def apply_T_alpha(self, i, alpha=1):
"""
Applies the matrix T_alpha = [1, alpha, 0, 1] to the i-th M-Symbol of
the list.
INPUT:
- ``i`` -- integer
- ``alpha`` -- element of the corresponding ring of integers(default 1)
OUTPUT:
integer -- the index of the M-Symbol obtained by the right action of
the matrix T_alpha = [1, alpha, 0, 1] on the i-th M-Symbol.
EXAMPLES::
sage: k.<a> = NumberField(x^3 + 11)
sage: N = k.ideal(5, a + 1)
sage: P = P1NFList(N)
sage: P.apply_T_alpha(4, a^ 2 - 2)
3
We test that T_a*T_b = T_(a+b):
::
sage: P.apply_T_alpha(3, a^2 - 2)==P.apply_T_alpha(P.apply_T_alpha(3,a^2),-2)
True
"""
c, d = self.__list[i].tuple()
t, j = search(self.__list, self.normalize(c, alpha * c + d))
return j
def apply_S(self, i):
"""
Applies the matrix S = [0, -1, 1, 0] to the i-th M-Symbol of the list.
INPUT:
- ``i`` -- integer
OUTPUT:
integer -- the index of the M-Symbol obtained by the right action of
the matrix S = [0, -1, 1, 0] on the i-th M-Symbol.
EXAMPLES::
sage: k.<a> = NumberField(x^3 + 11)
sage: N = k.ideal(5, a + 1)
sage: P = P1NFList(N)
sage: j = P.apply_S(P.index_of_normalized_pair(1, 0))
sage: P[j]
M-symbol (0: 1) of level Fractional ideal (5, a + 1)
We test that S has order 2:
::
sage: j = randint(0,len(P)-1)
sage: P.apply_S(P.apply_S(j))==j
True
"""
c, d = self.__list[i].tuple()
t, j = search(self.__list, self.normalize(d, -c))
return j
def apply_TS(self, i):
"""
Applies the matrix TS = [1, -1, 0, 1] to the i-th M-Symbol of the list.
INPUT:
- ``i`` -- integer
OUTPUT:
integer -- the index of the M-Symbol obtained by the right action of
the matrix TS = [1, -1, 0, 1] on the i-th M-Symbol.
EXAMPLES::
sage: k.<a> = NumberField(x^3 + 11)
sage: N = k.ideal(5, a + 1)
sage: P = P1NFList(N)
sage: P.apply_TS(3)
2
We test that TS has order 3:
::
sage: j = randint(0,len(P)-1)
sage: P.apply_TS(P.apply_TS(P.apply_TS(j)))==j
True
"""
c, d = self.__list[i].tuple()
t, j = search(self.__list, self.normalize(c + d, -c))
return j
def coefficient(self, P):
"""
Return the coefficient of a given point P in this divisor.
EXAMPLES::
sage: x,y = AffineSpace(2, GF(5), names='xy').gens()
sage: C = Curve(y^2 - x^9 - x)
sage: pts = C.rational_points(); pts
[(0, 0), (2, 2), (2, 3), (3, 1), (3, 4)]
sage: D = C.divisor(pts[0])
sage: D.coefficient(pts[0])
1
sage: D = C.divisor([(3,pts[0]), (-1,pts[1])]); D
3*(x, y) - (x - 2, y - 2)
sage: D.coefficient(pts[0])
3
sage: D.coefficient(pts[1])
-1
"""
P = self.parent().scheme()(P)
if not(P in self.support()):
return self.base_ring().zero()
t, i = search(self.support(), P)
assert t
try:
return self._points[i][0]
except AttributeError:
raise NotImplementedError
def apply_TS(self, i):
"""
Applies the matrix TS = [1, -1, 0, 1] to the i-th M-Symbol of the list.
INPUT:
- ``i`` -- integer
OUTPUT:
integer -- the index of the M-Symbol obtained by the right action of
the matrix TS = [1, -1, 0, 1] on the i-th M-Symbol.
