Lists of Manin symbols over number fields, elements of \(\mathbb{P}^1(R/N)\)

Lists of elements of \(\mathbb{P}^1(R/N)\) where \(R\) is the ring of integers of a number field \(K\) and \(N\) is an integral ideal.

AUTHORS:

• Maite Aranes (2009): Initial version

EXAMPLES:

We define a P1NFList:

sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^3 + 11)
sage: N = k.ideal(5, a^2 - a + 1)
sage: P = P1NFList(N); P
The projective line over the ring of integers modulo the Fractional ideal (5, a^2 - a + 1)

List operations with the P1NFList:

sage: len(P)
26
sage: [p for p in P]
[M-symbol (0: 1) of level Fractional ideal (5, a^2 - a + 1),
...
M-symbol (1: 2*a^2 + 2*a) of level Fractional ideal (5, a^2 - a + 1)]

The elements of the P1NFList are M-symbols:

sage: type(P[2])
<class 'sage.modular.modsym.p1list_nf.MSymbol'>

Definition of MSymbols:

sage: alpha = MSymbol(N, 3, a^2); alpha
M-symbol (3: a^2) of level Fractional ideal (5, a^2 - a + 1)

Find the index of the class of an M-Symbol \((c: d)\) in the list:

sage: i = P.index(alpha)
sage: P[i].c*alpha.d - P[i].d*alpha.c in N
True

Lift an MSymbol to a matrix in \(SL(2, R)\):

sage: alpha = MSymbol(N, a + 2, 3*a^2)
sage: alpha.lift_to_sl2_Ok()
[-1, 4*a^2 - 13*a + 23, a + 2, 5*a^2 + 3*a - 3]
sage: Ok = k.ring_of_integers()
sage: M = Matrix(Ok, 2, alpha.lift_to_sl2_Ok())
sage: det(M)
1
sage: M[1][1] - alpha.d in N
True

Lift an MSymbol from P1NFList to a matrix in \(SL(2, R)\)

sage: P[3]
M-symbol (1: -2*a) of level Fractional ideal (5, a^2 - a + 1)
sage: P.lift_to_sl2_Ok(3)
[0, -1, 1, -2*a]

class sage.modular.modsym.p1list_nf.MSymbol(N, c, d=None, check=True)

Bases: SageObject

The constructor for an M-symbol over a number field.

INPUT:

• N – integral ideal (the modulus or level).

• c – integral element of the underlying number field or an MSymbol of level N.

• d – (optional) when present, it must be an integral element such that <c> + <d> + N = R, where R is the corresponding ring of integers.

• check – bool (default True). If check=False the constructor does not check the condition <c> + <d> + N = R.

OUTPUT:

An M-symbol modulo the given ideal N, i.e. an element of the projective line \(\mathbb{P}^1(R/N)\), where R is the ring of integers of the underlying number field.

EXAMPLES:

sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^3 + 11)
sage: N = k.ideal(a + 1, 2)
sage: MSymbol(N, 3, a^2 + 1)
M-symbol (3: a^2 + 1) of level Fractional ideal (2, a + 1)

We can give a tuple as input:

sage: MSymbol(N, (1, 0))
M-symbol (1: 0) of level Fractional ideal (2, a + 1)

We get an error if <c>, <d> and N are not coprime:

sage: MSymbol(N, 2*a, a - 1)
Traceback (most recent call last):
...
ValueError: (2*a, a - 1) is not an element of P1(R/N).
sage: MSymbol(N, (0, 0))
Traceback (most recent call last):
...
ValueError: (0, 0) is not an element of P1(R/N).

Saving and loading works:

sage: alpha = MSymbol(N, 3, a^2 + 1)
sage: loads(dumps(alpha))==alpha
True

N()

Return the level or modulus of this MSymbol.

EXAMPLES:

sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^2 + 23)
sage: N = k.ideal(3, a - 1)
sage: alpha = MSymbol(N, 3, a)
sage: alpha.N()
Fractional ideal (3, 1/2*a - 1/2)

property c

Return the first coefficient of the M-symbol.

