Helper functions for local components

This module contains various functions relating to lifting elements of \(\SL_2(\ZZ / N\ZZ)\) to \(\SL_2(\ZZ)\), and other related problems.

sage.modular.local_comp.liftings.lift_for_SL(A, N=None)[source]

Lift a matrix \(A\) from \(SL_m(\ZZ / N\ZZ)\) to \(SL_m(\ZZ)\).

This follows [Shi1971], Lemma 1.38, p. 21.

INPUT:

  • A – a square matrix with coefficients in \(\ZZ / N\ZZ\) (or \(\ZZ\))

  • N – the modulus (optional) required only if the matrix A has coefficients in \(\ZZ\)

EXAMPLES:

sage: from sage.modular.local_comp.liftings import lift_for_SL
sage: A = matrix(Zmod(11), 4, 4, [6, 0, 0, 9, 1, 6, 9, 4, 4, 4, 8, 0, 4, 0, 0, 8])
sage: A.det()
1
sage: L = lift_for_SL(A)
sage: L.det()
1
sage: (L - A) == 0
True

sage: B = matrix(Zmod(19), 4, 4, [1, 6, 10, 4, 4, 14, 15, 4, 13, 0, 1, 15, 15, 15, 17, 10])
sage: B.det()
1
sage: L = lift_for_SL(B)
sage: L.det()
1
sage: (L - B) == 0
True

>>> from sage.all import *
>>> from sage.modular.local_comp.liftings import lift_for_SL
>>> A = matrix(Zmod(Integer(11)), Integer(4), Integer(4), [Integer(6), Integer(0), Integer(0), Integer(9), Integer(1), Integer(6), Integer(9), Integer(4), Integer(4), Integer(4), Integer(8), Integer(0), Integer(4), Integer(0), Integer(0), Integer(8)])
>>> A.det()
1
>>> L = lift_for_SL(A)
>>> L.det()
1
>>> (L - A) == Integer(0)
True

>>> B = matrix(Zmod(Integer(19)), Integer(4), Integer(4), [Integer(1), Integer(6), Integer(10), Integer(4), Integer(4), Integer(14), Integer(15), Integer(4), Integer(13), Integer(0), Integer(1), Integer(15), Integer(15), Integer(15), Integer(17), Integer(10)])
>>> B.det()
1
>>> L = lift_for_SL(B)
>>> L.det()
1
>>> (L - B) == Integer(0)
True
sage.modular.local_comp.liftings.lift_gen_to_gamma1(m, n)[source]

Return four integers defining a matrix in \(\SL_2(\ZZ)\) which is congruent to \(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \pmod m\) and lies in the subgroup \(\begin{pmatrix} 1 & * \\ 0 & 1 \end{pmatrix} \pmod n\).

This is a special case of lift_to_gamma1(), and is coded as such.

INPUT:

  • \(m\), \(n\) – coprime positive integers

EXAMPLES:

sage: from sage.modular.local_comp.liftings import lift_gen_to_gamma1
sage: A = matrix(ZZ, 2, lift_gen_to_gamma1(9, 8)); A
[441  62]
[ 64   9]
sage: A.change_ring(Zmod(9))
[0 8]
[1 0]
sage: A.change_ring(Zmod(8))
[1 6]
[0 1]
sage: type(lift_gen_to_gamma1(9, 8)[0])
<class 'sage.rings.integer.Integer'>

>>> from sage.all import *
>>> from sage.modular.local_comp.liftings import lift_gen_to_gamma1
>>> A = matrix(ZZ, Integer(2), lift_gen_to_gamma1(Integer(9), Integer(8))); A
[441  62]
[ 64   9]
>>> A.change_ring(Zmod(Integer(9)))
[0 8]
[1 0]
>>> A.change_ring(Zmod(Integer(8)))
[1 6]
[0 1]
>>> type(lift_gen_to_gamma1(Integer(9), Integer(8))[Integer(0)])
<class 'sage.rings.integer.Integer'>
sage.modular.local_comp.liftings.lift_matrix_to_sl2z(A, N)[source]

Given a list of length 4 representing a 2x2 matrix over \(\ZZ / N\ZZ\) with determinant 1 (mod \(N\)), lift it to a 2x2 matrix over \(\ZZ\) with determinant 1.

