def unimod_matrices_to_infty(r, s):
r"""
Return a list of matrices whose associated unimodular paths connect `0` to ``r/s``.
INPUT:
- ``r``, ``s`` -- rational numbers
OUTPUT:
- a list of matrices in `SL_2(\ZZ)`
EXAMPLES::
sage: v = sage.modular.pollack_stevens.manin_map.unimod_matrices_to_infty(19,23); v
[
[1 0] [ 0 1] [1 4] [-4 5] [ 5 19]
[0 1], [-1 1], [1 5], [-5 6], [ 6 23]
]
sage: [a.det() for a in v]
[1, 1, 1, 1, 1]
sage: sage.modular.pollack_stevens.manin_map.unimod_matrices_to_infty(11,25)
[
[1 0] [ 0 1] [1 3] [-3 4] [ 4 11]
[0 1], [-1 2], [2 7], [-7 9], [ 9 25]
]
ALGORITHM:
This is Manin's continued fraction trick, which gives an expression
`\{0,r/s\} = \{0,\infty\} + ... + \{a,b\} + ... + \{*,r/s\}`, where each `\{a,b\}` is
the image of `\{0,\infty\}` under a matrix in `SL_2(\ZZ)`.
"""
if s == 0:
return []
# the function contfrac_q in
# https://github.com/williamstein/psage/blob/master/psage/modform/rational/modular_symbol_map.pyx
# is very, very relevant to massively optimizing this.
L = convergents(r / s)
# Computes the continued fraction convergents of r/s
v = [M2Z([1, L[0].numerator(), 0, L[0].denominator()])]
# Initializes the list of matrices
for j in range(len(L) - 1):
a = L[j].numerator()
c = L[j].denominator()
b = L[j + 1].numerator()
d = L[j + 1].denominator()
v.append(M2Z([(-1) ** (j + 1) * a, b, (-1) ** (j + 1) * c, d]))
# The matrix connecting two consecutive convergents is added on
return v
def unimod_matrices_to_infty(r, s):
r"""
Return a list of matrices whose associated unimodular paths connect `0` to ``r/s``.
INPUT:
- ``r``, ``s`` -- rational numbers
OUTPUT:
- a list of matrices in `SL_2(\ZZ)`
EXAMPLES::
sage: v = sage.modular.pollack_stevens.manin_map.unimod_matrices_to_infty(19,23); v
[
[1 0] [ 0 1] [1 4] [-4 5] [ 5 19]
[0 1], [-1 1], [1 5], [-5 6], [ 6 23]
]
sage: [a.det() for a in v]
[1, 1, 1, 1, 1]
sage: sage.modular.pollack_stevens.manin_map.unimod_matrices_to_infty(11,25)
[
[1 0] [ 0 1] [1 3] [-3 4] [ 4 11]
[0 1], [-1 2], [2 7], [-7 9], [ 9 25]
]
ALGORITHM:
This is Manin's continued fraction trick, which gives an expression
`\{0,r/s\} = \{0,\infty\} + ... + \{a,b\} + ... + \{*,r/s\}`, where each `\{a,b\}` is
the image of `\{0,\infty\}` under a matrix in `SL_2(\ZZ)`.
"""
if s == 0:
return []
# the function contfrac_q in
# https://github.com/williamstein/psage/blob/master/psage/modform/rational/modular_symbol_map.pyx
# is very, very relevant to massively optimizing this.
L = convergents(r / s)
# Computes the continued fraction convergents of r/s
v = [M2Z([1, L[0].numerator(), 0, L[0].denominator()])]
# Initializes the list of matrices
for j in range(len(L) - 1):
a = L[j].numerator()
c = L[j].denominator()
b = L[j + 1].numerator()
d = L[j + 1].denominator()
v.append(M2Z([(-1)**(j + 1) * a, b, (-1)**(j + 1) * c, d]))
# The matrix connecting two consecutive convergents is added on
return v
def unimod_matrices_from_infty(r, s):
r"""
Return a list of matrices whose associated unimodular paths connect `\infty` to ``r/s``.
