def test_relations_plusminus(self):
I = SL2Z([1, 0, 0, 1])
S = SL2Z([0, -1, 1, 0])
U = SL2Z([1, -1, 1, 0])
U2 = SL2Z([0, -1, 1, -1])
lS = [(1, I), (1, S)]
lU = [(1, I), (1, U), (1, U2)]
ppp1 = {}
ppp2 = {}
ppp3 = {}
ppp4 = {}
for n in range(self._dim):
p1 = self.action_by_lin_comb(n, lS, sym=1)
p2 = self.action_by_lin_comb(n, lU, sym=1)
ppp1[n] = p1
ppp2[n] = p2
p1 = self.action_by_lin_comb(n, lS, sym=-1)
p2 = self.action_by_lin_comb(n, lU, sym=-1)
ppp3[n] = p1
ppp4[n] = p2
for pp in [ppp1, ppp2, ppp3, ppp4]:
maxv = 0
for p in pp.values():
if hasattr(p, "coeffs"):
if p.coefficients(sparse=False) <> []:
tmp = max(map(abs, p.coefficients(sparse=False)))
else:
tmp = 0
else:
tmp = abs(p)
if tmp > maxv:
maxv = tmp
print "maxv=", maxv
def _set_coset_reps(self):
if is_prime(self._level):
self._coset_reps[0] = SL2Z([1, 0, 0, 1])
for n in range(self._level):
self._coset_reps[n + 1] = SL2Z([0, -1, 1, n])
else:
G = Gamma0(self._level)
n = 0
for A in G.coset_reps():
self._coset_reps[n] = A
n += 1
def is_consistent(self, k):
r"""
Checks that v(-I)=(-1)^k,
"""
Z = SL2Z([-1, 0, 0, -1])
zi = CyclotomicField(4).gen()
v = self._action(Z)
if self._verbose > 0:
print("test consistency for k={0}".format(k))
print("v(Z)={0}".format(v))
if self._dim == 1:
if isinstance(k, Integer) or k.is_integral():
if is_even(k):
v1 = ZZ(1)
else:
v1 = ZZ(-1)
elif isinstance(k, Rational) and (k.denominator() == 2 or k == 0):
v1 = zi**(-QQ(2 * k))
if self._verbose > 0:
print("I**(-2k)={0}".format(v1))
else:
raise ValueError(
"Only integral and half-integral weight is currently supported! Got weight:{0} of type:{1}"
.format(k, type(k)))
else:
raise NotImplementedError(
"Override this function for vector-valued multipliers!")
return v1 == v
def petersson_norm(self, type=1):
r"""
Compute the Petersson norm of f.
ALGORITHM:
Uses
(f,f) = < rho_{f}^{+} | T - T^-1 , rho_f^{-}>
for k even and
(f,f) = < rho_{f}^{+} | T - T^-1 , rho_f^{+}>
for k odd.
See e.g. Thm 3.3. in Pasol - Popa "Modular forms and period polynomials"
"""
if self._peterson_norm <> 0:
return self._peterson_norm
T = SL2Z([1, 1, 0, 1])
Ti = SL2Z([1, -1, 0, 1])
norm = 0
for n in range(self._dim):
if type == 1:
p1 = self.slash_action(n, T, sym='plus')
p2 = self.slash_action(n, Ti, sym='plus')
else:
p1 = self.slash_action(n, T, sym='minus')
p2 = self.slash_action(n, Ti, sym='minus')
pL = p1 - p2
#print "Pl=",pL
if self.function().weight() % 2 == 1:
if type == 1:
pR = self.polynomial_plus(n)
else:
pR = self.polynomial_minus(n)
else:
if type == 1:
pR = self.polynomial_minus(n)
else:
pR = self.polynomial_plus(n)
#print "PR=",pR
t = self.pair_two_pols(pL, pR)
#print "t=",t
norm += t
CF = ComplexField(self._prec)
c2 = CF(3) * CF(0, 2)**(self._k - 1)
self._peterson_norm = -norm / c2
return self._peterson_norm
def __repr__(self):
s = "Multiplier system defined by action on the generators:"
if self._group == SL2Z:
S, T = SL2Z.gens()
Z = S * S
s += "r(S)=", self.vals[S]
s += "r(T)=", self.vals[T]
s += "r(Z)=", self.vals[Z]
else:
for g in self.vals.keys():
s += "r(", g, ")=", self.vals[g]
return s
def __repr__(self):
s="Multiplier system defined by action on the generators:"
if self._group==SL2Z:
S,T=SL2Z.gens()
Z=S*S
s+="r(S)=",self.vals[S]
s+="r(T)=",self.vals[T]
s+="r(Z)=",self.vals[Z]
else:
for g in self.vals.keys():
s+="r(",g,")=",self.vals[g]
return s
def _get_shifted_coset_m(self, n, A):
