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 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