INPUT:
- ``r``, ``s`` -- rational numbers
OUTPUT:
- a list of `SL_2(\ZZ)` matrices
EXAMPLES::
sage: v = sage.modular.pollack_stevens.manin_map.unimod_matrices_from_infty(19,23); v
[
[ 0 1] [-1 0] [-4 1] [-5 -4] [-19 5]
[-1 0], [-1 -1], [-5 1], [-6 -5], [-23 6]
]
sage: [a.det() for a in v]
[1, 1, 1, 1, 1]
sage: sage.modular.pollack_stevens.manin_map.unimod_matrices_from_infty(11,25)
[
[ 0 1] [-1 0] [-3 1] [-4 -3] [-11 4]
[-1 0], [-2 -1], [-7 2], [-9 -7], [-25 9]
]
ALGORITHM:
This is Manin's continued fraction trick, which gives an expression
`\{\infty,r/s\} = \{\infty,0\} + ... + \{a,b\} + ... + \{*,r/s\}`, where each
`\{a,b\}` is the image of `\{0,\infty\}` under a matrix in `SL_2(\ZZ)`.
"""
if s != 0:
L = convergents(r / s)
# Computes the continued fraction convergents of r/s
v = [M2Z([-L[0].numerator(), 1, -L[0].denominator(), 0])]
# Initializes the list of matrices
# the function contfrac_q in https://github.com/williamstein/psage/blob/master/psage/modform/rational/modular_symbol_map.pyx
# is very, very relevant to massively optimizing this.
for j in range(len(L) - 1):
a = L[j].numerator()
c = L[j].denominator()
b = L[j + 1].numerator()
d = L[j + 1].denominator()
v.append(M2Z([-b, (-1) ** (j + 1) * a, -d, (-1) ** (j + 1) * c]))
# The matrix connecting two consecutive convergents is added on
return v
else:
return []
def unimod_matrices_from_infty(r, s):
r"""
Return a list of matrices whose associated unimodular paths connect `\infty` to ``r/s``.
INPUT:
- ``r``, ``s`` -- rational numbers
OUTPUT:
- a list of `SL_2(\ZZ)` matrices
EXAMPLES::
sage: v = sage.modular.pollack_stevens.manin_map.unimod_matrices_from_infty(19,23); v
[
[ 0 1] [-1 0] [-4 1] [-5 -4] [-19 5]
[-1 0], [-1 -1], [-5 1], [-6 -5], [-23 6]
]
sage: [a.det() for a in v]
[1, 1, 1, 1, 1]
sage: sage.modular.pollack_stevens.manin_map.unimod_matrices_from_infty(11,25)
[
[ 0 1] [-1 0] [-3 1] [-4 -3] [-11 4]
[-1 0], [-2 -1], [-7 2], [-9 -7], [-25 9]
]
ALGORITHM:
This is Manin's continued fraction trick, which gives an expression
`\{\infty,r/s\} = \{\infty,0\} + ... + \{a,b\} + ... + \{*,r/s\}`, where each
`\{a,b\}` is the image of `\{0,\infty\}` under a matrix in `SL_2(\ZZ)`.
"""
if s != 0:
L = convergents(r / s)
# Computes the continued fraction convergents of r/s
v = [M2Z([-L[0].numerator(), 1, -L[0].denominator(), 0])]
# Initializes the list of matrices
# the function contfrac_q in https://github.com/williamstein/psage/blob/master/psage/modform/rational/modular_symbol_map.pyx
# is very, very relevant to massively optimizing this.
for j in range(len(L) - 1):
a = L[j].numerator()
c = L[j].denominator()
b = L[j + 1].numerator()
d = L[j + 1].denominator()
v.append(M2Z([-b, (-1)**(j + 1) * a, -d, (-1)**(j + 1) * c]))
# The matrix connecting two consecutive convergents is added on
return v
else:
return []