EXAMPLES::
sage: k.<a> = NumberField(x^3 + 11)
sage: N = k.ideal(5, a + 1)
sage: P = P1NFList(N)
sage: P.apply_TS(3)
2
We test that TS has order 3:
::
sage: j = randint(0,len(P)-1)
sage: P.apply_TS(P.apply_TS(P.apply_TS(j)))==j
True
"""
c, d = self.__list[i].tuple()
t, j = search(self.__list, self.normalize(c + d, -c))
return j
def apply_T_alpha(self, i, alpha=1):
"""
Applies the matrix T_alpha = [1, alpha, 0, 1] to the i-th M-Symbol of
the list.
INPUT:
- ``i`` -- integer
- ``alpha`` -- element of the corresponding ring of integers(default 1)
OUTPUT:
integer -- the index of the M-Symbol obtained by the right action of
the matrix T_alpha = [1, alpha, 0, 1] on the i-th M-Symbol.
EXAMPLES::
sage: k.<a> = NumberField(x^3 + 11)
sage: N = k.ideal(5, a + 1)
sage: P = P1NFList(N)
sage: P.apply_T_alpha(4, a^ 2 - 2)
3
We test that T_a*T_b = T_(a+b):
::
sage: P.apply_T_alpha(3, a^2 - 2)==P.apply_T_alpha(P.apply_T_alpha(3,a^2),-2)
True
"""
c, d = self.__list[i].tuple()
t, j = search(self.__list, self.normalize(c, alpha*c + d))
return j
def apply_S(self, i):
"""
Applies the matrix S = [0, -1, 1, 0] to the i-th M-Symbol of the list.
INPUT:
- ``i`` -- integer
OUTPUT:
integer -- the index of the M-Symbol obtained by the right action of
the matrix S = [0, -1, 1, 0] on the i-th M-Symbol.
EXAMPLES::
sage: k.<a> = NumberField(x^3 + 11)
sage: N = k.ideal(5, a + 1)
sage: P = P1NFList(N)
sage: j = P.apply_S(P.index_of_normalized_pair(1, 0))
sage: P[j]
M-symbol (0: 1) of level Fractional ideal (5, a + 1)
We test that S has order 2:
::
sage: j = randint(0,len(P)-1)
sage: P.apply_S(P.apply_S(j))==j
True
"""
c, d = self.__list[i].tuple()
t, j = search(self.__list, self.normalize(d, -c))
return j
def coefficient(self, P):
"""
Return the coefficient of a given point P in this divisor.
EXAMPLES::
sage: x,y = AffineSpace(2, GF(5), names='xy').gens()
sage: C = Curve(y^2 - x^9 - x)
sage: pts = C.rational_points(); pts
[(0, 0), (2, 2), (2, 3), (3, 1), (3, 4)]
sage: D = C.divisor(pts[0])
sage: D.coefficient(pts[0])
1
sage: D = C.divisor([(3,pts[0]), (-1,pts[1])]); D
3*(x, y) - (x - 2, y - 2)
sage: D.coefficient(pts[0])
3
sage: D.coefficient(pts[1])
-1
"""
P = self.parent().scheme()(P)
if not (P in self.support()):
return self.base_ring().zero()
t, i = search(self.support(), P)
assert t
try:
return self._points[i][0]
except AttributeError:
raise NotImplementedError
def index_of_normalized_pair(self, c, d=None):
r"""
Return the index of the class `(c, d)` in the fixed list of
representatives of `\mathbb(P)^1(R/N)`.
INPUT:
- ``c`` -- integral element of the corresponding number field, or a
normalized MSymbol.
- ``d`` -- (optional) when present, it must be an integral element of
the number field such that `(c, d)` defines a normalized M-symbol of
level `N`.
OUTPUT:
- ``i`` - the index of `(c, d)` in the list.
