def central_character(self):
r"""
Return the central character of this representation. This is the
restriction to `\QQ_p^\times` of the unique smooth character `\omega`
of `\mathbf{A}^\times / \QQ^\times` such that `\omega(\varpi_\ell) =
\ell^j \varepsilon(\ell)` for all primes `\ell \nmid Np`, where
`\varpi_\ell` is a uniformiser at `\ell`, `\varepsilon` is the
Nebentypus character of the newform `f`, and `j` is the twist factor
(see the documentation for :func:`~LocalComponent`).
EXAMPLES::
sage: LocalComponent(Newform('27a'), 3).central_character()
Character of Q_3*, of level 0, mapping 3 |--> 1
sage: LocalComponent(Newforms(Gamma1(5), 5, names='c')[0], 5).central_character()
Character of Q_5*, of level 1, mapping 2 |--> c0 + 1, 5 |--> 125
sage: LocalComponent(Newforms(DirichletGroup(24)([1, -1,-1]), 3, names='a')[0], 2).central_character()
Character of Q_2*, of level 3, mapping 7 |--> 1, 5 |--> -1, 2 |--> -2
"""
from sage.arith.all import crt
chi = self.newform().character()
f = self.prime() ** self.conductor()
N = self.newform().level() // f
G = DirichletGroup(f, self.coefficient_field())
chip = G([chi(crt(ZZ(x), 1, f, N)) for x in G.unit_gens()]).primitive_character()
a = crt(1, self.prime(), f, N)
if chip.conductor() == 1:
return SmoothCharacterGroupQp(self.prime(), self.coefficient_field()).character(0, [chi(a) * self.prime()**self.twist_factor()])
else:
return SmoothCharacterGroupQp(self.prime(), self.coefficient_field()).character(chip.conductor().valuation(self.prime()), list((~chip).values_on_gens()) + [chi(a) * self.prime()**self.twist_factor()])
def central_character(self):
r"""
Return the central character of this representation. This is the
restriction to `\QQ_p^\times` of the unique smooth character `\omega`
of `\mathbf{A}^\times / \QQ^\times` such that `\omega(\varpi_\ell) =
\ell^j \varepsilon(\ell)` for all primes `\ell \nmid Np`, where
`\varpi_\ell` is a uniformiser at `\ell`, `\varepsilon` is the
Nebentypus character of the newform `f`, and `j` is the twist factor
(see the documentation for :func:`~LocalComponent`).
EXAMPLES::
sage: LocalComponent(Newform('27a'), 3).central_character()
Character of Q_3*, of level 0, mapping 3 |--> 1
sage: LocalComponent(Newforms(Gamma1(5), 5, names='c')[0], 5).central_character()
Character of Q_5*, of level 1, mapping 2 |--> c0 + 1, 5 |--> 125
sage: LocalComponent(Newforms(DirichletGroup(24)([1, -1,-1]), 3, names='a')[0], 2).central_character()
Character of Q_2*, of level 3, mapping 7 |--> 1, 5 |--> -1, 2 |--> -2
"""
from sage.arith.all import crt
chi = self.newform().character()
f = self.prime() ** self.conductor()
N = self.newform().level() // f
G = DirichletGroup(f, self.coefficient_field())
chip = G([chi(crt(ZZ(x), 1, f, N)) for x in G.unit_gens()]).primitive_character()
a = crt(1, self.prime(), f, N)
if chip.conductor() == 1:
return SmoothCharacterGroupQp(self.prime(), self.coefficient_field()).character(0, [chi(a) * self.prime()**self.twist_factor()])
else:
return SmoothCharacterGroupQp(self.prime(), self.coefficient_field()).character(chip.conductor().valuation(self.prime()), list((~chip).values_on_gens()) + [chi(a) * self.prime()**self.twist_factor()])
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).
EXAMPLES::
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_ramified(g, p, u, n):
r"""
Given four integers `a,b,c,d` with `p \mid c` and `ad - bc = 1 \pmod{p^u}`,
find `a',b',c',d'` congruent to `a,b,c,d \pmod{p^u}`, with `c' = c
\pmod{p^{u+1}}`, such that `a'd' - b'c'` is exactly 1, and `\begin{pmatrix}
a & b \\ c & d \end{pmatrix}` is in `\Gamma_1(n)`.
Algorithm: Uses :func:`~lift_to_gamma1` to get a lifting modulo `p^u`, and
then adds an appropriate multiple of the top row to the bottom row in order
to get the bottom-left entry correct modulo `p^{u+1}`.
EXAMPLES::
sage: from sage.modular.local_comp.liftings import lift_ramified
sage: lift_ramified([2,2,3,2], 3, 1, 1)
[5, 8, 3, 5]
sage: lift_ramified([8,2,12,2], 3, 2, 23)
[323, 110, -133584, -45493]
sage: type(lift_ramified([8,2,12,2], 3, 2, 23)[0])
<type 'sage.rings.integer.Integer'>
"""
a,b,c,d = lift_to_gamma1(g, p**u, n)
r = crt( (c - g[2]) / p**u * inverse_mod(a, p), 0, p, n)
c = c - p**u * r * a
d = d - p**u * r * b
# assert (c - g[2]) % p**(u+1) == 0
return [a,b,c,d]
def parallel_function_combination(point_p_max):
r"""
Function used in parallel computation, computes rational
points lifted.
