def _brown_indecomposable(q, p):
r"""
Return the Brown invariant of the indecomposable form ``q``.
The values are taken from Table 2.1 in [Shim2016]_.
INPUT:
- ``q`` - an indecomposable quadratic form represented by a
rational `1 \times 1` or `2 \times 2` matrix
- ``p`` - a prime number
EXAMPLES::
sage: from sage.modules.torsion_quadratic_module import _brown_indecomposable
sage: q = Matrix(QQ, [1/3])
sage: _brown_indecomposable(q,3)
6
sage: q = Matrix(QQ, [2/3])
sage: _brown_indecomposable(q,3)
2
sage: q = Matrix(QQ, [5/4])
sage: _brown_indecomposable(q,2)
5
sage: q = Matrix(QQ, [7/4])
sage: _brown_indecomposable(q,2)
7
sage: q = Matrix(QQ, 2, [0,1,1,0])/2
sage: _brown_indecomposable(q,2)
0
sage: q = Matrix(QQ, 2, [2,1,1,2])/2
sage: _brown_indecomposable(q,2)
4
"""
v = q.denominator().valuation(p)
if p == 2:
# brown(U) = 0
if q.ncols() == 2:
if q[0, 0].valuation(2) > v + 1 and q[1, 1].valuation(2) > v + 1:
# type U
return mod(0, 8)
else:
# type V
return mod(4 * v, 8)
u = q[0, 0].numerator()
return mod(u + v * (u**2 - 1) / 2, 8)
if p % 4 == 1:
e = -1
if p % 4 == 3:
e = 1
if v % 2 == 1:
u = q[0, 0].numerator() // 2
if legendre_symbol(u, p) == 1:
return mod(1 + e, 8)
else:
return mod(-3 + e, 8)
return mod(0, 8)
def _brown_indecomposable(q, p):
r"""
Return the Brown invariant of the indecomposable form ``q``.
The values are taken from Table 2.1 in [Shim2016]_.
INPUT:
- ``q`` - an indecomposable quadratic form represented by a
rational `1 \times 1` or `2 \times 2` matrix
- ``p`` - a prime number
EXAMPLES::
sage: from sage.modules.torsion_quadratic_module import _brown_indecomposable
sage: q = Matrix(QQ, [1/3])
sage: _brown_indecomposable(q,3)
6
sage: q = Matrix(QQ, [2/3])
sage: _brown_indecomposable(q,3)
2
sage: q = Matrix(QQ, [5/4])
sage: _brown_indecomposable(q,2)
5
sage: q = Matrix(QQ, [7/4])
sage: _brown_indecomposable(q,2)
7
sage: q = Matrix(QQ, 2, [0,1,1,0])/2
sage: _brown_indecomposable(q,2)
0
sage: q = Matrix(QQ, 2, [2,1,1,2])/2
sage: _brown_indecomposable(q,2)
4
"""
v = q.denominator().valuation(p)
if p == 2:
# brown(U) = 0
if q.ncols() == 2:
if q[0,0].valuation(2)>v+1 and q[1,1].valuation(2)>v+1:
# type U
return mod(0, 8)
else:
# type V
return mod(4*v, 8)
u = q[0,0].numerator()
return mod(u + v*(u**2 - 1)/2, 8)
if p % 4 == 1:
e = -1
if p % 4 == 3:
e = 1
if v % 2 == 1:
u = q[0,0].numerator()//2
if legendre_symbol(u,p) == 1:
return mod(1 + e, 8)
else:
return mod(-3 + e, 8)
return mod(0, 8)
def quadratic_defect(self, a, p, check=True):
r"""
Return the valuation of the quadratic defect of `a` at `p`.
INPUT:
- ``a`` -- an element of ``self``
- ``p`` -- a prime ideal or a prime number
- ``check`` -- (default: ``True``); check if `p` is prime
REFERENCE:
[Kir2016]_
EXAMPLES::
sage: QQ.quadratic_defect(0, 7)
+Infinity
sage: QQ.quadratic_defect(5, 7)
0
sage: QQ.quadratic_defect(5, 2)
2
sage: QQ.quadratic_defect(5, 5)
1
"""
from sage.rings.all import Infinity
from sage.arith.misc import legendre_symbol
if not a in self:
raise TypeError(str(a) + " must be an element of " + str(self))
if p.parent() == ZZ.ideal_monoid():
p = p.gen()
if check and not p.is_prime():
raise ValueError(str(p) + " must be prime")
if a.is_zero():
return Infinity
v, u = self(a).val_unit(p)
if v % 2 == 1:
return v
if p != 2:
if legendre_symbol(u, p) == 1:
return Infinity
else:
return v
if p == 2:
if u % 8 == 1:
return Infinity
if u % 8 == 5:
return v + 2
if u % 8 in [3, 7]:
return v + 1
def quadratic_defect(self, a, p):
r"""
Return the valuation of the quadratic defect of `a` at `p`.
