Manin relations for overconvergent modular symbols

Code to create the Manin Relations class, which solves the “Manin relations”. That is, a description of $Di{v}^0(P^1(\mathbf{Q}))$ as a $\mathbf{Z}[{\mathrm{\Gamma }}_0(N)]$-module in terms of generators and relations is found. The method used is geometric, constructing a nice fundamental domain for ${\mathrm{\Gamma }}_0(N)$ and reading the relevant Manin relations off of that picture. The algorithm follows [PS2011].

AUTHORS:

• Robert Pollack, Jonathan Hanke (2012): initial version

sage.modular.pollack_stevens.fund_domain.M2Z(x)

Create an immutable $2\times 2$ integer matrix from x.

INPUT:

• x – anything that can be converted into a $2\times 2$ matrix

EXAMPLES:

Sage

sage: from sage.modular.pollack_stevens.fund_domain import M2Z
sage: M2Z([1,2,3,4])
[1 2]
[3 4]
sage: M2Z(1)
[1 0]
[0 1]

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.fund_domain import M2Z
>>> M2Z([Integer(1),Integer(2),Integer(3),Integer(4)])
[1 2]
[3 4]
>>> M2Z(Integer(1))
[1 0]
[0 1]

Sage Live

from sage.modular.pollack_stevens.fund_domain import M2Z
M2Z([1,2,3,4])
M2Z(1)

class sage.modular.pollack_stevens.fund_domain.ManinRelations(N)

Bases: PollackStevensModularDomain

This class gives a description of $Di{v}^0(P^1(\mathbf{Q}))$ as a $\mathbf{Z}[{\mathrm{\Gamma }}_0(N)]$-module.

INPUT:

• N – positive integer, the level of ${\mathrm{\Gamma }}_0(N)$ to work with

EXAMPLES:

Sage

sage: from sage.modular.pollack_stevens.fund_domain import ManinRelations
sage: ManinRelations(1)
Manin Relations of level 1
sage: ManinRelations(11)
Manin Relations of level 11

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.fund_domain import ManinRelations
>>> ManinRelations(Integer(1))
Manin Relations of level 1
>>> ManinRelations(Integer(11))
Manin Relations of level 11

Sage Live

from sage.modular.pollack_stevens.fund_domain import ManinRelations
ManinRelations(1)
ManinRelations(11)

Large values of N are not supported:

Sage

sage: ManinRelations(2^20)
Traceback (most recent call last):
...
OverflowError: Modulus is too large (must be <= 46340)

Python

>>> from sage.all import *
>>> ManinRelations(Integer(2)**Integer(20))
Traceback (most recent call last):
...
OverflowError: Modulus is too large (must be <= 46340)

Sage Live

ManinRelations(2^20)

fd_boundary(C)

Find matrices whose associated unimodular paths give the boundary of a fundamental domain.

Here the fundamental domain is for ${\mathrm{\Gamma }}_0(N)$. (In the case when ${\mathrm{\Gamma }}_0(N)$ has elements of order three the shape cut out by these unimodular matrices is a little smaller than a fundamental domain. See Section 2.5 of [PS2011].)

INPUT:

• C – list of rational numbers coming from self.form_list_of_cusps()

OUTPUT:

A list of $2\times 2$ integer matrices of determinant 1 whose associated unimodular paths give the boundary of a fundamental domain for ${\mathrm{\Gamma }}_0(N)$ (or nearly so in the case of 3-torsion).

EXAMPLES:

Sage

sage: from sage.modular.pollack_stevens.fund_domain import ManinRelations
sage: A = ManinRelations(11)
sage: C = A.form_list_of_cusps(); C
[-1, -2/3, -1/2, -1/3, 0]
sage: A.fd_boundary(C)
[
[1 0]  [ 1  1]  [ 0 -1]  [-1 -1]  [-1 -2]  [-2 -1]
[0 1], [-1  0], [ 1  3], [ 3  2], [ 2  3], [ 3  1]
]
sage: A = ManinRelations(13)
sage: C = A.form_list_of_cusps(); C
[-1, -2/3, -1/2, -1/3, 0]
sage: A.fd_boundary(C)
[
[1 0]  [ 1  1]  [ 0 -1]  [-1 -1]  [-1 -2]  [-2 -1]
[0 1], [-1  0], [ 1  3], [ 3  2], [ 2  3], [ 3  1]
]
sage: A = ManinRelations(101)
sage: C = A.form_list_of_cusps(); C
[-1, -6/7, -5/6, -4/5, -7/9, -3/4, -11/15, -8/11, -5/7, -7/10,
-9/13, -2/3, -5/8, -13/21, -8/13, -3/5, -7/12, -11/19, -4/7, -1/2,
-4/9, -3/7, -5/12, -7/17, -2/5, -3/8, -4/11, -1/3, -2/7, -3/11,
-1/4, -2/9, -1/5, -1/6, 0]
sage: A.fd_boundary(C)
[
[1 0]  [ 1  1]  [ 0 -1]  [-1 -1]  [-1 -2]  [-2 -1]  [-1 -3]  [-3 -2]
[0 1], [-1  0], [ 1  6], [ 6  5], [ 5  9], [ 9  4], [ 4 11], [11  7],