"""
rat_points = set()
for tupl in xmrange(len_modulo_points):
point = []
for k in range(N):
# lift all coordinates of given point using chinese remainder theorem
L = [
modulo_points[j][tupl[j]][k].lift()
for j in range(len_primes - 1)
]
L.append(point_p_max[k].lift())
point.append(crt(L, primes_list))
for i in range(num_comp):
for j in range(comp_dim_relative[i]):
m[i][j] = point[dim_prefix[i] + j]
# generating matrix to compute LLL reduction for each component
M = [matrix(ZZ, comp_dim_relative[i] + 1, comp_dim_relative[i], m[i]) \
for i in range(num_comp)]
A = [M[i].LLL() for i in range(num_comp)]
point = []
for i in range(num_comp):
point.extend(A[i][1])
# check if all coordinates of this point satisfy height bound
bound_satisfied = True
for coordinate in point:
if coordinate.abs() > bound:
bound_satisfied = False
break
if not bound_satisfied:
continue
try:
rat_points.add(
X(point)) # checks if this point lies on X or not
except:
pass
return [list(_) for _ in rat_points]
def group(self):
r"""
Return a `\Gamma_H` group which is the level of all of the relevant
twists of `f`.
EXAMPLE::
sage: from sage.modular.local_comp.type_space import example_type_space
sage: example_type_space().group()
Congruence Subgroup Gamma_H(98) with H generated by [57]
"""
p = self.prime()
r = self.conductor()
d = max(self.character_conductor(), r // 2)
n = self.tame_level()
chi = self.form().character()
tame_H = [i for i in chi.kernel() if (i % p**r) == 1]
wild_H = [crt(1 + p**d, 1, p**r, n)]
return GammaH(n * p**r, tame_H + wild_H)
def group(self):
r"""
Return a `\Gamma_H` group which is the level of all of the relevant
twists of `f`.
EXAMPLE::
sage: from sage.modular.local_comp.type_space import example_type_space
sage: example_type_space().group()
Congruence Subgroup Gamma_H(98) with H generated by [57]
"""
p = self.prime()
r = self.conductor()
d = max(self.character_conductor(), r//2)
n = self.tame_level()
chi = self.form().character()
tame_H = [i for i in chi.kernel() if (i % p**r) == 1]
wild_H = [crt(1 + p**d, 1, p**r, n)]
return GammaH(n * p**r, tame_H + wild_H)
def _unif_ramified(self):
r"""
Return the action of [0,-1,p,0], in the ramified (odd p-power level)
case.
EXAMPLES::
sage: from sage.modular.local_comp.type_space import example_type_space
sage: T = example_type_space(3)
sage: T._unif_ramified()
[-1 0]
[ 0 -1]
"""
p = self.prime()
k = self.form().weight()
return (self.t_space.atkin_lehner_operator(p).matrix().transpose() *
p**(-(k - 2) * self.u()) * self.t_space.diamond_bracket_matrix(
crt(1, p**self.u(), p**self.u(),
self.tame_level())).transpose())
def parallel_function_combination(point_p_max):
r"""
Function used in parallel computation, computes rational
points lifted.
"""
rat_points = set()
for tupl in xmrange(len_modulo_points):
point = []
for k in range(N + 1):
# lift all coordinates of given point using chinese remainder theorem
L = [
modulo_points[j][tupl[j]][k].lift()
for j in range(len_primes - 1)
]
L.append(point_p_max[k].lift())
point.append(crt(L, primes_list))
for i in range(N + 1):
m[i] = point[i]
M = matrix(ZZ, N + 2, N + 1, m)
A = M.LLL()
point = list(A[1])
# check if all coordinates of this point satisfy height bound
bound_satisfied = True
for coordinate in point:
if coordinate.abs() > bound:
bound_satisfied = False
break
if not bound_satisfied:
continue
try:
pt = X(list(A[1]))
except TypeError:
pass
else:
rat_points.add(pt)
return [list(_) for _ in rat_points]
def group(self):
r"""
Return a `\Gamma_H` group which is the level of all of the relevant
twists of `f`.
EXAMPLES::
sage: from sage.modular.local_comp.type_space import example_type_space
sage: example_type_space().group()
Congruence Subgroup Gamma_H(98) with H generated by [15, 29, 43]
"""
# Implementation here is not the most efficient but this is heavily not
# time-critical, and getting it wrong can lead to subtle bugs.
p = self.prime()
r = self.conductor()
d = max(self.character_conductor(), r // 2)
n = self.tame_level()
chi = self.form().character()
tame_H = [i for i in chi.kernel() if (i % p**r) == 1]
wild_H = [crt(x, 1, p**r, n) for x in range(p**r) if x % (p**d) == 1]
return GammaH(n * p**r, tame_H + wild_H)
def rho(self, g):
r"""
Calculate the action of the group element `g` on the type space.
