,c,d]`` is a list of integers defining a `2 \times 2` matrix
whose determinant is `1 \pmod m`, return a list of integers giving the
entries of a matrix which is congruent to `g \pmod m` and to
`\begin{pmatrix} 1 & * \\ 0 & 1 \end{pmatrix} \pmod n`. Here `m` and `n`
must be coprime.
Here `m` and `n` should be coprime positive integers. Either of `m` and `n`
can be `1`. If `n = 1`, this still makes perfect sense; this is what is
called by the function :func:`~lift_matrix_to_sl2z`. If `m = 1` this is a
rather silly question, so we adopt the convention of always returning the
identity matrix.
The result is always a list of Sage integers (unlike ``lift_to_sl2z``,
which tends to return Python ints).
EXAMPLE::
sage: from sage.modular.local_comp.liftings import lift_to_gamma1
sage: A = matrix(ZZ, 2, lift_to_gamma1([10, 11, 3, 11], 19, 5)); A
[371 68]
[ 60 11]
sage: A.det() == 1
True
sage: A.change_ring(Zmod(19))
[10 11]
[ 3 11]
sage: A.change_ring(Zmod(5))
[1 3]
[0 1]
sage: m = list(SL2Z.random_element())
sage: n = lift_to_gamma1(m, 11, 17)
sage: assert matrix(Zmod(11), 2, n) == matrix(Zmod(11),2,m)
sage: assert matrix(Zmod(17), 2, [n[0], 0, n[2], n[3]]) == 1
sage: type(lift_to_gamma1([10,11,3,11],19,5)[0])
<type 'sage.rings.integer.Integer'>
Tests with `m = 1` and with `n = 1`::
sage: lift_to_gamma1([1,1,0,1], 5, 1)
[1, 1, 0, 1]
sage: lift_to_gamma1([2,3,11,22], 1, 5)
[1, 0, 0, 1]
"""
if m == 1:
return [ZZ(1),ZZ(0),ZZ(0),ZZ(1)]
a,b,c,d = [ZZ(x) for x in g]
if not (a*d - b*c) % m == 1:
raise ValueError( "Determinant is {0} mod {1}, should be 1".format((a*d - b*c) % m, m) )
c2 = crt(c, 0, m, n)
d2 = crt(d, 1, m, n)
a3,b3,c3,d3 = map(ZZ, lift_to_sl2z(c2,d2,m*n))
r = (a3*b - b3*a) % m
return [a3 + r*c3, b3 + r*d3, c3, d3]
def galois_action(self, t, N):
from sage.modular.modsym.p1list import lift_to_sl2z
if self.is_infinity():
return self
if not isinstance(t, Integer):
t = Integer(t)
a = self._Cusp__a
b = self._Cusp__b * t.inverse_mod(N)
if b.gcd(a) != ZZ(1):
_,_,a,b = lift_to_sl2z(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):
from sage.modular.modsym.p1list import lift_to_sl2z
if self.is_infinity():
return self
if not isinstance(t, Integer):
t = Integer(t)
a = self._Cusp__a
b = self._Cusp__b * t.inverse_mod(N)
if b.gcd(a) != ZZ(1):
_,_,a,b = lift_to_sl2z(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 gamma0_coset_reps(self):
r"""
Return a set of coset representatives for self \\ Gamma0(N), where N is
the level of self.
EXAMPLE::
sage: GammaH(108, [1,-1]).gamma0_coset_reps()
[[1 0] [0 1], [-43 -45] [108 113], [ 31 33] [108 115], [-49 -54]
[108 119], [ 25 28] [108 121], [-19 -22] [108 125], [-17 -20] [108
127], [ 47 57] [108 131], [ 13 16] [108 133], [ 41 52] [108
137], [ 7 9] [108 139], [-37 -49] [108 143], [-35 -47] [108
145], [ 29 40] [108 149], [ -5 -7] [108 151], [ 23 33] [108
155], [-11 -16] [108 157], [ 53 79] [108 161]]
"""
from all import Gamma0
N = self.level()
G = Gamma0(N)
s = []
return [G(lift_to_sl2z(0, d.lift(), N)) for d in _GammaH_coset_helper(N, self._list_of_elements_in_H())]
def cusp_reduction_table(self):
r'''
Returns a dictionary and the set of cusps.