[-2 -1]  [-1 -4]  [-4 -3]  [-3 -2]  [-2 -7]  [-7 -5]  [-5 -3]  [-3 -4]
[ 7  3], [ 3 11], [11  8], [ 8  5], [ 5 17], [17 12], [12  7], [ 7  9],

[-4 -1]  [-1 -4]  [ -4 -11]  [-11  -7]  [-7 -3]  [-3 -8]  [ -8 -13]
[ 9  2], [ 2  7], [  7  19], [ 19  12], [12  5], [ 5 13], [ 13  21],

[-13  -5]  [-5 -2]  [-2 -9]  [-9 -7]  [-7 -5]  [-5 -8]  [ -8 -11]
[ 21   8], [ 8  3], [ 3 13], [13 10], [10  7], [ 7 11], [ 11  15],

[-11  -3]  [-3 -7]  [-7 -4]  [-4 -5]  [-5 -6]  [-6 -1]
[ 15   4], [ 4  9], [ 9  5], [ 5  6], [ 6  7], [ 7  1]
]

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.fund_domain import ManinRelations
>>> A = ManinRelations(Integer(11))
>>> C = A.form_list_of_cusps(); C
[-1, -2/3, -1/2, -1/3, 0]
>>> A.fd_boundary(C)
[
[1 0]  [ 1  1]  [ 0 -1]  [-1 -1]  [-1 -2]  [-2 -1]
[0 1], [-1  0], [ 1  3], [ 3  2], [ 2  3], [ 3  1]
]
>>> A = ManinRelations(Integer(13))
>>> C = A.form_list_of_cusps(); C
[-1, -2/3, -1/2, -1/3, 0]
>>> A.fd_boundary(C)
[
[1 0]  [ 1  1]  [ 0 -1]  [-1 -1]  [-1 -2]  [-2 -1]
[0 1], [-1  0], [ 1  3], [ 3  2], [ 2  3], [ 3  1]
]
>>> A = ManinRelations(Integer(101))
>>> C = A.form_list_of_cusps(); C
[-1, -6/7, -5/6, -4/5, -7/9, -3/4, -11/15, -8/11, -5/7, -7/10,
-9/13, -2/3, -5/8, -13/21, -8/13, -3/5, -7/12, -11/19, -4/7, -1/2,
-4/9, -3/7, -5/12, -7/17, -2/5, -3/8, -4/11, -1/3, -2/7, -3/11,
-1/4, -2/9, -1/5, -1/6, 0]
>>> A.fd_boundary(C)
[
[1 0]  [ 1  1]  [ 0 -1]  [-1 -1]  [-1 -2]  [-2 -1]  [-1 -3]  [-3 -2]
[0 1], [-1  0], [ 1  6], [ 6  5], [ 5  9], [ 9  4], [ 4 11], [11  7],
<BLANKLINE>
[-2 -1]  [-1 -4]  [-4 -3]  [-3 -2]  [-2 -7]  [-7 -5]  [-5 -3]  [-3 -4]
[ 7  3], [ 3 11], [11  8], [ 8  5], [ 5 17], [17 12], [12  7], [ 7  9],
<BLANKLINE>
[-4 -1]  [-1 -4]  [ -4 -11]  [-11  -7]  [-7 -3]  [-3 -8]  [ -8 -13]
[ 9  2], [ 2  7], [  7  19], [ 19  12], [12  5], [ 5 13], [ 13  21],
<BLANKLINE>
[-13  -5]  [-5 -2]  [-2 -9]  [-9 -7]  [-7 -5]  [-5 -8]  [ -8 -11]
[ 21   8], [ 8  3], [ 3 13], [13 10], [10  7], [ 7 11], [ 11  15],
<BLANKLINE>
[-11  -3]  [-3 -7]  [-7 -4]  [-4 -5]  [-5 -6]  [-6 -1]
[ 15   4], [ 4  9], [ 9  5], [ 5  6], [ 6  7], [ 7  1]
]