This is a special case of lift_to_gamma1(), and is coded as such.

INPUT:

  • A – list of 4 integers defining a \(2 \times 2\) matrix

  • N – positive integer

EXAMPLES:

sage: from sage.modular.local_comp.liftings import lift_matrix_to_sl2z
sage: lift_matrix_to_sl2z([10, 11, 3, 11], 19)
[29, 106, 3, 11]
sage: type(_[0])
<class 'sage.rings.integer.Integer'>
sage: lift_matrix_to_sl2z([2,0,0,1], 5)
Traceback (most recent call last):
...
ValueError: Determinant is 2 mod 5, should be 1

>>> from sage.all import *
>>> from sage.modular.local_comp.liftings import lift_matrix_to_sl2z
>>> lift_matrix_to_sl2z([Integer(10), Integer(11), Integer(3), Integer(11)], Integer(19))
[29, 106, 3, 11]
>>> type(_[Integer(0)])
<class 'sage.rings.integer.Integer'>
>>> lift_matrix_to_sl2z([Integer(2),Integer(0),Integer(0),Integer(1)], Integer(5))
Traceback (most recent call last):
...
ValueError: Determinant is 2 mod 5, should be 1
sage.modular.local_comp.liftings.lift_ramified(g, p, u, n)[source]

Given four integers \(a,b,c,d\) with \(p \mid c\) and \(ad - bc = 1 \pmod{p^u}\), find \(a',b',c',d'\) congruent to \(a,b,c,d \pmod{p^u}\), with \(c' = c \pmod{p^{u+1}}\), such that \(a'd' - b'c'\) is exactly 1, and \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) is in \(\Gamma_1(n)\).

Algorithm: Uses lift_to_gamma1() to get a lifting modulo \(p^u\), and then adds an appropriate multiple of the top row to the bottom row in order to get the bottom-left entry correct modulo \(p^{u+1}\).

EXAMPLES:

sage: from sage.modular.local_comp.liftings import lift_ramified
sage: lift_ramified([2,2,3,2], 3, 1, 1)
[-1, -1, 3, 2]
sage: lift_ramified([8,2,12,2], 3, 2, 23)
[323, 110, -133584, -45493]
sage: type(lift_ramified([8,2,12,2], 3, 2, 23)[0])
<class 'sage.rings.integer.Integer'>

>>> from sage.all import *
>>> from sage.modular.local_comp.liftings import lift_ramified
>>> lift_ramified([Integer(2),Integer(2),Integer(3),Integer(2)], Integer(3), Integer(1), Integer(1))
[-1, -1, 3, 2]
>>> lift_ramified([Integer(8),Integer(2),Integer(12),Integer(2)], Integer(3), Integer(2), Integer(23))
[323, 110, -133584, -45493]
>>> type(lift_ramified([Integer(8),Integer(2),Integer(12),Integer(2)], Integer(3), Integer(2), Integer(23))[Integer(0)])
<class 'sage.rings.integer.Integer'>
sage.modular.local_comp.liftings.lift_to_gamma1(g, m, n)[source]

If g = [a,b,c,d] is a list of integers defining a \(2 \times 2\) matrix whose determinant is \(1 \pmod m\), return a list of integers giving the entries of a matrix which is congruent to \(g \pmod m\) and to \(\begin{pmatrix} 1 & * \\ 0 & 1 \end{pmatrix} \pmod n\). Here \(m\) and \(n\) must be coprime.

INPUT:

  • g – list of 4 integers defining a \(2 \times 2\) matrix

  • \(m\), \(n\) – coprime positive integers

Here \(m\) and \(n\) should be coprime positive integers. Either of \(m\) and \(n\) can be \(1\). If \(n = 1\), this still makes perfect sense; this is what is called by the function lift_matrix_to_sl2z(). If \(m = 1\) this is a rather silly question, so we adopt the convention of always returning the identity matrix.