INPUT:
- ``a`` an element of ``self``
- ``p`` a prime ideal or a prime number
REFERENCE:
[Kir2016]_
EXAMPLES::
sage: QQ.quadratic_defect(0, 7)
+Infinity
sage: QQ.quadratic_defect(5, 7)
0
sage: QQ.quadratic_defect(5, 2)
2
sage: QQ.quadratic_defect(5, 5)
1
"""
from sage.rings.all import Infinity
from sage.arith.misc import legendre_symbol
if not a in self:
raise TypeError(str(a) + " must be an element of " + str(self))
if p.parent() == ZZ.ideal_monoid():
p = p.gen()
if not p.is_prime():
raise ValueError(str(p) + " must be prime")
if a.is_zero():
return Infinity
v = self(a).valuation(p)
if v % 2 == 1:
return v
a = a / (p**v)
if p != 2:
if legendre_symbol(a, p) == 1:
return Infinity
else:
return v
if p == 2:
if a % 8 == 1:
return Infinity
if a % 8 == 5:
return v + 2
if a % 8 in [3, 7]:
return v + 1
def is_genus(self, signature_pair, even=True):
r"""
Return ``True`` if there is a lattice with this signature and discriminant form.
.. TODO::
implement the same for odd lattices
INPUT:
- signature_pair -- a tuple of non negative integers ``(s_plus, s_minus)``
- even -- bool (default: ``True``)
EXAMPLES::
sage: L = IntegralLattice("D4").direct_sum(IntegralLattice(3 * Matrix(ZZ,2,[2,1,1,2])))
sage: D = L.discriminant_group()
sage: D.is_genus((6,0))
True
Let us see if there is a lattice in the genus defined by the same discriminant form
but with a different signature::
sage: D.is_genus((4,2))
False
sage: D.is_genus((16,2))
True
"""
s_plus = ZZ(signature_pair[0])
s_minus = ZZ(signature_pair[1])
if s_plus < 0 or s_minus < 0:
raise ValueError("signature invariants must be non negative")
rank = s_plus + s_minus
signature = s_plus - s_minus
D = self.cardinality()
det = (-1)**s_minus * D
if rank < len(self.invariants()):
return False
if even and self._modulus_qf != 2:
raise ValueError("the discriminant form of an even lattice has"
"values modulo 2.")
if (not even) and not (self._modulus == self._modulus_qf == 1):
raise ValueError("the discriminant form of an odd lattice has"
"values modulo 1.")
if not even:
raise NotImplementedError("at the moment sage knows how to do this only for even genera. " +
" Help us to implement this for odd genera.")
for p in D.prime_divisors():
# check the determinant conditions
Q_p = self.primary_part(p)
gram_p = Q_p.gram_matrix_quadratic()
length_p = len(Q_p.invariants())
u = det.prime_to_m_part(p)
up = gram_p.det().numerator().prime_to_m_part(p)
if p != 2 and length_p == rank:
if legendre_symbol(u, p) != legendre_symbol(up, p):
return False
if p == 2:
if rank % 2 != length_p % 2:
return False
n = (rank - length_p) / 2
if u % 4 != (-1)**(n % 2) * up % 4:
return False
if rank == length_p:
a = QQ(1) / QQ(2)
b = QQ(3) / QQ(2)
diag = gram_p.diagonal()
if not (a in diag or b in diag):
if u % 8 != up % 8:
return False
if self.brown_invariant() != signature:
return False
return True
def genus(self, signature_pair):
r"""
Return the genus defined by ``self`` and the ``signature_pair``.
If no such genus exists, raise a ``ValueError``.
REFERENCES:
[Nik1977]_ Corollary 1.9.4 and 1.16.3.
EXAMPLES::
sage: L = IntegralLattice("D4").direct_sum(IntegralLattice("A2"))
sage: D = L.discriminant_group()
sage: genus = D.genus(L.signature_pair())
sage: genus
Genus of
None
Signature: (6, 0)
Genus symbol at 2: 1^4:2^-2
Genus symbol at 3: 1^-5 3^-1
sage: genus == L.genus()
True
Let `H` be an even unimodular lattice of signature `(9, 1)`.