r"""
Get the index m s.t. A_n*A**-1 is in Gamma0(N)A_m
"""
An = self.coset_rep(n)
if A.det() == 1:
m = self._get_coset_n(An * A**-1)
elif A.det() == -1 and A[0, 1] == 0 and A[1, 0] == 0:
AE = SL2Z([An[0, 0], -An[0, 1], -An[1, 0], An[1, 1]])
m = self._get_coset_n(AE)
else:
raise ValueError, "Call with SL2Z element or [-1,0,1,0]. Got:{0}".format(
E)
return m
def test_relations(self):
I = SL2Z([1, 0, 0, 1])
S = SL2Z([0, -1, 1, 0])
U = SL2Z([1, -1, 1, 0])
U2 = SL2Z([0, -1, 1, -1])
lS = [(1, I), (1, S)]
lU = [(1, I), (1, U), (1, U2)]
pS = self * lS
pU = self * lU
maxS = 0
maxU = 0
if self._verbose > 0:
print "pS=", pS
print "pU=", pU
for p in pS.values():
if hasattr(p, "coeffs"):
if p.coefficients(sparse=False) <> []:
tmp = max(map(abs, p.coefficients(sparse=False)))
else:
tmp = 0
else:
tmp = abs(p)
if tmp > maxS:
maxS = tmp
for p in pU.values():
if hasattr(p, "coefficients"):
if p.coefficients(sparse=False) <> []:
tmp = max(map(abs, p.coefficients(sparse=False)))
else:
tmp = 0
else:
tmp = abs(p)
if tmp > maxU:
maxU = tmp
print "Max coeff of self|(1+S)=", maxS
print "Max coeff of self|(1+U+U^2)=", maxU
def get_rk(self, n, k):
r"""
Compute the coefficient r_k((f|A_n))
"""
if (n, k) not in self._rks:
S, T = SL2Z.gens()
nstar = self._get_shifted_coset_m(n, S)
kstar = self._k - 2 - k
i1 = self.get_integral_from_1_to_oo(n, k)
i2 = self.get_integral_from_1_to_oo(nstar, kstar)
if self._verbose > 0:
print("i1={0}".format(i1))
print("i2={0}".format(i2))
self._rks[(n, k)] = i1 + i2 * (-1)**(k + 1)
return self._rks[(n, k)]
def get_rk(self, n, k):
r"""
Compute the coefficient r_k((f|A_n))
"""
if not self._rks.has_key((n, k)):
S, T = SL2Z.gens()
nstar = self._get_shifted_coset_m(n, S)
kstar = self._k - 2 - k
i1 = self.get_integral_from_1_to_oo(n, k)
i2 = self.get_integral_from_1_to_oo(nstar, kstar)
if self._verbose > 0:
print "i1=", i1
print "i2=", i2
self._rks[(n, k)] = i1 + i2 * (-1)**(k + 1)
return self._rks[(n, k)]
def get_rk(self,n,k):
r"""
Compute the coefficient r_k((f|A_n))
"""
if not self._rks.has_key((n,k)):
S,T = SL2Z.gens()
nstar = self._get_shifted_coset_m(n,S)
kstar = self._k-2-k
i1 = self.get_integral_from_1_to_oo(n,k)
i2 = self.get_integral_from_1_to_oo(nstar,kstar)
if self._verbose>0:
print "i1=",i1
print "i2=",i2
self._rks[(n,k)] = i1+i2*(-1)**(k+1)
return self._rks[(n,k)]
def _action(self, A):
#if A not in self._group:
# raise ValueError,"Action is only defined for {0}!".format(self._group)
a, b, c, d = A
[z, n, l] = factor_matrix_in_sl2z(int(a), int(b), int(c), int(d))
#l,ep = factor_matrix_in_sl2z_in_S_and_T(A_in)
res = copy(self.T.parent().one())
if z == -1 and self.v != None:
tmp = self.v(SL2Z([-1, 0, 0, -1]))
for j in range(self._dim):
res[j, j] = tmp
if n != 0:
res = self.T**n
for i in range(len(l)):
res = res * self.S * self.T**l[i]
return res
def weight(self):
r"""
Return (modulo 2) which weight self is a multiplier system consistent with
"""
if self._is_trivial:
self._weight = 0
return self._weight
if self._dim == 1:
if self._weight is None:
Z = SL2Z([-1, 0, 0, -1])
v = self._action(Z)
if v == 1:
self._weight = QQ(0)
if v == -1:
self._weight = QQ(1)
elif v == -I:
self._weight = QQ(1) / QQ(2)
elif v == I:
self._weight = QQ(1) / QQ(2)
else:
raise NotImplementedError
return self._weight
def __init__(self,G,v=None,**kwargs):
r"""
G should be a subgroup of PSL(2,Z).