EXAMPLES::
sage: k.<a> = NumberField(x^2 + 31)
sage: N = k.ideal(5, a + 3)
sage: P = P1NFList(N)
sage: P.index_of_normalized_pair(1, 0)
3
sage: j = randint(0,len(P)-1)
sage: P.index_of_normalized_pair(P[j])==j
True
"""
if d is None:
try:
c = MSymbol(self.__N, c) # check that c is an MSymbol
except ValueError: # catch special case of wrong level
raise ValueError("The MSymbol is of a different level")
t, i = search(self.__list, c)
else:
t, i = search(self.__list, MSymbol(self.__N, c, d))
if t:
return i
return False
def index_of_normalized_pair(self, c, d=None):
"""
Returns the index of the class `(c, d)` in the fixed list of
representatives of `\mathbb(P)^1(R/N)`.
INPUT:
- ``c`` -- integral element of the corresponding number field, or a
normalized MSymbol.
- ``d`` -- (optional) when present, it must be an integral element of
the number field such that `(c, d)` defines a normalized M-symbol of
level `N`.
OUTPUT:
- ``i`` - the index of `(c, d)` in the list.
EXAMPLES::
sage: k.<a> = NumberField(x^2 + 31)
sage: N = k.ideal(5, a + 3)
sage: P = P1NFList(N)
sage: P.index_of_normalized_pair(1, 0)
3
sage: j = randint(0,len(P)-1)
sage: P.index_of_normalized_pair(P[j])==j
True
"""
if d is None:
try:
c = MSymbol(self.__N, c) # check that c is an MSymbol
except ValueError: # catch special case of wrong level
raise ValueError("The MSymbol is of a different level")
t, i = search(self.__list, c)
else:
t, i = search(self.__list, MSymbol(self.__N, c, d))
if t: return i
return False
def apply_J_epsilon(self, i, e1, e2=1):
r"""
Apply the matrix `J_{\epsilon}` = [e1, 0, 0, e2] to the i-th
M-Symbol of the list.
e1, e2 are units of the underlying number field.
INPUT:
- ``i`` -- integer
- ``e1`` -- unit
- ``e2`` -- unit (default 1)
OUTPUT:
integer -- the index of the M-Symbol obtained by the right action of
the matrix `J_{\epsilon}` = [e1, 0, 0, e2] on the i-th M-Symbol.
EXAMPLES::
sage: k.<a> = NumberField(x^3 + 11)
sage: N = k.ideal(5, a + 1)
sage: P = P1NFList(N)
sage: u = k.unit_group().gens_values(); u
[-1, 2*a^2 + 4*a - 1]
sage: P.apply_J_epsilon(4, -1)
2
sage: P.apply_J_epsilon(4, u[0], u[1])
1
::
sage: k.<a> = NumberField(x^4 + 13*x - 7)
sage: N = k.ideal(a + 1)
sage: P = P1NFList(N)
sage: u = k.unit_group().gens_values(); u
[-1, a^3 + a^2 + a + 12, a^3 + 3*a^2 - 1]
sage: P.apply_J_epsilon(3, u[2]^2)==P.apply_J_epsilon(P.apply_J_epsilon(3, u[2]),u[2])
True
"""
c, d = self.__list[i].tuple()
t, j = search(self.__list, self.normalize(c * e1, d * e2))
return j
def apply_J_epsilon(self, i, e1, e2=1):
"""
Applies the matrix `J_{\epsilon}` = [e1, 0, 0, e2] to the i-th
M-Symbol of the list.
e1, e2 are units of the underlying number field.
INPUT:
- ``i`` -- integer
- ``e1`` -- unit
- ``e2`` -- unit (default 1)
OUTPUT:
integer -- the index of the M-Symbol obtained by the right action of
the matrix `J_{\epsilon}` = [e1, 0, 0, e2] on the i-th M-Symbol.
