def setUp(self):
HomologyTestData.setUp(self)
# pre-compute the N1 matrices from the period matrices. note that
# Homology test data also contains N1 matrices "given" from the
# Kalla,Klein paper ("given" in quotes since only the H and Q matrices
# are actually produced)
R = involution_matrix(self.atrott, self.btrott)
S = integer_kernel_basis(R)
N1 = N1_matrix(self.atrott, self.btrott, S, tol=1e-3)
self.N1trott_from_periods = N1
R = involution_matrix(self.aklein, self.bklein, tol=1e-3)
S = integer_kernel_basis(R)
N1 = N1_matrix(self.aklein, self.bklein, S, tol=1e-3)
self.N1klein_from_periods = N1
R = involution_matrix(self.afermat, self.bfermat, tol=1e-3)
S = integer_kernel_basis(R)
N1 = N1_matrix(self.afermat, self.bfermat, S, tol=1e-3)
self.N1fermat_from_periods = N1
R = involution_matrix(self.a6, self.b6)
S = integer_kernel_basis(R)
N1 = N1_matrix(self.a6, self.b6, S)
self.N16_from_periods = N1
def test_integral(self):
# the integrality test is built into the function itself since the
# return type should be a matrix over ZZ
R = involution_matrix(self.atrott, self.btrott)
R = involution_matrix(self.aklein, self.bklein, tol=1e-3)
R = involution_matrix(self.afermat, self.bfermat, tol=1e-3)
R = involution_matrix(self.a6, self.b6)
def setUp(self):
HomologyTestData.setUp(self)
# pre-compute the N1 matrices from the period matrices. note that
# Homology test data also contains N1 matrices "given" from the
# Kalla,Klein paper ("given" in quotes since only the H and Q matrices
# are actually produced)
R = involution_matrix(self.atrott, self.btrott)
S = integer_kernel_basis(R)
N1 = N1_matrix(self.atrott, self.btrott, S, tol=1e-3)
self.N1trott_from_periods = N1
R = involution_matrix(self.aklein, self.bklein, tol=1e-3)
S = integer_kernel_basis(R)
N1 = N1_matrix(self.aklein, self.bklein, S, tol=1e-3)
self.N1klein_from_periods = N1
R = involution_matrix(self.afermat, self.bfermat, tol=1e-3)
S = integer_kernel_basis(R)
N1 = N1_matrix(self.afermat, self.bfermat, S, tol=1e-3)
self.N1fermat_from_periods = N1
R = involution_matrix(self.a6, self.b6)
S = integer_kernel_basis(R)
N1 = N1_matrix(self.a6, self.b6, S)
self.N16_from_periods = N1
def test_integral(self):
# the integrality test is built into the function itself since the
# return type should be a matrix over ZZ
R = involution_matrix(self.atrott, self.btrott)
R = involution_matrix(self.aklein, self.bklein, tol=1e-3)
R = involution_matrix(self.afermat, self.bfermat, tol=1e-3)
R = involution_matrix(self.a6, self.b6)
def test_symplectic(self):
# the symmetric transformation matrix should be symplectic
Pa = self.atrott
Pb = self.btrott
R = involution_matrix(Pa, Pb)
S = integer_kernel_basis(R)
N1 = N1_matrix(Pa, Pb, S)
H, Q = symmetric_block_diagonalize(N1)
Gamma = symmetric_transformation_matrix(Pa, Pb, S, H, Q, tol=1e-4)
g, g = Pa.dimensions()
J = zero_matrix(ZZ, 2 * g, 2 * g)
Ig = identity_matrix(ZZ, g, g)
J[:g, g:] = Ig
J[g:, :g] = -Ig
self.assertEqual(Gamma.T * J * Gamma, J)
Pa = self.aklein
Pb = self.bklein
R = involution_matrix(Pa, Pb, tol=1e-3)
S = integer_kernel_basis(R)
N1 = N1_matrix(Pa, Pb, S)
H, Q = symmetric_block_diagonalize(N1)
