def is_quotient(M, sym, rank):
symbol = GenusSymbol_global_ring(MatrixSpace(ZZ, rank, rank).one())
symbol._local_symbols = [
Genus_Symbol_p_adic_ring(p, syms) for p, syms in sym.iteritems()
]
s = get_symbol_string(symbol)
print s
N = FiniteQuadraticModule(s)
t = N.order() / M.order()
if not Integer(t).is_square():
return False
else:
t = sqrt(t)
for p in Integer(N.order()).prime_factors():
if not N.signature(p) == M.signature(p):
return False
# if not N.signature() == M.signature():
# return False
for G in N.subgroups():
if G.order() == t and G.is_isotropic():
Q = G.quotient()
if Q.is_isomorphic(M):
print Q
return N
else:
del Q
del N
return False
def is_quotient(M, sym, rank):
symbol = GenusSymbol_global_ring(MatrixSpace(ZZ, rank, rank).one())
symbol._local_symbols = [
Genus_Symbol_p_adic_ring(p, syms) for p, syms in sym.iteritems()]
s = get_symbol_string(symbol)
print s
N = FiniteQuadraticModule(s)
t = N.order() / M.order()
if not Integer(t).is_square():
return False
else:
t = sqrt(t)
for p in Integer(N.order()).prime_factors():
if not N.signature(p) == M.signature(p):
return False
# if not N.signature() == M.signature():
# return False
for G in N.subgroups():
if G.order() == t and G.is_isotropic():
Q = G.quotient()
if Q.is_isomorphic(M):
print Q
return N
else:
del Q
del N
return False
def dict_to_genus_symbol_string(sym):
symbol = GenusSymbol_global_ring(MatrixSpace(ZZ, 1, 1).one())
symbol._local_symbols = [
Genus_Symbol_p_adic_ring(p, syms) for p, syms in sym.iteritems()
]
s = get_symbol_string(symbol)
return s
def search_global_symbols(n, D):
rank = 2 + n
sign = 2 - n
csymbols = list() # a list of canonical symbols to avoid duplicates
D = (-1)**n * D
symbols = all_symbols(sign, rank, D)
global_symbols = []
for sym in symbols:
#print sym
for p in sym.keys():
prank = sum([s[1] for s in sym[p]])
v = sum([s[0] * s[1] for s in sym[p]])
Dp = D // (p**v)
if prank != rank:
eps = (Dp * Integer(prod([s[2] for s in sym[p]]))).kronecker(p)
if p == 2:
if eps == -1:
eps = 3
sym[p].insert(0, [0, rank - prank, eps, 0, 0])
else:
if eps == -1:
for x in Zmod(p):
if not x.is_square():
eps = x
break
sym[p].insert(0, [0, rank - prank, eps])
symbol = GenusSymbol_global_ring(MatrixSpace(ZZ, rank, rank).one())
symbol._local_symbols = [
Genus_Symbol_p_adic_ring(p, syms) for p, syms in sym.iteritems()
]
symbol._signature = (2, n)
#print symbol._local_symbols
isglob = is_GlobalGenus(symbol)
if isglob:
#print "GLOBAL SYMBOL:"
#print symbol._local_symbols
#return symbol
#for s in symbol._local_symbols:
# s = s.canonical_symbol()
append = True
for j, s in enumerate(symbol._local_symbols):
if s._prime == 2:
sc = deepcopy(symbol)
sc._local_symbols[j] = sc._local_symbols[
j].canonical_symbol()
if csymbols.count(sc) > 0:
append = False
else:
csymbols.append(sc)
break
if append:
global_symbols.append(symbol)
return global_symbols
def search_global_symbols(n,D):
rank=2+n
sign=2-n
csymbols=list() # a list of canonical symbols to avoid duplicates
D=(-1)**n*D
symbols = all_symbols(sign,rank,D)
global_symbols = []
for sym in symbols:
#print sym
for p in sym.keys():
prank = sum([s[1] for s in sym[p]])
v = sum([ s[0]*s[1] for s in sym[p] ])
Dp=D//(p**v)
if prank != rank:
eps= (Dp*Integer(prod([ s[2] for s in sym[p] ]))).kronecker(p)
if p==2:
if eps==-1:
eps=3
sym[p].insert(0,[0,rank - prank, eps,0,0])
else:
if eps==-1:
for x in Zmod(p):
if not x.is_square():
eps=x
break
sym[p].insert(0,[0,rank - prank, eps])
symbol=GenusSymbol_global_ring(MatrixSpace(ZZ,rank,rank).one())
symbol._local_symbols=[Genus_Symbol_p_adic_ring(p,syms) for p,syms in sym.iteritems()]
symbol._signature=(2,n)
#print symbol._local_symbols
isglob=is_GlobalGenus(symbol)
if isglob:
#print "GLOBAL SYMBOL:"
#print symbol._local_symbols
#return symbol
#for s in symbol._local_symbols:
# s = s.canonical_symbol()
append=True
for j,s in enumerate(symbol._local_symbols):
if s._prime==2:
sc=deepcopy(symbol)
sc._local_symbols[j]=sc._local_symbols[j].canonical_symbol()
if csymbols.count(sc)>0:
append=False
else:
csymbols.append(sc)
break
if append:
global_symbols.append(symbol)
return global_symbols
def is_global(M, r, s, return_symbol=False):
r"""
Test if the FiniteQuadraticModule M can be represented by a Z-lattice
of signature ``(r,s)``.
