def weight_one_half_dim(FQM,
use_reduction=True,
proof=False,
debug=0,
local=True):
N = Integer(FQM.level())
if not N % 4 == 0:
return 0
m = Integer(N / Integer(4))
d = 0
for l in m.divisors():
if is_squarefree(m / l):
if debug > 1: print "l = {0}".format(l)
TM = FiniteQuadraticModule([2 * l], [-1 / Integer(4 * l)])
if local:
dd = [0, 0] # eigenvalue 1, -1 multiplicity
for p, n in lcm(FQM.level(), 4 * l).factor():
N = None
TN = None
J = FQM.jordan_decomposition()
L = TM.jordan_decomposition()
for j in xrange(1, n + 1):
C = J.constituent(p**j)[0]
D = L.constituent(p**j)[0]
if debug > 1: print "C = {0}, D = {1}".format(C, D)
if N == None and C.level() != 1:
N = C
elif C.level() != 1:
N = N + C
if TN == None and D.level() != 1:
TN = D
elif D.level() != 1:
TN = TN + D
dd1 = invariants_eps(N, TN, use_reduction, proof, debug)
if debug > 1: print "dd1 = {}".format(dd1)
if dd1 == [0, 0]:
# the result is multiplicative
# a single [0,0] as a local result
# yields [0,0] in the end
# and we're done here
dd = [0, 0]
break
if dd == [0, 0]:
# this is the first prime
dd = dd1
else:
# some basic arithmetic ;-)
# 1 = 1*1 = (-1)(-1)
# -1 = 1*(-1) = (-1)*1
ddtmp = copy(dd)
ddtmp[0] = dd[0] * dd1[0] + dd[1] * dd1[1]
ddtmp[1] = dd[0] * dd1[1] + dd[1] * dd1[0]
dd = ddtmp
if debug > 1: print "dd = {0}".format(dd)
d += dd[0]
else:
d += invariants_eps(FQM, TM, use_reduction, proof, debug)[0]
return d
def weight_one_half_dim(FQM, use_reduction = True, proof = False, debug = 0, local=True):
N = Integer(FQM.level())
if not N % 4 == 0:
return 0
m = Integer(N/Integer(4))
d = 0
for l in m.divisors():
if is_squarefree(m/l):
if debug > 1: print "l = {0}".format(l)
TM = FiniteQuadraticModule([2*l],[-1/Integer(4*l)])
if local:
dd = [0,0] # eigenvalue 1, -1 multiplicity
for p,n in lcm(FQM.level(),4*l).factor():
N = None
TN = None
J = FQM.jordan_decomposition()
L = TM.jordan_decomposition()
for j in xrange(1,n+1):
C = J.constituent(p**j)[0]
D = L.constituent(p**j)[0]
if debug > 1: print "C = {0}, D = {1}".format(C,D)
if N == None and C.level() != 1:
N = C
elif C.level() != 1:
N = N + C
if TN == None and D.level() != 1:
TN = D
elif D.level() != 1:
TN = TN + D
dd1 = invariants_eps(N, TN, use_reduction, proof, debug)
if debug > 1: print "dd1 = {}".format(dd1)
if dd1 == [0,0]:
# the result is multiplicative
# a single [0,0] as a local result
# yields [0,0] in the end
# and we're done here
dd = [0,0]
break
if dd == [0,0]:
# this is the first prime
dd = dd1
else:
# some basic arithmetic ;-)
# 1 = 1*1 = (-1)(-1)
# -1 = 1*(-1) = (-1)*1
ddtmp = copy(dd)
ddtmp[0] = dd[0]*dd1[0] + dd[1]*dd1[1]
ddtmp[1] = dd[0]*dd1[1] + dd[1]*dd1[0]
dd = ddtmp
if debug > 1: print "dd = {0}".format(dd)
d += dd[0]
else:
d += invariants_eps(FQM, TM, use_reduction, proof, debug)[0]
return d
def compute_charpolies():
"""Compute the characteristic polynomials of homology actions of finite
order mapping classes.
"""
from sage.all import Integer
data = {}
for genus in finite_order_primitives.keys():
data[genus] = []
count = 0
for order, f in finite_order_primitives[genus]:
count += 1
order = Integer(order)
for divisor in order.divisors():
if divisor == order:
continue
g = f**divisor
action = g.action_on_homology()
char_poly = action.charpoly()
data[genus].append(
(order // divisor, char_poly.factor(), order))
data[genus].sort()
return data