Assumes we have a finite set surjecting to the cusps (namely, P^1(O_F/N)). Runs through
and computes a subset which represents the cusps, and shows how to go from any element
of P^1(O_F/N) to the chosen equivalent cusp.
Takes as input the object representing P^1(O_F/N), where F is a number field
(that is possibly Q), and N is some ideal in the field. Runs the following algorithm:
- take a remaining element C = (c:d) of P^1(O_F/N);
- add this to the set of cusps, declaring it to be our chosen rep;
- run through every translate C' = (c':d') of C under the stabiliser of infinity, and
remove this translate from the set of remaining elements;
- store the matrix T in the stabiliser such that C' * T = C (as elements in P^1)
in the dictionary, with key C'.
'''
P = self.get_P1List()
if hasattr(P.N(),'number_field'):
K = P.N().number_field()
else:
K = QQ
# from sage.modular.modsym.p1list_nf import lift_to_sl2_Ok
from .my_p1list_nf import lift_to_sl2_Ok
from sage.modular.modsym.p1list import lift_to_sl2z
## Define new function on the fly to pick which of Q/more general field we work in
## lift_to_matrix takes parameters c,d, then lifts (c:d) to a 2X2 matrix over the NF representing it
lift_to_matrix = lambda c, d: lift_to_sl2z(c,d,P.N()) if K.degree() == 1 else lift_to_sl2_Ok(P.N(), c, d)
## Put all the points of P^1(O_F/N) into a list; these will corr. to our dictionary keys
remaining_points = set(list(P)) if K == QQ else set([c.tuple() for c in P])
reduction_table = {}
cusp_set = []
initial_points = len(remaining_points)
## Loop over all points of P^1(O_F/N)
while len(remaining_points) > 0:
## Pick a new cusp representative
c = remaining_points.pop()
update_progress(1 - float(len(remaining_points)) / float(initial_points), "Finding cusps...")
## c is an MSymbol so not hashable. Create tuple that is
## Represent the cusp as a matrix, add to list of cusps, and add to dictionary
new_cusp = Matrix(2,2,lift_to_matrix(c[0], c[1]))
new_cusp.set_immutable()
cusp_set.append(new_cusp)
reduction_table[c]=(new_cusp,matrix(2,2,1)) ## Set the value to I_2
## Now run over the whole orbit of this point under the stabiliser at infinity.
## For each elt of the orbit, explain how to reduce to the chosen cusp.
## Run over lifts of elements of O_F/N:
if K == QQ:
residues = Zmod(P.N())
units = [1, -1]
else:
residues = P.N().residues()
units = K.roots_of_unity()
for hh in residues:
h = K(hh) ## put into the number field
## Run over all finite order units in the number field
for u in units:
## Now have the matrix (u,h; 0,u^-1).
## Compute the action of this matrix on c
new_c = P.normalize(u * c[0], u**-1 * c[1] + h * c[0])
if K != QQ:
new_c = new_c.tuple()
if new_c not in reduction_table:
## We've not seen this point before! But it's equivalent to c, so kill it!
## (and also store the matrix we used to get to it)
remaining_points.remove(new_c)
T = matrix(2,2,[u,h,0,u**-1]) ## we used this matrix to get from c to new_c
reduction_table[new_c]=(new_cusp, T) ## update dictionary with the new_c + the matrix
if K != QQ:
assert P.normalize(*(vector(c) * T)).tuple() == new_c ## sanity check
else:
assert P.normalize(*(vector(c) * T)) == new_c ## sanity check
return reduction_table, cusp_set