INPUT:
-``M`` -- FiniteQuadraticModule
- ``r`` -- positive integer
- ``s`` -- positive integer
OUTPUT:
- boolean
"""
J = M.jordan_decomposition()
symbols = {}
n = r + s
sig = r - s
for A in J:
p = A[1][0]
if not symbols.has_key(p):
symbols[p] = list()
sym = list(A[1][1:len(A[1])])
if p == 2:
if len(A[1]) == 4:
sym.append(0)
sym.append(0)
else:
if sym[3].kronecker(2) == sym[2]:
det = sym[3] % 8
else:
if sym[2] == -1:
det = 3
else:
det = 1
sym = [sym[0], sym[1], det, 1, sym[3] % 8]
#print sym
#if sym[1]==1:
# if sym[2].kronecker(2)==sym[4].kronecker(2):
# sym[2]=sym[4]
# else:
# return False
#print p, sym
symbols[p].append(sym)
D = M.order() * (-1)**s
print D
for p in symbols.keys():
prank = sum([sym[1] for sym in symbols[p]])
v = sum([sym[0] * sym[1] for sym in symbols[p]])
Dp = D // (p**v)
print Dp
if prank != n:
eps = (Dp).kronecker(p) * Integer(
prod([sym[2] for sym in symbols[p]]))
print eps
if p == 2:
if eps == -1:
eps = 3
symbols[p].append([0, n - prank, eps, 0, 0])
else:
#if eps==-1:
# for x in Zmod(p):
# if not x.is_square():
# eps=x
# break
symbols[p].append([0, n - prank, eps])
symbol = GenusSymbol_global_ring(MatrixSpace(ZZ, r + s, r + s).one())
symbol._local_symbols = [
Genus_Symbol_p_adic_ring(p, syms) for p, syms in symbols.iteritems()
]
symbol._signature = (r, s)
#print r,s, symbol._local_symbols
isglob = is_GlobalGenus(symbol)
if return_symbol:
return symbol, isglob
else:
return isglob
def search_global_symbols(n, D):
rank = 2 + n
sign = 2 - n
csymbols = list() # a list of canonical symbols to avoid duplicates
symbols = list()
#print D
D = (-1)**n * D
fac = Integer(D).factor()
symbols = list()
for p, v in fac:
psymbols = list()
parts = partitions(v)
Dp = D // (p**v)
for vs in parts:
#print "partition:", vs
l = list() # list of p-symbols corresponding to the partition vs
if len(vs) <= rank:
exponents = Set(list(vs))
# now we set up a list ll for each vv in the partition vs
# that contains an entry for each possibility
# and then update l with ll (see below)
if p == 2:
for vv in exponents:
mult = vs.count(vv)
ll = list()
for t in [0, 1]: # even(0) or odd(1) type
for det in [1, 3, 5,
7]: # the possible determinants
if mult % 2 == 0 and t == 0:
ll.append([vv, mult, det, 0, 0])
if mult == 1:
odds = [det]
elif mult == 2:
if det in [1, 7]:
odds = [0, 2, 6]
else:
odds = [2, 4, 6]
else:
odds = [
o for o in range(8)
if o % 2 == mult % 2
]
for oddity in odds:
if t == 1:
ll.append([vv, mult, det, 1, oddity])
#else:
#ll.append([vv,1,det,0,0])
#if mult % 2 == 0 and mult>2:
# for x in range(1,Integer(mult)/Integer(2)):
# if mult-2*x==2 and det in [1,7] and oddity not in [0,2,6]:
# continue
# elif mult-2*x==2 and det in [3,5] and oddity not in [2,4,6]:
# continue
# ll.append([[vv,2*x,det,0,0],[vv,mult-2*x,det,1,oddity]])
#print "ll:\n",ll
if len(l) == 0:
for t in ll:
if type(t[0]) == list:
l.append({p: t})
else:
l.append({p: [t]})
else:
newl = list()
for t in ll:
for sym in l:
newsym = deepcopy(sym)
#print newsym
if type(t[0]) == list:
newsym[p] = newsym[p] + t
else:
newsym[p].append(t)
#print newsym
newl.append(newsym)
#print l
l = newl
#print "l:\n",l
else:
for vv in exponents:
ll = [[vv, vs.count(vv), 1], [vv, vs.count(vv), -1]]
if len(l) == 0:
for t in ll:
l.append({p: [t]})
else:
newl = list()
for t in ll:
for sym in l:
sym[p].append(t)
newl.append(sym)
l = newl
#print "l=\n",l
#print "psymbols=\n",psymbols
#print psymbols+l
psymbols = psymbols + l
if len(symbols) == 0:
symbols = psymbols
else:
symbols_new = list()
for sym in symbols:
for psym in psymbols:
newsym = deepcopy(sym)
newsym.update(psym)
symbols_new.append(newsym)
symbols = symbols_new
global_symbols = []
for sym in symbols:
#print sym
for p in sym.keys():
prank = sum([s[1] for s in sym[p]])
v = sum([s[0] * s[1] for s in sym[p]])
Dp = D // (p**v)
if prank != rank:
eps = (Dp * Integer(prod([s[2] for s in sym[p]]))).kronecker(p)
if p == 2:
if eps == -1:
eps = 3
sym[p].insert(0, [0, rank - prank, eps, 0, 0])
else:
if eps == -1:
for x in Zmod(p):
if not x.is_square():
eps = x
break
sym[p].insert(0, [0, rank - prank, eps])
symbol = GenusSymbol_global_ring(MatrixSpace(ZZ, rank, rank).one())
symbol._local_symbols = [
Genus_Symbol_p_adic_ring(p, syms) for p, syms in sym.items()
]
symbol._signature = (2, n)
#print symbol._local_symbols
isglob = is_GlobalGenus(symbol)
if isglob:
#print "GLOBAL SYMBOL:"
#print symbol._local_symbols
#return symbol
#for s in symbol._local_symbols:
# s = s.canonical_symbol()
append = True
for j, s in enumerate(symbol._local_symbols):
if s._prime == 2:
sc = deepcopy(symbol)
sc._local_symbols[j] = sc._local_symbols[
j].canonical_symbol()
if csymbols.count(sc) > 0:
append = False
else:
csymbols.append(sc)
break
if append:
global_symbols.append(symbol)
return global_symbols
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_global(M,r,s,return_symbol=False):
r"""
Test if the FiniteQuadraticModule M can be represented by a Z-lattice
of signature ``(r,s)``.
INPUT:
-``M`` -- FiniteQuadraticModule
- ``r`` -- positive integer
- ``s`` -- positive integer
OUTPUT:
- boolean
"""
J=M.jordan_decomposition()
symbols={}
n=r+s
sig=r-s
for A in J:
p=A[1][0]
if not symbols.has_key(p):
symbols[p]=list()
sym=list(A[1][1:len(A[1])])
if p==2:
if len(A[1])==4:
sym.append(0)
sym.append(0)
else:
if sym[3].kronecker(2)==sym[2]:
det=sym[3] % 8
else:
if sym[2]==-1:
det=3
else:
det=1
sym = [sym[0], sym[1], det, 1, sym[3] % 8]
#print sym
#if sym[1]==1:
# if sym[2].kronecker(2)==sym[4].kronecker(2):
# sym[2]=sym[4]
# else:
# return False
#print p, sym
symbols[p].append(sym)
D=M.order()*(-1)**s
print D
for p in symbols.keys():
prank = sum([sym[1] for sym in symbols[p]])
v = sum([ sym[0]*sym[1] for sym in symbols[p] ])
Dp=D//(p**v)
print Dp
if prank != n:
eps= (Dp).kronecker(p)*Integer(prod([ sym[2] for sym in symbols[p] ]))
print eps
if p==2:
if eps==-1:
eps=3
symbols[p].append([0,n - prank, eps,0,0])
else:
#if eps==-1:
# for x in Zmod(p):
# if not x.is_square():
# eps=x
# break
symbols[p].append([0,n - prank, eps])
symbol=GenusSymbol_global_ring(MatrixSpace(ZZ,r+s,r+s).one())
symbol._local_symbols=[Genus_Symbol_p_adic_ring(p,syms) for p,syms in symbols.iteritems()]
symbol._signature=(r,s)
#print r,s, symbol._local_symbols
isglob=is_GlobalGenus(symbol)
if return_symbol:
return symbol, isglob
else:
return isglob
def search_global_symbols(n,D):
rank=2+n
sign=2-n
csymbols=list() # a list of canonical symbols to avoid duplicates
symbols=list()
#print D
D=(-1)**n*D