Gamma = symmetric_transformation_matrix(Pa, Pb, S, H, Q, tol=1e-3)
g, g = Pa.dimensions()
J = zero_matrix(ZZ, 2 * g, 2 * g)
Ig = identity_matrix(ZZ, g, g)
J[:g, g:] = Ig
J[g:, :g] = -Ig
self.assertEqual(Gamma.T * J * Gamma, J)
Pa = self.afermat
Pb = self.bfermat
R = involution_matrix(Pa, Pb, tol=1e-3)
S = integer_kernel_basis(R)
N1 = N1_matrix(Pa, Pb, S)
H, Q = symmetric_block_diagonalize(N1)
Gamma = symmetric_transformation_matrix(Pa, Pb, S, H, Q, tol=1e-3)
g, g = Pa.dimensions()
J = zero_matrix(ZZ, 2 * g, 2 * g)
Ig = identity_matrix(ZZ, g, g)
J[:g, g:] = Ig
J[g:, :g] = -Ig
self.assertEqual(Gamma.T * J * Gamma, J)
Pa = self.a6
Pb = self.b6
R = involution_matrix(Pa, Pb)
S = integer_kernel_basis(R)
N1 = N1_matrix(Pa, Pb, S)
H, Q = symmetric_block_diagonalize(N1)
Gamma = symmetric_transformation_matrix(Pa, Pb, S, H, Q, tol=1e-4)
g, g = Pa.dimensions()
J = zero_matrix(ZZ, 2 * g, 2 * g)
Ig = identity_matrix(ZZ, g, g)
J[:g, g:] = Ig
J[g:, :g] = -Ig
self.assertEqual(Gamma.T * J * Gamma, J)
def test_symplectic(self):
# the symmetric transformation matrix should be symplectic
Pa = self.atrott
Pb = self.btrott
R = involution_matrix(Pa, Pb)
S = integer_kernel_basis(R)
N1 = N1_matrix(Pa, Pb, S)
H,Q = symmetric_block_diagonalize(N1)
Gamma = symmetric_transformation_matrix(Pa, Pb, S, H, Q, tol=1e-4)
g,g = Pa.dimensions()
J = zero_matrix(ZZ,2*g,2*g)
Ig = identity_matrix(ZZ, g, g)
J[:g,g:] = Ig
J[g:,:g] = -Ig
self.assertEqual(Gamma.T*J*Gamma,J)
Pa = self.aklein
Pb = self.bklein
R = involution_matrix(Pa, Pb, tol=1e-3)
S = integer_kernel_basis(R)
N1 = N1_matrix(Pa, Pb, S)
H,Q = symmetric_block_diagonalize(N1)
Gamma = symmetric_transformation_matrix(Pa, Pb, S, H, Q, tol=1e-3)
g,g = Pa.dimensions()
J = zero_matrix(ZZ,2*g,2*g)
Ig = identity_matrix(ZZ, g, g)
J[:g,g:] = Ig
J[g:,:g] = -Ig
self.assertEqual(Gamma.T*J*Gamma,J)
Pa = self.afermat
Pb = self.bfermat
R = involution_matrix(Pa, Pb, tol=1e-3)
S = integer_kernel_basis(R)
N1 = N1_matrix(Pa, Pb, S)
H,Q = symmetric_block_diagonalize(N1)
Gamma = symmetric_transformation_matrix(Pa, Pb, S, H, Q, tol=1e-3)
g,g = Pa.dimensions()
J = zero_matrix(ZZ,2*g,2*g)
Ig = identity_matrix(ZZ, g, g)
J[:g,g:] = Ig
J[g:,:g] = -Ig
self.assertEqual(Gamma.T*J*Gamma,J)
Pa = self.a6
Pb = self.b6
R = involution_matrix(Pa, Pb)
S = integer_kernel_basis(R)
N1 = N1_matrix(Pa, Pb, S)
H,Q = symmetric_block_diagonalize(N1)
Gamma = symmetric_transformation_matrix(Pa, Pb, S, H, Q, tol=1e-4)
g,g = Pa.dimensions()
J = zero_matrix(ZZ,2*g,2*g)
Ig = identity_matrix(ZZ, g, g)
J[:g,g:] = Ig
J[g:,:g] = -Ig
self.assertEqual(Gamma.T*J*Gamma,J)
def test_diagonalizable(self):
# the involution matrix should be diagonalizable
R = involution_matrix(self.atrott, self.btrott)
RQQ = R.change_ring(QQ)
self.assertTrue(RQQ.is_diagonalizable())
R = involution_matrix(self.aklein, self.bklein, tol=1e-3)
RQQ = R.change_ring(QQ)
self.assertTrue(RQQ.is_diagonalizable())
R = involution_matrix(self.afermat, self.bfermat, tol=1e-3)
RQQ = R.change_ring(QQ)
self.assertTrue(RQQ.is_diagonalizable())
R = involution_matrix(self.a6, self.b6)
RQQ = R.change_ring(QQ)
self.assertTrue(RQQ.is_diagonalizable())
def test_eigenvalues(self):