EXAMPLES:

sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^3 + 11)
sage: N = k.ideal(a + 1, 2)
sage: alpha = MSymbol(N, 3, a^2 + 1)
sage: alpha.c # indirect doctest
3

property d

Return the second coefficient of the M-symbol.

EXAMPLES:

sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^3 + 11)
sage: N = k.ideal(a + 1, 2)
sage: alpha = MSymbol(N, 3, a^2 + 1)
sage: alpha.d # indirect doctest
a^2 + 1

lift_to_sl2_Ok()

Lift the MSymbol to an element of \(SL(2, Ok)\), where \(Ok\) is the ring of integers of the corresponding number field.

OUTPUT:

A list of integral elements \([a, b, c', d']\) that are the entries of a 2x2 matrix with determinant 1. The lower two entries are congruent (modulo the level) to the coefficients \(c, d\) of the MSymbol self.

EXAMPLES:

sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^2 + 23)
sage: N = k.ideal(3, a - 1)
sage: alpha = MSymbol(N, 3*a + 1, a)
sage: alpha.lift_to_sl2_Ok()
[0, -1, 1, a]

normalize(with_scalar=False)

Return a normalized MSymbol (a canonical representative of an element of \(\mathbb{P}^1(R/N)\) ) equivalent to self.

INPUT:

• with_scalar – bool (default False)

OUTPUT:

• (only if with_scalar=True) a transforming scalar \(u\), such that \((u*c', u*d')\) is congruent to \((c: d)\) (mod \(N\)), where \((c: d)\) are the coefficients of self and \(N\) is the level.

• a normalized MSymbol (c’: d’) equivalent to self.

EXAMPLES:

sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^2 + 23)
sage: N = k.ideal(3, a - 1)
sage: alpha1 = MSymbol(N, 3, a); alpha1
M-symbol (3: a) of level Fractional ideal (3, 1/2*a - 1/2)
sage: alpha1.normalize()
M-symbol (0: 1) of level Fractional ideal (3, 1/2*a - 1/2)
sage: alpha2 = MSymbol(N, 4, a + 1)
sage: alpha2.normalize()
M-symbol (1: -a) of level Fractional ideal (3, 1/2*a - 1/2)

We get the scaling factor by setting with_scalar=True:

sage: alpha1.normalize(with_scalar=True)
(a, M-symbol (0: 1) of level Fractional ideal (3, 1/2*a - 1/2))
sage: r, beta1 = alpha1.normalize(with_scalar=True)
sage: r*beta1.c - alpha1.c in N
True
sage: r*beta1.d - alpha1.d in N
True
sage: r, beta2 = alpha2.normalize(with_scalar=True)
sage: r*beta2.c - alpha2.c in N
True
sage: r*beta2.d - alpha2.d in N
True

tuple()

Return the MSymbol as a list (c, d).

EXAMPLES:

sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^2 + 23)
sage: N = k.ideal(3, a - 1)
sage: alpha = MSymbol(N, 3, a); alpha
M-symbol (3: a) of level Fractional ideal (3, 1/2*a - 1/2)
sage: alpha.tuple()
(3, a)

class sage.modular.modsym.p1list_nf.P1NFList(N)

Bases: SageObject

The class for \(\mathbb{P}^1(R/N)\), the projective line modulo \(N\), where \(R\) is the ring of integers of a number field \(K\) and \(N\) is an integral ideal.

INPUT:

• N - integral ideal (the modulus or level).

OUTPUT:

A P1NFList object representing \(\mathbb{P}^1(R/N)\).

EXAMPLES:

sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^3 + 11)
sage: N = k.ideal(5, a + 1)
sage: P = P1NFList(N); P
The projective line over the ring of integers modulo the Fractional ideal (5, a + 1)

Saving and loading works.

sage: loads(dumps(P)) == P
True

N()

Return the level or modulus of this P1NFList.