The result is always a list of Sage integers (unlike lift_to_sl2z, which tends to return Python ints).

EXAMPLES:

sage: from sage.modular.local_comp.liftings import lift_to_gamma1
sage: A = matrix(ZZ, 2, lift_to_gamma1([10, 11, 3, 11], 19, 5)); A
[371  68]
[ 60  11]
sage: A.det() == 1
True
sage: A.change_ring(Zmod(19))
[10 11]
[ 3 11]
sage: A.change_ring(Zmod(5))
[1 3]
[0 1]
sage: m = list(SL2Z.random_element())
sage: n = lift_to_gamma1(m, 11, 17)
sage: assert matrix(Zmod(11), 2, n) == matrix(Zmod(11),2,m)
sage: assert matrix(Zmod(17), 2, [n[0], 0, n[2], n[3]]) == 1
sage: type(lift_to_gamma1([10,11,3,11],19,5)[0])
<class 'sage.rings.integer.Integer'>

>>> from sage.all import *
>>> from sage.modular.local_comp.liftings import lift_to_gamma1
>>> A = matrix(ZZ, Integer(2), lift_to_gamma1([Integer(10), Integer(11), Integer(3), Integer(11)], Integer(19), Integer(5))); A
[371  68]
[ 60  11]
>>> A.det() == Integer(1)
True
>>> A.change_ring(Zmod(Integer(19)))
[10 11]
[ 3 11]
>>> A.change_ring(Zmod(Integer(5)))
[1 3]
[0 1]
>>> m = list(SL2Z.random_element())
>>> n = lift_to_gamma1(m, Integer(11), Integer(17))
>>> assert matrix(Zmod(Integer(11)), Integer(2), n) == matrix(Zmod(Integer(11)),Integer(2),m)
>>> assert matrix(Zmod(Integer(17)), Integer(2), [n[Integer(0)], Integer(0), n[Integer(2)], n[Integer(3)]]) == Integer(1)
>>> type(lift_to_gamma1([Integer(10),Integer(11),Integer(3),Integer(11)],Integer(19),Integer(5))[Integer(0)])
<class 'sage.rings.integer.Integer'>

Tests with \(m = 1\) and with \(n = 1\):

sage: lift_to_gamma1([1,1,0,1], 5, 1)
[1, 1, 0, 1]
sage: lift_to_gamma1([2,3,11,22], 1, 5)
[1, 0, 0, 1]

>>> from sage.all import *
>>> lift_to_gamma1([Integer(1),Integer(1),Integer(0),Integer(1)], Integer(5), Integer(1))
[1, 1, 0, 1]
>>> lift_to_gamma1([Integer(2),Integer(3),Integer(11),Integer(22)], Integer(1), Integer(5))
[1, 0, 0, 1]
sage.modular.local_comp.liftings.lift_uniformiser_odd(p, u, n)[source]

Construct a matrix over \(\ZZ\) whose determinant is \(p\), and which is congruent to \(\begin{pmatrix} 0 & -1 \\ p & 0 \end{pmatrix} \pmod{p^u}\) and to \(\begin{pmatrix} p & 0 \\ 0 & 1\end{pmatrix} \pmod n\).

This is required for the local components machinery in the “ramified” case (when the exponent of \(p\) dividing the level is odd).

EXAMPLES:

sage: from sage.modular.local_comp.liftings import lift_uniformiser_odd
sage: lift_uniformiser_odd(3, 2, 11)
[432, 377, 165, 144]
sage: type(lift_uniformiser_odd(3, 2, 11)[0])
<class 'sage.rings.integer.Integer'>

>>> from sage.all import *
>>> from sage.modular.local_comp.liftings import lift_uniformiser_odd
>>> lift_uniformiser_odd(Integer(3), Integer(2), Integer(11))
[432, 377, 165, 144]
>>> type(lift_uniformiser_odd(Integer(3), Integer(2), Integer(11))[Integer(0)])
<class 'sage.rings.integer.Integer'>