EXAMPLE::
sage: from sage.modular.local_comp.type_space import example_type_space
sage: T = example_type_space(2)
sage: m = T.rho([2,0,0,1]); m
[ 1 -2 1 0]
[ 1 -1 0 1]
[ 1 0 -1 1]
[ 0 1 -2 1]
sage: v = T.eigensymbol_subspace().basis()[0]
sage: m * v == v
True
We test that it is a left action::
sage: T = example_type_space(0)
sage: a = [0,5,4,3]; b = [0,2,3,5]; ab = [1,4,2,2]
sage: T.rho(ab) == T.rho(a) * T.rho(b)
True
An odd level example::
sage: from sage.modular.local_comp.type_space import TypeSpace
sage: T = TypeSpace(Newform('54a'), 3)
sage: a = [0,1,3,0]; b = [2,1,0,1]; ab = [0,1,6,3]
sage: T.rho(ab) == T.rho(a) * T.rho(b)
True
"""
if not self.is_minimal():
raise NotImplementedError(
"Group action on non-minimal type space not implemented")
if self.u() == 0:
# silly special case: rep is principal series or special, so SL2
# action on type space is trivial
raise ValueError("Representation is not supercuspidal")
p = self.prime()
f = p**self.u()
g = [ZZ(_) for _ in g]
d = (g[0] * g[3] - g[2] * g[1])
# g is in S(K_0) (easy case)
if d % f == 1:
return self._rho_s(g)
# g is in K_0, but not in S(K_0)
if d % p != 0:
try:
a = self._a
except AttributeError:
self._discover_torus_action()
a = self._a
i = 0
while (d * a**i) % f != 1:
i += 1
if i > f: raise ArithmeticError
return self._rho_s([a**i * g[0], g[1], a**i * g[2], g[3]
]) * self._amat**(-i)
# funny business
if (self.conductor() % 2 == 0):
if all([x.valuation(p) > 0 for x in g]):
eps = self.form().character()(crt(1, p, f, self.tame_level()))
return ~eps * self.rho([x // p for x in g])
else:
raise ArithmeticError("g(={0}) not in K".format(g))
else:
m = matrix(ZZ, 2, g)
s = m.det().valuation(p)
mm = (matrix(QQ, 2, [0, -1, p, 0])**(-s) * m).change_ring(ZZ)
return self._unif_ramified()**s * self.rho(mm.list())
def rho(self, g):
r"""
Calculate the action of the group element `g` on the type space.
EXAMPLE::
sage: from sage.modular.local_comp.type_space import example_type_space
sage: T = example_type_space(2)
sage: m = T.rho([2,0,0,1]); m
[ 1 -2 1 0]
[ 1 -1 0 1]
[ 1 0 -1 1]
[ 0 1 -2 1]
sage: v = T.eigensymbol_subspace().basis()[0]
sage: m * v == v
True
We test that it is a left action::
sage: T = example_type_space(0)
sage: a = [0,5,4,3]; b = [0,2,3,5]; ab = [1,4,2,2]
sage: T.rho(ab) == T.rho(a) * T.rho(b)
True
An odd level example::
sage: from sage.modular.local_comp.type_space import TypeSpace
sage: T = TypeSpace(Newform('54a'), 3)
sage: a = [0,1,3,0]; b = [2,1,0,1]; ab = [0,1,6,3]
sage: T.rho(ab) == T.rho(a) * T.rho(b)
True
"""
if not self.is_minimal():
raise NotImplementedError( "Group action on non-minimal type space not implemented" )
if self.u() == 0:
# silly special case: rep is principal series or special, so SL2
# action on type space is trivial
raise ValueError( "Representation is not supercuspidal" )
p = self.prime()
f = p**self.u()
g = [ZZ(_) for _ in g]
d = (g[0]*g[3] - g[2]*g[1])
# g is in S(K_0) (easy case)
if d % f == 1:
return self._rho_s(g)
# g is in K_0, but not in S(K_0)
if d % p != 0:
try:
a = self._a
except AttributeError:
self._discover_torus_action()
a = self._a
i = 0
while (d * a**i) % f != 1:
i += 1
if i > f: raise ArithmeticError
return self._rho_s([a**i*g[0], g[1], a**i*g[2], g[3]]) * self._amat**(-i)
# funny business
if (self.conductor() % 2 == 0):
if all([x.valuation(p) > 0 for x in g]):
eps = self.form().character()(crt(1, p, f, self.tame_level()))
return ~eps * self.rho([x // p for x in g])
else:
raise ArithmeticError( "g(={0}) not in K".format(g) )
else:
m = matrix(ZZ, 2, g)
s = m.det().valuation(p)
mm = (matrix(QQ, 2, [0, -1, p, 0])**(-s) * m).change_ring(ZZ)
return self._unif_ramified()**s * self.rho(mm.list())