Then `L = D_4 + A_2` is primitively embedded in `H`. We compute the discriminant
form of the orthogonal complement of `L` in `H`::
sage: DK = D.twist(-1)
sage: DK
Finite quadratic module over Integer Ring with invariants (2, 6)
Gram matrix of the quadratic form with values in Q/2Z:
[ 1 1/2]
[1/2 1/3]
We know that `K` has signature `(5, 1)` and thus we can compute
the genus of `K` as::
sage: DK.genus((3,1))
Genus of
None
Signature: (3, 1)
Genus symbol at 2: 1^2:2^-2
Genus symbol at 3: 1^-3 3^1
We can also compute the genus of an odd lattice
from its discriminant form::
sage: L = IntegralLattice(matrix.diagonal(range(1,5)))
sage: D = L.discriminant_group()
sage: D.genus((4,0))
Genus of
None
Signature: (4, 0)
Genus symbol at 2: [1^-2 2^1 4^1]_6
Genus symbol at 3: 1^-3 3^1
TESTS::
sage: L.genus() == D.genus((4,0))
True
sage: D.genus((1,0))
Traceback (most recent call last):
...
ValueError: this discriminant form and signature do not define a genus
A systematic test of lattices of small ranks and determinants::
sage: from sage.quadratic_forms.genera.genus import genera
sage: signatures = [(1,0),(1,1),(1,2),(3,0),(0,4)]
sage: dets = range(1,33)
sage: genera = flatten([genera(s, d, even=False) for d in dets for s in signatures]) # long time
sage: all(g == g.discriminant_form().genus(g.signature_pair()) for g in genera) # long time
True
"""
from sage.quadratic_forms.genera.genus import (Genus_Symbol_p_adic_ring,
GenusSymbol_global_ring,
p_adic_symbol,
is_GlobalGenus,
_blocks)
from sage.misc.misc_c import prod
s_plus = signature_pair[0]
s_minus = signature_pair[1]
rank = s_plus + s_minus
if len(self.invariants()) > rank:
raise ValueError("this discriminant form and " +
"signature do not define a genus")
disc = self.cardinality()
determinant = (-1)**s_minus * disc
local_symbols = []
for p in (2 * disc).prime_divisors():
D = self.primary_part(p)
if len(D.invariants()) != 0:
G_p = D.gram_matrix_quadratic().inverse()
# get rid of denominators without changing the local equivalence class
G_p *= G_p.denominator()**2
G_p = G_p.change_ring(ZZ)
local_symbol = p_adic_symbol(G_p, p, D.invariants()[-1].valuation(p))
else:
local_symbol = []
rk = rank - len(D.invariants())
if rk > 0:
if p == 2:
det = determinant.prime_to_m_part(2)
det *= prod([di[2] for di in local_symbol])
det = det % 8
local_symbol.append([ZZ(0), rk, det, ZZ(0), ZZ(0)])
else:
det = legendre_symbol(determinant.prime_to_m_part(p), p)
det = (det * prod([di[2] for di in local_symbol]))
local_symbol.append([ZZ(0), rk, det])
local_symbol.sort()
local_symbol = Genus_Symbol_p_adic_ring(p, local_symbol)
local_symbols.append(local_symbol)
# This genus has the right discriminant group
# but it may be empty
genus = GenusSymbol_global_ring(signature_pair, local_symbols)
sym2 = local_symbols[0].symbol_tuple_list()
if sym2[0][0] != 0:
sym2 = [[ZZ(0), ZZ(0), ZZ(1), ZZ(0), ZZ(0)]] + sym2
if len(sym2) <= 1 or sym2[1][0] != 1:
sym2 = sym2[:1] + [[ZZ(1), ZZ(0), ZZ(1), ZZ(0), ZZ(0)]] + sym2[1:]
if len(sym2) <= 2 or sym2[2][0] != 2:
sym2 = sym2[:2] + [[ZZ(2), ZZ(0), ZZ(1), ZZ(0), ZZ(0)]] + sym2[2:]
if self.value_module_qf().n == 1:
# in this case the blocks of scales 1, 2, 4 are under determined
# make sure the first 3 symbols are of scales 1, 2, 4
# i.e. their valuations are 0, 1, 2
# the form is odd
block0 = [b for b in _blocks(sym2[0]) if b[3] == 1]
o = sym2[1][3]
# no restrictions on determinant and
# oddity beyond existence
# but we know if even or odd
block1 = [b for b in _blocks(sym2[1]) if b[3] == o]
d = sym2[2][2]
o = sym2[2][3]
t = sym2[2][4]