EXAMPLES:

sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^2 + 31)
sage: N = k.ideal(5, a + 3)
sage: P = P1NFList(N)
sage: P.N()
Fractional ideal (5, 1/2*a + 3/2)

apply_J_epsilon(i, e1, e2=1)

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: x = polygen(QQ, 'x')
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])
5

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

apply_S(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: x = polygen(QQ, 'x')
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

apply_TS(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: x = polygen(QQ, 'x')
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

apply_T_alpha(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: x = polygen(QQ, 'x')
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

index(c, d=None, with_scalar=False)

Return 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: x = polygen(QQ, 'x')
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

index_of_normalized_pair(c, d=None)

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: x = polygen(QQ, 'x')
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

lift_to_sl2_Ok(i)

Lift the \(i\)-th element of this P1NFList to an element of \(SL(2, R)\), where \(R\) is the ring of integers of the corresponding number field.

INPUT:

• i - integer (index of the element to lift)

OUTPUT:

If the \(i\)-th element is \((c : d)\), the function returns a list of integral elements \([a, b, c', d']\) that defines a 2x2 matrix with determinant 1 and such that \(c=c'\) (mod \(N\)) and \(d=d'\) (mod \(N\)).

EXAMPLES:

sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^2 + 23)
sage: N = k.ideal(3)
sage: P = P1NFList(N)
sage: len(P)
16
sage: P[5]
M-symbol (1/2*a + 1/2: -a) of level Fractional ideal (3)
sage: P.lift_to_sl2_Ok(5)
[-a, 2*a - 2, 1/2*a + 1/2, -a]

sage: Ok = k.ring_of_integers()
sage: L = [Matrix(Ok, 2, P.lift_to_sl2_Ok(i)) for i in range(len(P))]
sage: all(det(L[i]) == 1 for i in range(len(L)))
True

list()

Return the underlying list of this P1NFList object.

EXAMPLES:

sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^3 + 11)
sage: N = k.ideal(5, a+1)
sage: P = P1NFList(N)
sage: type(P)
<class 'sage.modular.modsym.p1list_nf.P1NFList'>
sage: type(P.list())
<... 'list'>

normalize(c, d=None, with_scalar=False)

Return a normalised element of \(\mathbb{P}^1(R/N)\).

INPUT:

• c – integral element of the underlying 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:

• (only if with_scalar=True) a transforming scalar \(u\), such that \((u*c', u*d')\) is congruent to \((c: d)\) (mod \(N\)).

• a normalized MSymbol (c’: d’) equivalent to \((c: d)\).

EXAMPLES:

sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^2 + 31)
sage: N = k.ideal(5, a + 3)
sage: P = P1NFList(N)
sage: P.normalize(3, a)
M-symbol (1: 2*a) of level Fractional ideal (5, 1/2*a + 3/2)

We can use an MSymbol as input:

sage: alpha = MSymbol(N, 3, a)
sage: P.normalize(alpha)
M-symbol (1: 2*a) of level Fractional ideal (5, 1/2*a + 3/2)

If we are interested in the normalizing scalar:

sage: P.normalize(alpha, with_scalar=True)
(-a, M-symbol (1: 2*a) of level Fractional ideal (5, 1/2*a + 3/2))
sage: r, beta = P.normalize(alpha, with_scalar=True)
sage: (r*beta.c - alpha.c in N) and (r*beta.d - alpha.d in N)
True

sage.modular.modsym.p1list_nf.P1NFList_clear_level_cache()

Clear the global cache of data for the level ideals.

EXAMPLES:

sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^3 + 11)
sage: N = k.ideal(a+1)
sage: alpha = MSymbol(N, 2*a^2, 5)
sage: alpha.normalize()
M-symbol (-4*a^2: 5*a^2) of level Fractional ideal (a + 1)
sage: sage.modular.modsym.p1list_nf._level_cache
{Fractional ideal (a + 1): (...)}
sage: sage.modular.modsym.p1list_nf.P1NFList_clear_level_cache()
sage: sage.modular.modsym.p1list_nf._level_cache
{}

sage.modular.modsym.p1list_nf.lift_to_sl2_Ok(N, c, d)

Lift a pair (c, d) to an element of \(SL(2, O_k)\), where \(O_k\) is the ring of integers of the corresponding number field.

