def siegel_series_dim2(q, p):
det_4 = q.Gram_det() * ZZ(4)
c = q.content_order()
fd = fundamental_discriminant(-det_4)
f = (valuation(det_4, p) - valuation(fd, p)) / ZZ(2)
return (_siegel_series_dim2(p, c, f + 1) -
kronecker_symbol(fd, p) * p * X * _siegel_series_dim2(p, c, f))
def test_jordan_decomposition_odd_p(self):
for _ in range(100):
S = random_even_symm_mat(5)
for p in [3, 5, 7]:
self.assertEqual(
kronecker_symbol(
mul(_jordan_decomposition_odd_p(S, p)) / S.det(), p),
1)
def _chi_p_odd(a, p):
p = ZZ(p)
r = valuation(a, p)
if r % 2 == 0:
c = a // (p ** r)
return kronecker_symbol(c, p)
else:
return 0
def IsSquareInQp(x, p):
if x == 0:
return true
# Check parity of valuation
v = valuation(x, p)
if v % 2:
return false
# Renormalise to get a unit
x //= p**v
# Reduce mod p and conclude
if p == 2:
return Mod(x, 8) == 1
else:
return kronecker_symbol(x, p) == 1
def assert_jordan_blcs(self, p, mat):
p = ZZ(p)
if p == 2:
blcs = jordan_blocks_2(mat)
else:
blcs = jordan_blocks_odd(mat, p)
q = QuadraticForm(ZZ, ZZ(2) * mat)
q1 = _blocks_to_quad_form(blcs.blocks, p)
if p == 2:
self.assertTrue((q.det() / q1.det()) % 8 == 1)
else:
self.assertTrue(kronecker_symbol(q.det() / q1.det(), p) == 1)
self.assertEqual(
q.hasse_invariant__OMeara(p), q1.hasse_invariant__OMeara(p))
def IsSquareInQp(x,p):
if x==0:
return true
# Check parity of valuation
v=valuation(x,p)
if v%2:
return false
# Renormalise to get a unit
x//=p**v
# Reduce mod p and conclude
if p==2:
return Mod(x,8)==1
else:
return kronecker_symbol(x,p)==1
def assert_jordan_blocks_method(self, p, mat):
p = ZZ(p)
if p == 2:
blcs = jordan_blocks_2(mat)
else:
blcs = jordan_blocks_odd(mat, p)
q = QuadraticForm(ZZ, 2 * mat)
if p == 2:
should1 = (q.Gram_det() / blcs.Gram_det()) % 8
else:
should1 = kronecker_symbol((q.Gram_det() / blcs.Gram_det()), p)
self.assertEqual(should1, 1)
self.assertEqual(
q.hasse_invariant__OMeara(p), blcs.hasse_invariant__OMeara())
self.assertEqual(q.dim(), blcs.dim())
self.assertEqual(q.content().valuation(p), blcs.content_order())
def check_inert_primes(self, rec, verbose=False):
"""
for each discriminant D in self_twist_discs, check that for
each prime p not dividing the level for which (D/p) = -1,
check that traces[p] = 0 (we could also check values in
mf_hecke_nf and/or mf_hecke_cc, but this would be far more
costly)
"""
# TIME about 3600s for full table
N = rec['level']
traces = [0] + rec['traces'] # shift so indexing correct
primes = [p for p in prime_range(len(traces)) if N % p != 0]
for D in rec['self_twist_discs']:
for p in primes:
if kronecker_symbol(D, p) == -1 and traces[p] != 0:
if verbose:
print("CM failure", D, p, traces[p])
return False
return True
def check_inert_primes(self, rec, verbose=False):
"""
for each discriminant D in self_twist_discs, check that for
each prime p not dividing the level for which (D/p) = -1,
check that traces[p] = 0 (we could also check values in
mf_hecke_nf and/or mf_hecke_cc, but this would be far more
costly)
"""
# TIME about 3600s for full table
N = rec['level']
traces = [0] + rec['traces'] # shift so indexing correct
primes = [p for p in prime_range(len(traces)) if N % p != 0]
for D in rec['self_twist_discs']:
for p in primes:
if kronecker_symbol(D, p) == -1 and traces[p] != 0:
if verbose:
print "CM failure", D, p, traces[p]
return False
return True