EXAMPLE::
sage: te=TestMultiplier(Gamma0(2),weight=1/2)
sage: r=InducedRepresentation(Gamma0(2),v=te)
"""
dim = len(list(G.coset_reps()))
MultiplierSystem.__init__(self,Gamma0(1),dimension=dim)
self._induced_from=G
# setup the action on S and T (should be faster...)
self.v = v
if v<>None:
k = v.order()
if k>2:
K = CyclotomicField(k)
else:
K=ZZ
self.S=matrix(K,dim,dim)
self.T=matrix(K,dim,dim)
else:
self.S=matrix(dim,dim)
self.T=matrix(dim,dim)
S,T=SL2Z.gens()
if hasattr(G,"coset_reps"):
if isinstance(G.coset_reps(),list):
Vl=G.coset_reps()
else:
Vl=list(G.coset_reps())
elif hasattr(G,"_G"):
Vl=list(G._G.coset_reps())
else:
raise ValueError,"Could not get coset representatives from {0}!".format(G)
self.repsT=dict()
self.repsS=dict()
for i in range(dim):
Vi=Vl[i]
for j in range(dim):
Vj=Vl[j]
BS = Vi*S*Vj**-1
BT = Vi*T*Vj**-1
#print "i,j
#print "ViSVj^-1=",BS
#print "ViTVj^-1=",BT
if BS in G:
if v<>None:
vS=v(BS)
else:
vS=1
self.S[i,j]=vS
self.repsS[(i,j)]=BS
if BT in G:
if v<>None:
vT=v(BT)
else:
vT=1
self.T[i,j]=vT
self.repsT[(i,j)]=BT
def rho(self,M,silent=0,numeric=0,prec=-1):
r""" The Weil representation acting on SL(2,Z).
INPUT::
-``M`` -- element of SL2Z
- ''numeric'' -- set to 1 to return a Matrix_complex_dense with prec=prec instead of exact
- ''prec'' -- precision
EXAMPLES::
sage: WR=WeilRepDiscriminantForm(1,dual=False)
sage: S,T=SL2Z.gens()
sage: WR.rho(S)
[
[-zeta8^3 -zeta8^3]
[-zeta8^3 zeta8^3], sqrt(1/2)
]
sage: WR.rho(T)
[
[ 1 0]
[ 0 -zeta8^2], 1
]
sage: A=SL2Z([-1,1,-4,3]); WR.rho(A)
[
[zeta8^2 0]
[ 0 1], 1
]
sage: A=SL2Z([41,77,33,62]); WR.rho(A)
[
[-zeta8^3 zeta8^3]
[ zeta8 zeta8], sqrt(1/2)
]
"""
N=self._N; D=2*N; D2=2*D
if numeric==0:
K=CyclotomicField (lcm(4*self._N,8))
z=K(CyclotomicField(4*self._N).gen())
rho=matrix(K,D)
else:
CF = MPComplexField(prec)
RF = CF.base()
MS = MatrixSpace(CF,int(D),int(D))
rho = Matrix_complex_dense(MS)
#arg = RF(2)*RF.pi()/RF(4*self._N)
z = CF(0,RF(2)*RF.pi()/RF(4*self._N)).exp()
[a,b,c,d]=M
fak=1; sig=1
if c<0:
# need to use the reflection
# r(-A)=r(Z)r(A)sigma(Z,A) where sigma(Z,A)=-1 if c>0
sig=-1
if numeric==0:
fz=CyclotomicField(4).gen() # = i
else:
fz=CF(0,1)
# the factor is rho(Z) sigma(Z,-A)
#if(c < 0 or (c==0 and d>0)):
# fak=-fz
#else:
#sig=1
#fz=1
fak=fz
a=-a; b=-b; c=-c; d=-d;
A=SL2Z([a,b,c,d])
if numeric==0:
chi=self.xi(A)
else:
chi=CF(self.xi(A).complex_embedding(prec))
if(silent>0):
print("fz={0}".format(fz))
print("chi={0}".format(chi))
elif c == 0: # then we use the simple formula
if d < 0:
sig=-1
if numeric == 0:
fz=CyclotomicField(4).gen()
else:
fz=CF(0,1)
fak=fz
a=-a; b=-b; c=-c; d=-d;
else:
fak=1
for alpha in range(D):
arg=(b*alpha*alpha ) % D2
if(sig==-1):
malpha = (D - alpha) % D