EXAMPLES::
sage: k.<a> = NumberField(x^3 + 11)
sage: N = k.ideal(5, a + 1)
sage: P = P1NFList(N)
sage: u = k.unit_group().gens_values(); u
[-1, 2*a^2 + 4*a - 1]
sage: P.apply_J_epsilon(4, -1)
2
sage: P.apply_J_epsilon(4, u[0], u[1])
1
::
sage: k.<a> = NumberField(x^4 + 13*x - 7)
sage: N = k.ideal(a + 1)
sage: P = P1NFList(N)
sage: u = k.unit_group().gens_values(); u
[-1, a^3 + a^2 + a + 12, a^3 + 3*a^2 - 1]
sage: P.apply_J_epsilon(3, u[2]^2)==P.apply_J_epsilon(P.apply_J_epsilon(3, u[2]),u[2])
True
"""
c, d = self.__list[i].tuple()
t, j = search(self.__list, self.normalize(c*e1, d*e2))
return j
def gens_to_basis_matrix(syms, relation_matrix, mod, field, sparse):
"""
Compute echelon form of 3-term relation matrix, and read off each
generator in terms of basis.
INPUT:
- ``syms`` - a ManinSymbols object
- ``relation_matrix`` - as output by
``__compute_T_relation_matrix(self, mod)``
- ``mod`` - quotient of modular symbols modulo the
2-term S (and possibly I) relations
- ``field`` - base field
- ``sparse`` - (bool): whether or not matrix should be
sparse
OUTPUT:
- ``matrix`` - a matrix whose ith row expresses the
Manin symbol generators in terms of a basis of Manin symbols
(modulo the S, (possibly I,) and T rels) Note that the entries of
the matrix need not be integers.
- ``list`` - integers i, such that the Manin symbols `x_i` are a basis.
EXAMPLE::
sage: from sage.modular.modsym.relation_matrix import *
sage: L = sage.modular.modsym.manin_symbols.ManinSymbolList_gamma1(4, 3)
sage: modS = sparse_2term_quotient(modS_relations(L), 24, GF(3))
sage: gens_to_basis_matrix(L, T_relation_matrix_wtk_g0(L, modS, GF(3), 24), modS, GF(3), True)
(24 x 2 sparse matrix over Finite Field of size 3, [13, 23])
"""
if not sage.matrix.all.is_Matrix(relation_matrix):
raise TypeError, "relation_matrix must be a matrix"
if not isinstance(mod, list):
raise TypeError, "mod must be a list"
misc.verbose(str(relation_matrix.parent()))
try:
h = relation_matrix.height()
except AttributeError:
h = 9999999
tm = misc.verbose("putting relation matrix in echelon form (height = %s)"%h)
if h < 10:
A = relation_matrix.echelon_form(algorithm='multimodular', height_guess=1)
else:
A = relation_matrix.echelon_form()
A.set_immutable()
tm = misc.verbose('finished echelon', tm)
tm = misc.verbose("Now creating gens --> basis mapping")
basis_set = set(A.nonpivots())
pivots = A.pivots()
basis_mod2 = set([j for j,c in mod if c != 0])
basis_set = basis_set.intersection(basis_mod2)
basis = list(basis_set)
basis.sort()
ONE = field(1)
misc.verbose("done doing setup",tm)
tm = misc.verbose("now forming quotient matrix")
M = matrix_space.MatrixSpace(field, len(syms), len(basis), sparse=sparse)
B = M(0)
cols_index = dict([(basis[i], i) for i in range(len(basis))])
for i in basis_mod2:
t, l = search(basis, i)
if t:
B[i,l] = ONE
else:
_, r = search(pivots, i) # so pivots[r] = i
# Set row i to -(row r of A), but where we only take
# the non-pivot columns of A:
B._set_row_to_negative_of_row_of_A_using_subset_of_columns(i, A, r, basis, cols_index)
misc.verbose("done making quotient matrix",tm)
# The following is very fast (over Q at least).
tm = misc.verbose('now filling in the rest of the matrix')
k = 0
for i in range(len(mod)):
j, s = mod[i]
if j != i and s != 0: # ignored in the above matrix
k += 1
B.set_row_to_multiple_of_row(i, j, s)
misc.verbose("set %s rows"%k)
tm = misc.verbose("time to fill in rest of matrix", tm)
return B, basis
def index(self, c, d=None, with_scalar=False):
"""
Returns the index of the class of the pair `(c, d)` in the fixed list
of representatives of `\mathbb{P}^1(R/N)`.