# the eigenvalues of the involution matrix should be equal to -1 or 1
R = involution_matrix(self.atrott, self.btrott)
evals = R.eigenvalues()
self.assertSetEqual(set(evals), {1.0, -1.0})
R = involution_matrix(self.aklein, self.bklein, tol=1e-3)
evals = R.eigenvalues()
self.assertSetEqual(set(evals), {1.0, -1.0})
R = involution_matrix(self.afermat, self.bfermat, tol=1e-3)
evals = R.eigenvalues()
self.assertSetEqual(set(evals), {1.0, -1.0})
R = involution_matrix(self.a6, self.b6)
evals = R.eigenvalues()
self.assertSetEqual(set(evals), {1.0, -1.0})
def test_diagonalizable(self):
# the involution matrix should be diagonalizable
R = involution_matrix(self.atrott, self.btrott)
RQQ = R.change_ring(QQ)
self.assertTrue(RQQ.is_diagonalizable())
R = involution_matrix(self.aklein, self.bklein, tol=1e-3)
RQQ = R.change_ring(QQ)
self.assertTrue(RQQ.is_diagonalizable())
R = involution_matrix(self.afermat, self.bfermat, tol=1e-3)
RQQ = R.change_ring(QQ)
self.assertTrue(RQQ.is_diagonalizable())
R = involution_matrix(self.a6, self.b6)
RQQ = R.change_ring(QQ)
self.assertTrue(RQQ.is_diagonalizable())
def test_eigenvalues(self):
# the eigenvalues of the involution matrix should be equal to -1 or 1
R = involution_matrix(self.atrott, self.btrott)
evals = R.eigenvalues()
self.assertSetEqual(set(evals), {1.0, -1.0})
R = involution_matrix(self.aklein, self.bklein, tol=1e-3)
evals = R.eigenvalues()
self.assertSetEqual(set(evals), {1.0, -1.0})
R = involution_matrix(self.afermat, self.bfermat, tol=1e-3)
evals = R.eigenvalues()
self.assertSetEqual(set(evals), {1.0, -1.0})
R = involution_matrix(self.a6, self.b6)
evals = R.eigenvalues()
self.assertSetEqual(set(evals), {1.0, -1.0})
def test_symmetric(self):
# the N1 matrix should be symmetric
R = involution_matrix(self.atrott, self.btrott)
S = integer_kernel_basis(R)
N1 = N1_matrix(self.atrott, self.btrott, S)
self.assertEqual(N1, N1.T)
R = involution_matrix(self.aklein, self.bklein, tol=1e-3)
S = integer_kernel_basis(R)
N1 = N1_matrix(self.aklein, self.bklein, S, tol=1e-3)
self.assertEqual(N1, N1.T)
R = involution_matrix(self.afermat, self.bfermat, tol=1e-3)
S = integer_kernel_basis(R)
N1 = N1_matrix(self.afermat, self.bfermat, S, tol=1e-3)
self.assertEqual(N1, N1.T)
R = involution_matrix(self.a6, self.b6)
S = integer_kernel_basis(R)
N1 = N1_matrix(self.a6, self.b6, S)
self.assertEqual(N1, N1.T)
def test_symmetric(self):
# the N1 matrix should be symmetric
R = involution_matrix(self.atrott, self.btrott)
S = integer_kernel_basis(R)
N1 = N1_matrix(self.atrott, self.btrott, S)
self.assertEqual(N1, N1.T)
R = involution_matrix(self.aklein, self.bklein, tol=1e-3)
S = integer_kernel_basis(R)
N1 = N1_matrix(self.aklein, self.bklein, S, tol=1e-3)
self.assertEqual(N1, N1.T)
R = involution_matrix(self.afermat, self.bfermat, tol=1e-3)
S = integer_kernel_basis(R)
N1 = N1_matrix(self.afermat, self.bfermat, S, tol=1e-3)
self.assertEqual(N1, N1.T)
R = involution_matrix(self.a6, self.b6)
S = integer_kernel_basis(R)
N1 = N1_matrix(self.a6, self.b6, S)
self.assertEqual(N1, N1.T)
def test_recover_action(self):
# see equation (28) of Kalla,Klein
def compute_R(Gamma, H):
H = H.change_ring(ZZ)