rho[malpha,alpha]=fak*z**arg
else:
#print "D2=",D2
#print "b=",b
#print "arg=",arg
rho[alpha,alpha]=z**arg
return [rho,1]
else:
if numeric==0:
chi=self.xi(M)
else:
chi=CF(self.xi(M).complex_embedding(prec))
Nc=gcd(Integer(D),Integer(c))
#chi=chi*sqrt(CF(Nc)/CF(D))
if(valuation(Integer(c),2)==valuation(Integer(D),2)):
xc=Integer(N)
else:
xc=0
if silent>0:
print("c={0}".format(c))
print("xc={0}".format(xc))
print("chi={0}".format(chi))
for alpha in range(D):
al=QQ(alpha)/QQ(D)
for beta in range(D):
be=QQ(beta)/QQ(D)
c_div=False
if(xc==0):
alpha_minus_dbeta=(alpha-d*beta) % D
else:
alpha_minus_dbeta=(alpha-d*beta-xc) % D
if silent > 0: # and alpha==7 and beta == 7):
print("alpha,beta={0},{1}".format(alpha,beta))
print("c,d={0},{1}".format(c,d))
print("alpha-d*beta={0}".format(alpha_minus_dbeta))
invers=0
for r in range(D):
if (r*c - alpha_minus_dbeta) % D ==0:
c_div=True
invers=r
break
if c_div and silent > 0:
print("invers={0}".format(invers))
print(" inverse(alpha-d*beta) mod c={0}".format(invers))
elif(silent>0):
print(" no inverse!")
if(c_div):
y=invers
if xc==0:
argu=a*c*y**2+b*d*beta**2+2*b*c*y*beta
else:
argu=a*c*y**2+2*xc*(a*y+b*beta)+b*d*beta**2+2*b*c*y*beta
argu = argu % D2
tmp1=z**argu # exp(2*pi*I*argu)
if silent>0:# and alpha==7 and beta==7):
print("a,b,c,d={0},{1},{2},{3}".format(a,b,c,d))
print("xc={0}".format(xc))
print("argu={0}".format(argu))
print("exp(...)={0}".format(tmp1))
print("chi={0}".format(chi))
print("sig={0}".format(sig))
if sig == -1:
minus_alpha = (D - alpha) % D
rho[minus_alpha,beta]=tmp1*chi
else:
rho[alpha,beta]=tmp1*chi
#print "fak=",fak
if numeric==0:
return [fak*rho,sqrt(QQ(Nc)/QQ(D))]
else:
return [CF(fak)*rho,RF(sqrt(QQ(Nc)/QQ(D)))]
def nice_coset_reps(G):
r"""
Compute a better/nicer list of right coset representatives [V_j]
i.e. SL2Z = \cup G V_j
Use this routine for known congruence subgroups.
EXAMPLES::
sage: G=MySubgroup(Gamma0(5))
sage: G._get_coset_reps_from_G(Gamma0(5))
[[1 0]
[0 1], [ 0 -1]
[ 1 0], [ 0 -1]
[ 1 1], [ 0 -1]
[ 1 -1], [ 0 -1]
[ 1 2], [ 0 -1]
[ 1 -2]]
"""
cl = list()
S, T = SL2Z.gens()
lvl = G.generalised_level()
# Start with identity rep.
cl.append(SL2Z([1, 0, 0, 1]))
if (not S in G):
cl.append(S)
# If the original group is given as a Gamma0 then
# the reps are not the one we want
# I.e. we like to have a fundamental domain in
# -1/2 <=x <= 1/2 for Gamma0, Gamma1, Gamma
for j in range(1, ZZ(ceil(RR(lvl / 2.0)) + 2)):
for ep in [1, -1]:
if (len(cl) >= G.index()):
break
# The ones about 0 are all of this form
A = SL2Z([0, -1, 1, ep * j])
# just make sure they are inequivalent
try:
for V in cl:
if ((A <> V and A * V**-1 in G) or cl.count(A) > 0):
raise StopIteration()
cl.append(A)
except StopIteration:
pass
# We now addd the rest of the "flips" of these reps.
# So that we end up with a connected domain
i = 1
while (True):
lold = len(cl)
for V in cl:
for A in [S, T, T**-1]:
B = V * A
try:
for W in cl:
if ((B * W**-1 in G) or cl.count(B) > 0):
raise StopIteration()
cl.append(B)
except StopIteration:
pass
if (len(cl) >= G.index() or lold >= len(cl)):
# If we either did not addd anything or if we addded enough
# we exit
break
# If we missed something (which is unlikely)
if (len(cl) <> G.index()):
print "cl=", cl
raise ValueError, "Problem getting coset reps! Need %s and got %s" % (
G.index(), len(cl))
return cl
def nice_coset_reps(G):
r"""
Compute a better/nicer list of right coset representatives [V_j]
i.e. SL2Z = \cup G V_j
Use this routine for known congruence subgroups.
EXAMPLES::
sage: G=MySubgroup(Gamma0(5))
sage: G._get_coset_reps_from_G(Gamma0(5))
[[1 0]
[0 1], [ 0 -1]
[ 1 0], [ 0 -1]
[ 1 1], [ 0 -1]
[ 1 -1], [ 0 -1]
[ 1 2], [ 0 -1]
[ 1 -2]]
"""
cl=list()
S,T=SL2Z.gens()
lvl=G.generalised_level()
# Start with identity rep.
cl.append(SL2Z([1 ,0 ,0 ,1 ]))
if(not S in G):
cl.append(S)
# If the original group is given as a Gamma0 then
# the reps are not the one we want
# I.e. we like to have a fundamental domain in
# -1/2 <=x <= 1/2 for Gamma0, Gamma1, Gamma
for j in range(1 , ZZ( ceil(RR(lvl/2.0))+2)):
for ep in [1 ,-1 ]:
if(len(cl)>=G.index()):
break
# The ones about 0 are all of this form
A=SL2Z([0 ,-1 ,1 ,ep*j])
# just make sure they are inequivalent
try:
for V in cl:
if((A<>V and A*V**-1 in G) or cl.count(A)>0 ):
raise StopIteration()
cl.append(A)
except StopIteration:
pass
# We now addd the rest of the "flips" of these reps.
# So that we end up with a connected domain
i=1
while(True):
lold=len(cl)
for V in cl:
for A in [S,T,T**-1 ]:
B=V*A
try:
for W in cl:
if( (B*W**-1 in G) or cl.count(B)>0 ):
raise StopIteration()
cl.append(B)
except StopIteration:
pass
if(len(cl)>=G.index() or lold>=len(cl)):
# If we either did not addd anything or if we addded enough
# we exit
break
# If we missed something (which is unlikely)
if(len(cl)<>G.index()):
print "cl=",cl
raise ValueError,"Problem getting coset reps! Need %s and got %s" %(G.index(),len(cl))
return cl
def __init__(self, G, v=None, **kwargs):
r"""
G should be a subgroup of PSL(2,Z).
EXAMPLE::
sage: te=TestMultiplier(Gamma0(2),weight=1/2)
sage: r=InducedRepresentation(Gamma0(2),v=te)
"""
dim = len(list(G.coset_reps()))
MultiplierSystem.__init__(self, Gamma0(1), dimension=dim)
self._induced_from = G
# setup the action on S and T (should be faster...)
self.v = v
if v != None:
k = v.order()
if k > 2:
K = CyclotomicField(k)
else:
K = ZZ
self.S = matrix(K, dim, dim)
self.T = matrix(K, dim, dim)
else:
self.S = matrix(dim, dim)
self.T = matrix(dim, dim)
S, T = SL2Z.gens()
if hasattr(G, "coset_reps"):
if isinstance(G.coset_reps(), list):
Vl = G.coset_reps()
else:
Vl = list(G.coset_reps())
elif hasattr(G, "_G"):
Vl = list(G._G.coset_reps())
else:
raise ValueError(
"Could not get coset representatives from {0}!".format(G))
self.repsT = dict()
self.repsS = dict()
for i in range(dim):
Vi = Vl[i]
for j in range(dim):
Vj = Vl[j]
BS = Vi * S * Vj**-1
BT = Vi * T * Vj**-1
#print "i,j
#print "ViSVj^-1=",BS
#print "ViTVj^-1=",BT
if BS in G:
if v != None:
vS = v(BS)
else:
vS = 1
self.S[i, j] = vS
self.repsS[(i, j)] = BS
if BT in G:
if v != None:
vT = v(BT)
else:
vT = 1
self.T[i, j] = vT
self.repsT[(i, j)] = BT