INPUT:
- ``c`` -- integral element of the corresponding number field, or an
MSymbol.
- ``d`` -- (optional) when present, it must be an integral element of
the number field such that `(c, d)` defines an M-symbol of level `N`.
- ``with_scalar`` -- bool (default False)
OUTPUT:
- ``u`` - the normalizing scalar (only if ``with_scalar=True``)
- ``i`` - the index of `(c, d)` in the list.
EXAMPLES::
sage: k.<a> = NumberField(x^2 + 31)
sage: N = k.ideal(5, a + 3)
sage: P = P1NFList(N)
sage: P.index(3,a)
5
sage: P[5]==MSymbol(N, 3, a).normalize()
True
We can give an MSymbol as input:
::
sage: alpha = MSymbol(N, 3, a)
sage: P.index(alpha)
5
We cannot look for the class of an MSymbol of a different level:
::
sage: M = k.ideal(a + 1)
sage: beta = MSymbol(M, 0, 1)
sage: P.index(beta)
Traceback (most recent call last):
...
ValueError: The MSymbol is of a different level
If we are interested in the transforming scalar:
::
sage: alpha = MSymbol(N, 3, a)
sage: P.index(alpha, with_scalar=True)
(-a, 5)
sage: u, i = P.index(alpha, with_scalar=True)
sage: (u*P[i].c - alpha.c in N) and (u*P[i].d - alpha.d in N)
True
"""
if d is None:
try:
c = MSymbol(self.__N, c) # check that c is an MSymbol
except ValueError: # catch special case of wrong level
raise ValueError("The MSymbol is of a different level")
if with_scalar:
u, norm_c = c.normalize(with_scalar=True)
else:
norm_c = c.normalize()
else:
if with_scalar:
u, norm_c = MSymbol(self.__N, c, d).normalize(with_scalar=True)
else:
norm_c = MSymbol(self.__N, c, d).normalize()
t, i = search(self.__list, norm_c)
if t:
if with_scalar:
return u, i
else:
return i
return False
def gens_to_basis_matrix(syms, relation_matrix, mod, field, sparse):
"""
Compute echelon form of 3-term relation matrix, and read off each
generator in terms of basis.
INPUT:
- ``syms`` - a ManinSymbols object
- ``relation_matrix`` - as output by
``__compute_T_relation_matrix(self, mod)``
- ``mod`` - quotient of modular symbols modulo the
2-term S (and possibly I) relations
- ``field`` - base field
- ``sparse`` - (bool): whether or not matrix should be
sparse
OUTPUT:
- ``matrix`` - a matrix whose ith row expresses the
Manin symbol generators in terms of a basis of Manin symbols
(modulo the S, (possibly I,) and T rels) Note that the entries of
the matrix need not be integers.
- ``list`` - integers i, such that the Manin symbols `x_i` are a basis.
