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from sage.all import *
import itertools
from ..theta_structures.theta_helpers_dim4 import multindex_to_index
def bloc_decomposition(M):
I1=[0,1,2,3]
I2=[4,5,6,7]
A=M[I1,I1]
B=M[I2,I1]
C=M[I1,I2]
D=M[I2,I2]
return A,B,C,D
def mat_prod_vect(A,I):
J=[]
for i in range(4):
J.append(0)
for k in range(4):
J[i]+=A[i,k]*I[k]
return tuple(J)
def add_tuple(I,J):
K=[]
for k in range(4):
K.append(I[k]+J[k])
return tuple(K)
def scal_prod_tuple(I,J):
s=0
for k in range(4):
s+=I[k]*J[k]
return s
def red_mod_2(I):
J=[]
for x in I:
J.append(ZZ(x)%2)
return tuple(J)
def choose_non_vanishing_index(C,D,zeta):
for I0 in itertools.product([0,1],repeat=4):
L=[0 for k in range(16)]
for J in itertools.product([0,1],repeat=4):
CJ=mat_prod_vect(C,J)
DJ=mat_prod_vect(D,J)
e=-scal_prod_tuple(CJ,DJ)-2*scal_prod_tuple(I0,DJ)
I0pDJ=add_tuple(I0,CJ)
L[multindex_to_index(red_mod_2(I0pDJ))]+=zeta**(ZZ(e))
for k in range(16):
if L[k]!=0:
return I0,L
def base_change_theta_dim4(M,zeta):
Z4=Integers(4)
A,B,C,D=bloc_decomposition(M.change_ring(Z4))
I0,L0=choose_non_vanishing_index(C,D,zeta)
N=[L0]+[[0 for j in range(16)] for i in range(15)]
for I in itertools.product([0,1],repeat=4):
if I!=(0,0,0,0):
AI=mat_prod_vect(A,I)
BI=mat_prod_vect(B,I)
for J in itertools.product([0,1],repeat=4):
CJ=mat_prod_vect(C,J)
DJ=mat_prod_vect(D,J)
AIpCJ=add_tuple(AI,CJ)
BIpDJ=add_tuple(BI,DJ)
e=scal_prod_tuple(I,J)-scal_prod_tuple(AIpCJ,BIpDJ)-2*scal_prod_tuple(I0,BIpDJ)
N[multindex_to_index(I)][multindex_to_index(red_mod_2(add_tuple(AIpCJ,I0)))]+=zeta**(ZZ(e))
Fp2=zeta.parent()
return matrix(Fp2,N)
def apply_base_change_theta_dim4(N,P):
Q=[]
for i in range(16):
Q.append(0)
for j in range(16):
Q[i]+=N[i,j]*P[j]
return Q
def complete_symplectic_matrix_dim4(C,D,n=4):
Zn=Integers(n)
Col_I4=[matrix(Zn,[[1],[0],[0],[0]]),matrix(Zn,[[0],[1],[0],[0],[0]]),
matrix(Zn,[[0],[0],[1],[0],[0],[0]]),matrix(Zn,[[0],[0],[0],[1],[0],[0],[0]])]
L_DC=block_matrix([[D.transpose(),-C.transpose()]])
Col_AB_i=L_DC.solve_right(Col_I4[0])
A_t=Col_AB_i[[0,1,2,3],0].transpose()
B_t=Col_AB_i[[4,5,6,7],0].transpose()
for i in range(1,4):
F=block_matrix(2,1,[L_DC,block_matrix(1,2,[B_t,-A_t])])
Col_AB_i=F.solve_right(Col_I4[i])
A_t=block_matrix(2,1,[A_t,Col_AB_i[[0,1,2,3],0].transpose()])
B_t=block_matrix(2,1,[B_t,Col_AB_i[[4,5,6,7],0].transpose()])
A=A_t.transpose()
B=B_t.transpose()
M=block_matrix([[A,C],[B,D]])
return M
def random_symmetric_matrix(n,r):
Zn=Integers(n)
M=zero_matrix(Zn,r,r)
for i in range(r):
M[i,i]=randint(0,n-1)
for i in range(r):
for j in range(i+1,r):
M[i,j]=randint(0,n-1)
M[j,i]=M[i,j]
return M
def random_symplectic_matrix(n):
Zn=Integers(n)
C=random_symmetric_matrix(n,4)
Cp=random_symmetric_matrix(n,4)
A=random_matrix(Zn,4,4)
while not A.is_invertible():
A=random_matrix(Zn,4,4)
tA_inv=A.inverse().transpose()
ACp=A*Cp
return block_matrix([[ACp,A],[C*ACp-tA_inv,C*A]])
def is_symplectic_matrix_dim4(M):
A,B,C,D=bloc_decomposition(M)
if B.transpose()*A!=A.transpose()*B:
return False
if C.transpose()*D!=D.transpose()*C:
return False
if A.transpose()*D-B.transpose()*C!=identity_matrix(4):
return False
return True
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