# if the jordan block of scale 2 is even we know it
if o == 0:
block2 = [sym2[2]]
# if it is odd we know det and oddity mod 4 at least
else:
block2 = [b for b in _blocks(sym2[2]) if b[3] == o
and (b[2] - d) % 4 == 0
and (b[4] - t) % 4 == 0
and (b[2] - d) % 8 == (b[4] - t) % 8 # if the oddity is altered by 4 then so is the determinant
]
elif self.value_module_qf().n == 2:
# the form is even
block0 = [b for b in _blocks(sym2[0]) if b[3] == 0]
# if the jordan block of scale 2 is even we know it
d = sym2[1][2]
o = sym2[1][3]
t = sym2[1][4]
if o == 0:
block1 = [sym2[1]]
else:
# the block is odd and we know det and oddity mod 4
block1 = [b for b in _blocks(sym2[1])
if b[3] == o
and (b[2] - d) % 4 == 0
and (b[4] - t) % 4 == 0
and (b[2] - d) % 8 == (b[4] - t) % 8 # if the oddity is altered by 4 then so is the determinant
]
# this is completely determined
block2 = [sym2[2]]
else:
raise ValueError("this is not a discriminant form")
# figure out which symbol defines a genus and return that
for b0 in block0:
for b1 in block1:
for b2 in block2:
sym2[:3] = [b0, b1, b2]
local_symbols[0] = Genus_Symbol_p_adic_ring(2, sym2)
genus = GenusSymbol_global_ring(signature_pair, local_symbols)
if is_GlobalGenus(genus):
# make the symbol sparse again.
i = 0
k = 0
while i < 3:
if sym2[k][1] == 0:
sym2.pop(k)
else:
k = k + 1
i = i + 1
local_symbols[0] = Genus_Symbol_p_adic_ring(2, sym2)
genus = GenusSymbol_global_ring(signature_pair, local_symbols)
return genus
raise ValueError("this discriminant form and signature do not define a genus")
def is_genus(self, signature_pair, even=True):
r"""
Return ``True`` if there is a lattice with this signature and discriminant form.
TODO:
- implement the same for odd lattices
INPUT:
- signature_pair -- a tuple of non negative integers ``(s_plus, s_minus)``
- even -- bool (default: ``True``)
EXAMPLES::
sage: L = IntegralLattice("D4").direct_sum(IntegralLattice(3 * Matrix(ZZ,2,[2,1,1,2])))
sage: D = L.discriminant_group()
sage: D.is_genus((6,0))
True
Let us see if there is a lattice in the genus defined by the same discriminant form
but with a different signature::
sage: D.is_genus((4,2))
False
sage: D.is_genus((16,2))
True
"""
s_plus = ZZ(signature_pair[0])
s_minus = ZZ(signature_pair[1])
if s_plus<0 or s_minus<0:
raise ValueError("signature invariants must be non negative")
rank = s_plus + s_minus
signature = s_plus - s_minus
D = self.cardinality()
det = (-1)**s_minus * D
if rank < len(self.invariants()):
return False
if even and self._modulus_qf != 2:
raise ValueError("the discriminant form of an even lattice has"
"values modulo 2.")
if (not even) and not (self.modulus == self._modulus_qf == 1):
raise ValueError("the discriminant form of an odd lattice has"
"values modulo 1.")
if not even:
raise NotImplementedError("at the moment sage knows how to do this only for even genera. " +
" Help us to implement this for odd genera.")
for p in D.prime_divisors():
# check the determinat conditions
Q_p = self.primary_part(p)
gram_p = Q_p.gram_matrix_quadratic()
length_p = len(Q_p.invariants())
u = det.prime_to_m_part(p)
up = gram_p.det().numerator().prime_to_m_part(p)
if p!=2 and length_p==rank:
if legendre_symbol(u, p) != legendre_symbol(up, p):
return False
if p == 2:
if rank % 2 != length_p % 2:
return False
n = (rank - length_p) / 2
if u % 4 != (-1)**(n % 2) * up % 4:
return False
if rank == length_p:
a = QQ(1) / QQ(2)
b = QQ(3) / QQ(2)
diag = gram_p.diagonal()
if not (a in diag or b in diag):
if u % 8 != up % 8:
return False
if self.brown_invariant() != signature:
return False
return True