EXAMPLE::
sage: from sage.modular.modsym.relation_matrix import *
sage: L = sage.modular.modsym.manin_symbols.ManinSymbolList_gamma1(4, 3)
sage: modS = sparse_2term_quotient(modS_relations(L), 24, GF(3))
sage: gens_to_basis_matrix(L, T_relation_matrix_wtk_g0(L, modS, GF(3), 24), modS, GF(3), True)
(24 x 2 sparse matrix over Finite Field of size 3, [13, 23])
"""
from sage.matrix.matrix import is_Matrix
if not is_Matrix(relation_matrix):
raise TypeError("relation_matrix must be a matrix")
if not isinstance(mod, list):
raise TypeError("mod must be a list")
misc.verbose(str(relation_matrix.parent()))
try:
h = relation_matrix.height()
except AttributeError:
h = 9999999
tm = misc.verbose("putting relation matrix in echelon form (height = %s)"%h)
if h < 10:
A = relation_matrix.echelon_form(algorithm='multimodular', height_guess=1)
else:
A = relation_matrix.echelon_form()
A.set_immutable()
tm = misc.verbose('finished echelon', tm)
tm = misc.verbose("Now creating gens --> basis mapping")
basis_set = set(A.nonpivots())
pivots = A.pivots()
basis_mod2 = set([j for j,c in mod if c != 0])
basis_set = basis_set.intersection(basis_mod2)
basis = sorted(basis_set)
ONE = field(1)
misc.verbose("done doing setup",tm)
tm = misc.verbose("now forming quotient matrix")
M = matrix_space.MatrixSpace(field, len(syms), len(basis), sparse=sparse)
B = M(0)
cols_index = dict([(basis[i], i) for i in range(len(basis))])
for i in basis_mod2:
t, l = search(basis, i)
if t:
B[i,l] = ONE
else:
_, r = search(pivots, i) # so pivots[r] = i
# Set row i to -(row r of A), but where we only take
# the non-pivot columns of A:
B._set_row_to_negative_of_row_of_A_using_subset_of_columns(i, A, r, basis, cols_index)
misc.verbose("done making quotient matrix",tm)
# The following is very fast (over Q at least).
tm = misc.verbose('now filling in the rest of the matrix')
k = 0
for i in range(len(mod)):
j, s = mod[i]
if j != i and s != 0: # ignored in the above matrix
k += 1
B.set_row_to_multiple_of_row(i, j, s)
misc.verbose("set %s rows"%k)
tm = misc.verbose("time to fill in rest of matrix", tm)
return B, basis
def index(self, c, d=None, with_scalar=False):
"""
Returns the index of the class of the pair `(c, d)` in the fixed list
of representatives of `\mathbb{P}^1(R/N)`.
INPUT:
- ``c`` -- integral element of the corresponding number field, or an
MSymbol.
- ``d`` -- (optional) when present, it must be an integral element of
the number field such that `(c, d)` defines an M-symbol of level `N`.
- ``with_scalar`` -- bool (default False)
OUTPUT:
- ``u`` - the normalizing scalar (only if ``with_scalar=True``)
- ``i`` - the index of `(c, d)` in the list.
EXAMPLES::
sage: k.<a> = NumberField(x^2 + 31)
sage: N = k.ideal(5, a + 3)
sage: P = P1NFList(N)
sage: P.index(3,a)
5
sage: P[5]==MSymbol(N, 3, a).normalize()
True
We can give an MSymbol as input:
::
sage: alpha = MSymbol(N, 3, a)
sage: P.index(alpha)
5
We cannot look for the class of an MSymbol of a different level:
::
sage: M = k.ideal(a + 1)
sage: beta = MSymbol(M, 0, 1)
sage: P.index(beta)
Traceback (most recent call last):
...
ValueError: The MSymbol is of a different level
If we are interested in the transforming scalar:
::
sage: alpha = MSymbol(N, 3, a)
sage: P.index(alpha, with_scalar=True)
(-a, 5)
sage: u, i = P.index(alpha, with_scalar=True)
sage: (u*P[i].c - alpha.c in N) and (u*P[i].d - alpha.d in N)
True
"""
if d is None:
try:
c = MSymbol(self.__N, c) # check that c is an MSymbol
except ValueError: # catch special case of wrong level
raise ValueError("The MSymbol is of a different level")
if with_scalar:
u, norm_c = c.normalize(with_scalar=True)
else:
norm_c = c.normalize()
else:
if with_scalar:
u, norm_c = MSymbol(self.__N, c, d).normalize(with_scalar=True)
else:
norm_c = MSymbol(self.__N, c, d).normalize()
t, i = search(self.__list, norm_c)
if t:
if with_scalar:
return u, i
else:
return i
return False