INPUT:

• N – number field ideal

• c – integral element of the number field

• d – integral element of the number field

OUTPUT:

A list [a, b, c’, d’] of integral elements that are the entries of a 2x2 matrix with determinant 1. The lower two entries are congruent to c, d modulo the ideal \(N\).

EXAMPLES:

sage: from sage.modular.modsym.p1list_nf import lift_to_sl2_Ok
sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^2 + 23)
sage: Ok = k.ring_of_integers()
sage: N = k.ideal(3)
sage: M = Matrix(Ok, 2, lift_to_sl2_Ok(N, 1, a))
sage: det(M)
1
sage: M = Matrix(Ok, 2, lift_to_sl2_Ok(N, 0, a))
sage: det(M)
1
sage: (M[1][0] in N) and (M[1][1] - a in N)
True
sage: M = Matrix(Ok, 2, lift_to_sl2_Ok(N, 0, 0))
Traceback (most recent call last):
...
ValueError: Cannot lift (0, 0) to an element of Sl2(Ok).

sage: k.<a> = NumberField(x^3 + 11)
sage: Ok = k.ring_of_integers()
sage: N = k.ideal(3, a - 1)
sage: M = Matrix(Ok, 2, lift_to_sl2_Ok(N, 2*a, 0))
sage: det(M)
1
sage: (M[1][0] - 2*a in N) and (M[1][1] in N)
True
sage: M = Matrix(Ok, 2, lift_to_sl2_Ok(N, 4*a^2, a + 1))
sage: det(M)
1
sage: (M[1][0] - 4*a^2 in N) and (M[1][1] - (a+1) in N)
True

sage: k.<a> = NumberField(x^4 - x^3 -21*x^2 + 17*x + 133)
sage: Ok = k.ring_of_integers()
sage: N = k.ideal(7, a)
sage: M = Matrix(Ok, 2, lift_to_sl2_Ok(N, 0, a^2 - 1))
sage: det(M)
1
sage: (M[1][0] in N) and (M[1][1] - (a^2-1) in N)
True
sage: M = Matrix(Ok, 2, lift_to_sl2_Ok(N, 0, 7))
Traceback (most recent call last):
...
ValueError: <0> + <7> and the Fractional ideal (7, a) are not coprime.

sage.modular.modsym.p1list_nf.make_coprime(N, c, d)

Return (c, d’) so d’ is congruent to d modulo N, and such that c and d’ are coprime (<c> + <d’> = R).

INPUT:

• N – number field ideal

• c – integral element of the number field

• d – integral element of the number field

OUTPUT:

A pair (c, d’) where c, d’ are integral elements of the corresponding number field, with d’ congruent to d mod N, and such that <c> + <d’> = R (R being the corresponding ring of integers).

EXAMPLES:

sage: from sage.modular.modsym.p1list_nf import make_coprime
sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^2 + 23)
sage: N = k.ideal(3, a - 1)
sage: c = 2*a; d = a + 1
sage: N.is_coprime(k.ideal(c, d))
True
sage: k.ideal(c).is_coprime(d)
False
sage: c, dp = make_coprime(N, c, d)
sage: k.ideal(c).is_coprime(dp)
True

sage.modular.modsym.p1list_nf.p1NFlist(N)

Return a list of the normalized elements of \(\mathbb{P}^1(R/N)\), where \(N\) is an integral ideal.

INPUT:

• N - integral ideal (the level or modulus).

EXAMPLES:

sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^2 + 23)
sage: N = k.ideal(3)
sage: from sage.modular.modsym.p1list_nf import p1NFlist, psi
sage: len(p1NFlist(N))==psi(N)
True

sage.modular.modsym.p1list_nf.psi(N)

The index \([\Gamma : \Gamma_0(N)]\), where \(\Gamma = GL(2, R)\) for \(R\) the corresponding ring of integers, and \(\Gamma_0(N)\) standard congruence subgroup.

EXAMPLES:

sage: from sage.modular.modsym.p1list_nf import psi
sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^2 + 23)
sage: N = k.ideal(3, a - 1)
sage: psi(N)
4

sage: k.<a> = NumberField(x^2 + 23)
sage: N = k.ideal(5)
sage: psi(N)
26