def __init__(self,
parent,
matrix,
inhomogeneous,
is_zero=lambda p: False,
relations=[]):
## Checking the input of matrix and vector
if (not SAGE_element.is_Matrix(matrix)):
matrix = Matrix(matrix)
if (not SAGE_element.is_Vector(inhomogeneous)):
inhomogeneous = vector(inhomogeneous)
if (isinstance(parent, Wrap_w_Sequence_Ring)):
parent = parent.base()
## Building the parent of the matrix and the vector
pmatrix = matrix.parent().base()
pinhom = inhomogeneous.parent().base()
## Setting the variables for the matrix, the parent and the vector
self.__parent = pushout(pmatrix, pinhom)
self.__matrix = matrix.change_ring(self.__parent)
self.__inhomogeneous = inhomogeneous.change_ring(self.__parent)
## Setting the parent for the solutions
if (not pushout(parent, self.__parent) == parent):
try:
self.__matrix = self.__matrix.change_ring(parent)
self.__inhomogeneous = self.__inhomogeneous.change_ring(parent)
self.__parent = parent
except:
raise TypeError(
"The parent for the solutions must be an extension of the parent for the input"
)
self.__solution_parent = parent
## Setting other variables
if (relations is None):
relations = []
self.__relations = [self.__parent(el) for el in relations]
try:
self.__gb = ideal(self.parent(), self.__relations).groebner_basis()
except AttributeError:
try:
self.__gb = [ideal(self.parent(), self.__relations).gen()]
except:
self.__gb = [0]
self.__is_zero = is_zero
## Creating the variables for the echelon form
self.__echelon = None
self.__transformation = None
## Creating the variables for the solutions
self.__solution = None
self.__syzygy = None
def invariants_eps(FQM, TM, use_reduction=True, proof=False, debug=0):
r"""
Computes the invariants of a direct summand in the decomposition for weight
one-half modular forms. Such a summand is of the form
\[
\mathbb{C}[(\mathbb{Z}/2N\mathbb{Z}, -x^2/4N)]^\epsilon \otimes \mathbb{C}[M],
\]
where $M$ is a given finite quadratic module and $\epsilon$ acts on the first
factor via $\mathfrak{e}_\mu \mapsto \mathfrak{e}_{-\mu}$.
INPUT:
- FQM: A given finite quadratic module, referred to as $M$ above
- TM: A cyclic module of the form as given abve (the first factor)
NOTE:
We do not check that TM is of the stated form. The function is an auxiliary function
usually only called by ``weight_one_half_dim``.
EXAMPLES:
NONE
"""
eps = True
if TM != None and FQM != None:
TMM = TM + FQM
elif TM != None:
TMM = TM
else:
TMM = FQM
eps = False
debug2 = 0
if debug > 1:
print "FQM = {0}, TM = {1}, TMM = {2}".format(FQM, TM, TMM)
debug2 = 1
if debug > 2:
debug2 = debug
inv = invariants(TMM, use_reduction, proof=proof, debug=debug2)
if debug > 1: print inv
if type(inv) in [list, tuple]:
V = inv[1]
else:
V = inv
d = [0, 0]
if V.dimension() != 0:
el = list()
M = Matrix(V.base_ring(), V.ambient_module().dimension())
if eps:
f = 1 if TMM.signature() % 4 == 0 else -1
for v in inv[0]:
#Get the coordinate of this isotropic element
vv = v.c_list()
#change the first coordinate to its negative (the eps-action)
vv[0] = -vv[0]
vv = TMM(vv, can_coords=True)
#since the isotropic elements are taken up to the action of +-1, we need to check
#if we have this element (vv) or its negative (-vv) in the list
#we append the index of the match, together with a sign to the list `el`,
#where the sign is -1 if -vv is in inv[0] and the signature is 2 mod 4
#(i.e. the std generator of the center acts as -1)
if inv[0].count(vv) > 0:
el.append((inv[0].index(vv), 1))
else:
el.append((inv[0].index(-vv), f))
#We create the entries of the matrix M
#which acts as eps on the space spanned by the isotropic vectors (mod +-1)
for i in range(len(el)):
M[el[i][0], i] = el[i][1]
#import pdb; pdb.set_trace()
if debug > 1: print "M={0}, V={1}".format(M, V)
try:
KM = (M - M.parent().one()).kernel_on(V)
if debug > 1: print "KM for ev 1 = {0}".format(KM)
d[0] = KM.dimension()
KM = (M + M.parent().one()).kernel_on(V)
if debug > 1: print "KM for ev -1 = {0}".format(KM)
d[1] = KM.dimension()
except Exception as e:
raise RuntimeError(
"Error occured for {0}, {1}".format(
FQM.jordan_decomposition().genus_symbol(), e), M, V)
else:
d = [V.dimension(), 0]
if debug > 1: print d
return d
def invariants_eps(FQM, TM, use_reduction = True, proof = False, debug = 0):
r"""
Computes the invariants of a direct summand in the decomposition for weight
one-half modular forms. Such a summand is of the form
\[
\mathbb{C}[(\mathbb{Z}/2N\mathbb{Z}, -x^2/4N)]^\epsilon \otimes \mathbb{C}[M],
\]
where $M$ is a given finite quadratic module and $\epsilon$ acts on the first
factor via $\mathfrak{e}_\mu \mapsto \mathfrak{e}_{-\mu}$.