fac = Integer(D).factor()
symbols=list()
for p, v in fac:
psymbols=list()
parts=partitions(v)
Dp=D//(p**v)
for vs in parts:
#print "partition:", vs
l=list() # list of p-symbols corresponding to the partition vs
if len(vs) <= rank:
exponents=Set(list(vs))
# now we set up a list ll for each vv in the partition vs
# that contains an entry for each possibility
# and then update l with ll (see below)
if p==2:
for vv in exponents:
mult=vs.count(vv)
ll=list()
for t in [0,1]: # even(0) or odd(1) type
for det in [1,3,5,7]: # the possible determinants
if mult % 2 == 0 and t==0:
ll.append([vv,mult,det,0,0])
if mult==1:
odds=[det]
elif mult==2:
if det in [1,7]:
odds=[0,2,6]
else:
odds=[2,4,6]
else:
odds=[o for o in range(8) if o%2==mult%2]
for oddity in odds:
if t==1:
ll.append([vv,mult,det,1,oddity])
#else:
#ll.append([vv,1,det,0,0])
#if mult % 2 == 0 and mult>2:
# for x in range(1,Integer(mult)/Integer(2)):
# if mult-2*x==2 and det in [1,7] and oddity not in [0,2,6]:
# continue
# elif mult-2*x==2 and det in [3,5] and oddity not in [2,4,6]:
# continue
# ll.append([[vv,2*x,det,0,0],[vv,mult-2*x,det,1,oddity]])
#print "ll:\n",ll
if len(l)==0:
for t in ll:
if type(t[0])==list:
l.append({p: t})
else:
l.append({p: [t]})
else:
newl=list()
for t in ll:
for sym in l:
newsym = deepcopy(sym)
#print newsym
if type(t[0])==list:
newsym[p]=newsym[p]+t
else:
newsym[p].append(t)
#print newsym
newl.append(newsym)
#print l
l=newl
#print "l:\n",l
else:
for vv in exponents:
ll=[[vv,vs.count(vv),1],[vv,vs.count(vv),-1]]
if len(l)==0:
for t in ll:
l.append({p: [t]})
else:
newl=list()
for t in ll:
for sym in l:
sym[p].append(t)
newl.append(sym)
l=newl
#print "l=\n",l
#print "psymbols=\n",psymbols
#print psymbols+l
psymbols=psymbols+l
if len(symbols)==0:
symbols=psymbols
else:
symbols_new=list()
for sym in symbols:
for psym in psymbols:
newsym=deepcopy(sym)
newsym.update(psym)
symbols_new.append(newsym)
symbols=symbols_new
global_symbols = []
for sym in symbols:
#print sym
for p in sym.keys():
prank = sum([s[1] for s in sym[p]])
v = sum([ s[0]*s[1] for s in sym[p] ])
Dp=D//(p**v)
if prank != rank:
eps= (Dp*Integer(prod([ s[2] for s in sym[p] ]))).kronecker(p)
if p==2:
if eps==-1:
eps=3
sym[p].insert(0,[0,rank - prank, eps,0,0])
else:
if eps==-1:
for x in Zmod(p):
if not x.is_square():
eps=x
break
sym[p].insert(0,[0,rank - prank, eps])
symbol=GenusSymbol_global_ring(MatrixSpace(ZZ,rank,rank).one())
symbol._local_symbols=[Genus_Symbol_p_adic_ring(p,syms) for p,syms in sym.iteritems()]
symbol._signature=(2,n)
#print symbol._local_symbols
isglob=is_GlobalGenus(symbol)
if isglob:
#print "GLOBAL SYMBOL:"
#print symbol._local_symbols
#return symbol
#for s in symbol._local_symbols:
# s = s.canonical_symbol()
append=True
for j,s in enumerate(symbol._local_symbols):
if s._prime==2:
sc=deepcopy(symbol)
sc._local_symbols[j]=sc._local_symbols[j].canonical_symbol()
if csymbols.count(sc)>0:
append=False
else:
csymbols.append(sc)
break
if append:
global_symbols.append(symbol)
return global_symbols
def dict_to_genus_symbol_string(sym):
symbol = GenusSymbol_global_ring(MatrixSpace(ZZ,1,1).one())
symbol._local_symbols=[Genus_Symbol_p_adic_ring(p,syms) for p,syms in sym.iteritems()]
s = get_symbol_string(symbol)
return s