g, g = H.dimensions()
A = Gamma[:g, :g]
B = Gamma[:g, g:]
C = Gamma[g:, :g]
D = Gamma[g:, g:]
Ig = identity_matrix(ZZ, g)
R = zero_matrix(ZZ, 2 * g, 2 * g)
R[:g, :g] = (2 * C.T * B - A.T * H * B + Ig).T
R[:g, g:] = 2 * D.T * B - B.T * H * B
R[g:, :g] = -2 * C.T * A + A.T * H * A
R[g:, g:] = -(2 * C.T * B - A.T * H * B + Ig)
return R
# trott curve
Pa = self.atrott
Pb = self.btrott
R = involution_matrix(Pa, Pb)
S = integer_kernel_basis(R)
N1 = N1_matrix(Pa, Pb, S)
H, Q = symmetric_block_diagonalize(N1)
Gamma = symmetric_transformation_matrix(Pa, Pb, S, H, Q, tol=1e-4)
Ralt = compute_R(Gamma, H)
self.assertEqual(R, Ralt)
# klein curve
Pa = self.aklein
Pb = self.bklein
R = involution_matrix(Pa, Pb, tol=1e-3)
S = integer_kernel_basis(R)
N1 = N1_matrix(Pa, Pb, S, tol=1e-3)
H, Q = symmetric_block_diagonalize(N1)
Gamma = symmetric_transformation_matrix(Pa, Pb, S, H, Q, tol=1e-4)
Ralt = compute_R(Gamma, H)
self.assertEqual(R, Ralt)
# fermat curve
Pa = self.afermat
Pb = self.bfermat
R = involution_matrix(Pa, Pb, tol=1e-3)
S = integer_kernel_basis(R)
N1 = N1_matrix(Pa, Pb, S, tol=1e-3)
H, Q = symmetric_block_diagonalize(N1)
Gamma = symmetric_transformation_matrix(Pa, Pb, S, H, Q, tol=1e-4)
Ralt = compute_R(Gamma, H)
self.assertEqual(R, Ralt)
# genus 6 curve
Pa = self.a6
Pb = self.b6
R = involution_matrix(Pa, Pb)
S = integer_kernel_basis(R)
N1 = N1_matrix(Pa, Pb, S)
H, Q = symmetric_block_diagonalize(N1)
Gamma = symmetric_transformation_matrix(Pa, Pb, S, H, Q, tol=1e-4)
Ralt = compute_R(Gamma, H)
self.assertEqual(R, Ralt)
def test_recover_action(self):
# see equation (28) of Kalla,Klein
def compute_R(Gamma, H):
H = H.change_ring(ZZ)
g,g = H.dimensions()
A = Gamma[:g,:g]
B = Gamma[:g,g:]
C = Gamma[g:,:g]
D = Gamma[g:,g:]
Ig = identity_matrix(ZZ,g)
R = zero_matrix(ZZ,2*g,2*g)
R[:g,:g] = (2*C.T*B - A.T*H*B + Ig).T
R[:g,g:] = 2*D.T*B - B.T*H*B
R[g:,:g] = -2*C.T*A + A.T*H*A
R[g:,g:] = -(2*C.T*B - A.T*H*B + Ig)
return R
# trott curve
Pa = self.atrott
Pb = self.btrott
R = involution_matrix(Pa, Pb)
S = integer_kernel_basis(R)
N1 = N1_matrix(Pa, Pb, S)
H,Q = symmetric_block_diagonalize(N1)
Gamma = symmetric_transformation_matrix(Pa, Pb, S, H, Q, tol=1e-4)
Ralt = compute_R(Gamma, H)
self.assertEqual(R, Ralt)
# klein curve
Pa = self.aklein
Pb = self.bklein
R = involution_matrix(Pa, Pb, tol=1e-3)
S = integer_kernel_basis(R)
N1 = N1_matrix(Pa, Pb, S, tol=1e-3)
H,Q = symmetric_block_diagonalize(N1)
Gamma = symmetric_transformation_matrix(Pa, Pb, S, H, Q, tol=1e-4)
Ralt = compute_R(Gamma, H)
self.assertEqual(R, Ralt)
# fermat curve
Pa = self.afermat
Pb = self.bfermat
R = involution_matrix(Pa, Pb, tol=1e-3)
S = integer_kernel_basis(R)
N1 = N1_matrix(Pa, Pb, S, tol=1e-3)
H,Q = symmetric_block_diagonalize(N1)
Gamma = symmetric_transformation_matrix(Pa, Pb, S, H, Q, tol=1e-4)
Ralt = compute_R(Gamma, H)
self.assertEqual(R, Ralt)
# genus 6 curve
Pa = self.a6
Pb = self.b6
R = involution_matrix(Pa, Pb)
S = integer_kernel_basis(R)
N1 = N1_matrix(Pa, Pb, S)
H,Q = symmetric_block_diagonalize(N1)
Gamma = symmetric_transformation_matrix(Pa, Pb, S, H, Q, tol=1e-4)
Ralt = compute_R(Gamma, H)
self.assertEqual(R, Ralt)