INPUT:
- FQM: A given finite quadratic module, referred to as $M$ above
- TM: A cyclic module of the form as given abve (the first factor)
NOTE:
We do not check that TM is of the stated form. The function is an auxiliary function
usually only called by ``weight_one_half_dim``.
EXAMPLES:
NONE
"""
eps = True
if TM != None and FQM != None:
TMM = TM+FQM
elif TM != None:
TMM = TM
else:
TMM = FQM
eps = False
debug2 = 0
if debug > 1:
print "FQM = {0}, TM = {1}, TMM = {2}".format(FQM, TM, TMM)
debug2 = 1
if debug > 2:
debug2=debug
inv = invariants(TMM, use_reduction, proof=proof, debug=debug2)
if debug > 1: print inv
if type(inv) in [list,tuple]:
V = inv[1]
else:
V = inv
d = [0,0]
if V.dimension() != 0:
el = list()
M = Matrix(V.base_ring(), V.ambient_module().dimension())
if eps:
f = 1 if TMM.signature() % 4 == 0 else -1
for v in inv[0]:
#Get the coordinate of this isotropic element
vv = v.c_list()
#change the first coordinate to its negative (the eps-action)
vv[0] = -vv[0]
vv = TMM(vv,can_coords=True)
#since the isotropic elements are taken up to the action of +-1, we need to check
#if we have this element (vv) or its negative (-vv) in the list
#we append the index of the match, together with a sign to the list `el`,
#where the sign is -1 if -vv is in inv[0] and the signature is 2 mod 4
#(i.e. the std generator of the center acts as -1)
if inv[0].count(vv) > 0:
el.append((inv[0].index(vv),1))
else:
el.append((inv[0].index(-vv),f))
#We create the entries of the matrix M
#which acts as eps on the space spanned by the isotropic vectors (mod +-1)
for i in range(len(el)):
M[el[i][0],i] = el[i][1]
#import pdb; pdb.set_trace()
if debug > 1: print "M={0}, V={1}".format(M, V)
try:
KM = (M-M.parent().one()).kernel_on(V)
if debug > 1: print "KM for ev 1 = {0}".format(KM)
d[0] = KM.dimension()
KM = (M+M.parent().one()).kernel_on(V)
if debug > 1: print "KM for ev -1 = {0}".format(KM)
d[1] = KM.dimension()
except Exception as e:
raise RuntimeError("Error occured for {0}, {1}".format(FQM.jordan_decomposition().genus_symbol(), e), M, V)
else:
d = [V.dimension(), 0]
if debug > 1: print d
return d
def invariants_eps(FQM, TM, use_reduction = True, proof = False, debug = 0):
r"""
Computes the invariants of a direct summand in the decomposition for weight
one-half modular forms. Such a summand is of the form
\[
\mathbb{C}[(\mathbb{Z}/2N\mathbb{Z}, -x^2/4N)]^\epsilon \otimes \mathbb{C}[M],
\]
where $M$ is a given finite quadratic module and $\epsilon$ acts on the first
factor via $\mathfrak{e}_\mu \mapsto \mathfrak{e}_{-\mu}$.
INPUT:
- FQM: A given finite quadratic module, referred to as $M$ above
- TM: A cyclic module of the form as given abve (the first factor)
NOTE:
We do not check that TM is of the stated form. The function is an auxiliary function
usually only called by ``weight_one_half_dim``.
EXAMPLES:
NONE
"""
eps = True
if TM != None and FQM != None:
TMM = TM+FQM
elif TM != None:
TMM = TM
else:
TMM = FQM
eps = False
if debug > 1: print "FQM = {0}, TM = {1}, TMM = {2}".format(FQM, TM, TMM)
if debug > 2:
debug2=debug
else:
debug2=0
inv = invariants(TMM, use_reduction, proof, debug2)
if debug > 1: print inv
if type(inv) in [list,tuple]:
V = inv[1]
else:
V = inv
d = [0,0]
if V.dimension() != 0:
el = list()
M = Matrix(V.base_ring(), V.ambient_module().dimension())
if eps:
f = 1 if TMM.signature() % 4 == 0 else -1
for v in inv[0]:
vv = v.c_list()
vv[0] = -vv[0]
vv = TMM(vv,can_coords=True)
if inv[0].count(vv) > 0:
el.append((inv[0].index(vv),1))
else:
el.append((inv[0].index(-vv),f))
for i in range(len(el)):
M[el[i][0],i] = el[i][1]
if debug > 1: print "M={0}, V={1}".format(M, V)
try:
KM = (M-M.parent().one()).kernel_on(V)
if debug > 1: print "KM for ev 1 = {0}".format(KM)
d[0] = KM.dimension()
KM = (M+M.parent().one()).kernel_on(V)
if debug > 1: print "KM for ev -1 = {0}".format(KM)
d[1] = KM.dimension()
except:
raise RuntimeError("Error occured for {0}".format(FQM.jordan_decomposition().genus_symbol()), M, V)
else:
d = [V.dimension(), 0]
if debug > 1: print d
return d