def degeneracy_matrix(self, p=None):
"""
Map from self to QuaterniocModule of level self/p.
EXAMPLES::
sage: from sage.modular.hilbert.sqrt5_hmf import F, QuaternionicModule
sage: H = QuaternionicModule(2*F.prime_above(31)); H
Quaternionic module of dimension 4, level 10*a-4 (of norm 124=2^2*31) over QQ(sqrt(5))
sage: H.degeneracy_matrix(2)
[1 0]
[0 1]
[0 1]
[0 1]
sage: H.degeneracy_matrix(F.prime_above(31))
[1]
[1]
[1]
[1]
sage: H.degeneracy_matrix()
[1 0 1]
[0 1 1]
[0 1 1]
[0 1 1]
sage: H.degeneracy_matrix() is H.degeneracy_matrix()
False
sage: H.degeneracy_matrix(2) is H.degeneracy_matrix(2)
True
"""
if self.level().is_prime():
return matrix(QQ, self.dimension(), 0, sparse=True)
if self._degeneracy_matrices.has_key(p):
return self._degeneracy_matrices[p]
if p is None:
A = None
for p in prime_divisors(self._level):
A = self.degeneracy_matrix(p) if A is None else A.augment(self.degeneracy_matrix(p))
A.set_immutable()
self._degeneracy_matrices[None] = A
return A
p = sqrt5_ideal(p)
if self._degeneracy_matrices.has_key(p):
return self._degeneracy_matrices[p]
d = self._icosians_mod_p1.degeneracy_matrix(p)
d.set_immutable()
self._degeneracy_matrices[p] = d
return d
def __init__(self, level):
"""
INPUT:
- ``level`` -- an ideal or element of ZZ[(1+sqrt(5))/2].
TESTS::
sage: H = sage.modular.hilbert.sqrt5_hmf.QuaternionicModule(3); H
Quaternionic module of dimension 1, level 3 (of norm 9=3^2) over QQ(sqrt(5))
sage: loads(dumps(H)) == H
True
"""
self._level = sqrt5_ideal(level)
self._gen = self._level.gens_reduced()[0]
self._icosians_mod_p1 = IcosiansModP1ModN(self._level)
self._dimension = self._icosians_mod_p1.cardinality()
self._vector_space = QQ**self._dimension
self._hecke_matrices = {}
self._degeneracy_matrices = {}
def hecke_matrix(self, n):
"""
Return the matrix of the `n`-th Hecke operator.
This is only implemented when `n` is a prime ideal that is
coprime to the level, though the notion of Hecke operator
is defined for any nonzero ideal `n`.
INPUT:
- `n` -- nonzero prime ideal of ring of integers of
QQ(sqrt(5))
OUTPUT:
- a matrix over the rational numbers with integer entries
EXAMPLES::
sage: from sage.modular.hilbert.sqrt5_hmf import F, QuaternionicModule
sage: H = QuaternionicModule(F.prime_above(31)); H
Quaternionic module of dimension 2, level 5*a-2 (of norm 31=31) over QQ(sqrt(5))
sage: H.hecke_matrix(2)
[0 5]
[3 2]
sage: P = F.prime_above(5); P
Fractional ideal (-2*a + 1)
sage: H.hecke_matrix(P)
[1 5]
[3 3]
At least the prime does not have to be coprime to the norm of the level::
sage: v = F.primes_above(31)
sage: H = QuaternionicModule(v[0]); H
Quaternionic module of dimension 2, level 5*a-2 (of norm 31=31) over QQ(sqrt(5))
sage: H.hecke_matrix(v[1])
[17 15]
[ 9 23]
The input must be nonzero, or you get a ValueError::
sage: H.hecke_matrix(F.ideal(0))
Traceback (most recent call last):
...
ValueError: n must be nonzero
We illustrate some shortcomings of this function::
sage: H.hecke_matrix(P^2)
Traceback (most recent call last):
...
NotImplementedError: n must be prime
sage: H.hecke_matrix(F.prime_above(31))
Traceback (most recent call last):
...
NotImplementedError: n must be coprime to the level
You may also use T as an alias for hecke_matrix::
sage: H.T(3)
[5 5]
[3 7]
"""
# I'm not using @cached_method, since I want to ensure that
# the input "n" is properly normalized. I also want it
# to be transparent to see which matrices have been computed,
# to clear the cache, etc.
n = sqrt5_ideal(n)
if n.is_zero():
raise ValueError, "n must be nonzero"
if not n.is_prime():
raise NotImplementedError, "n must be prime"
if not self.level().is_coprime(n):
raise NotImplementedError, "n must be coprime to the level"
if self._hecke_matrices.has_key(n):
return self._hecke_matrices[n]
t = self._icosians_mod_p1.hecke_matrix(n)
t.set_immutable()
self._hecke_matrices[n] = t
return t