Sage Live

from sage.modular.pollack_stevens.fund_domain import ManinRelations
A = ManinRelations(11)
C = A.form_list_of_cusps(); C
A.fd_boundary(C)
A = ManinRelations(13)
C = A.form_list_of_cusps(); C
A.fd_boundary(C)
A = ManinRelations(101)
C = A.form_list_of_cusps(); C
A.fd_boundary(C)

form_list_of_cusps()

Return the intersection of a fundamental domain for ${\mathrm{\Gamma }}_0(N)$ with the real axis.

The construction of this fundamental domain follows the arguments of [PS2011] Section 2. The boundary of this fundamental domain consists entirely of unimodular paths when ${\mathrm{\Gamma }}_0(N)$ has no elements of order 3. (See [PS2011] Section 2.5 for the case when there are elements of order 3.)

OUTPUT:

A sorted list of rational numbers marking the intersection of a fundamental domain for ${\mathrm{\Gamma }}_0(N)$ with the real axis.

EXAMPLES:

Sage

sage: from sage.modular.pollack_stevens.fund_domain import ManinRelations
sage: A = ManinRelations(11)
sage: A.form_list_of_cusps()
[-1, -2/3, -1/2, -1/3, 0]
sage: A = ManinRelations(13)
sage: A.form_list_of_cusps()
[-1, -2/3, -1/2, -1/3, 0]
sage: A = ManinRelations(101)
sage: A.form_list_of_cusps()
[-1, -6/7, -5/6, -4/5, -7/9, -3/4, -11/15, -8/11, -5/7, -7/10,
-9/13, -2/3, -5/8, -13/21, -8/13, -3/5, -7/12, -11/19, -4/7, -1/2,
-4/9, -3/7, -5/12, -7/17, -2/5, -3/8, -4/11, -1/3, -2/7, -3/11,
-1/4, -2/9, -1/5, -1/6, 0]

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.fund_domain import ManinRelations
>>> A = ManinRelations(Integer(11))
>>> A.form_list_of_cusps()
[-1, -2/3, -1/2, -1/3, 0]
>>> A = ManinRelations(Integer(13))
>>> A.form_list_of_cusps()
[-1, -2/3, -1/2, -1/3, 0]
>>> A = ManinRelations(Integer(101))
>>> A.form_list_of_cusps()
[-1, -6/7, -5/6, -4/5, -7/9, -3/4, -11/15, -8/11, -5/7, -7/10,
-9/13, -2/3, -5/8, -13/21, -8/13, -3/5, -7/12, -11/19, -4/7, -1/2,
-4/9, -3/7, -5/12, -7/17, -2/5, -3/8, -4/11, -1/3, -2/7, -3/11,
-1/4, -2/9, -1/5, -1/6, 0]

Sage Live

from sage.modular.pollack_stevens.fund_domain import ManinRelations
A = ManinRelations(11)
A.form_list_of_cusps()
A = ManinRelations(13)
A.form_list_of_cusps()
A = ManinRelations(101)
A.form_list_of_cusps()

indices_with_three_torsion()

A list of indices of coset representatives whose associated unimodular path contains a point fixed by a ${\mathrm{\Gamma }}_0(N)$ element of order 3 in the ideal triangle directly below that path (the order is computed in $PS{L}_2(\mathbf{Z})$).

EXAMPLES:

Sage

sage: from sage.modular.pollack_stevens.fund_domain import ManinRelations
sage: MR = ManinRelations(11)
sage: MR.indices_with_three_torsion()
[]
sage: MR = ManinRelations(13)
sage: MR.indices_with_three_torsion()
[2, 5]
sage: B = MR.reps(2); B
[ 0 -1]
[ 1  3]

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.fund_domain import ManinRelations
>>> MR = ManinRelations(Integer(11))
>>> MR.indices_with_three_torsion()
[]
>>> MR = ManinRelations(Integer(13))
>>> MR.indices_with_three_torsion()
[2, 5]
>>> B = MR.reps(Integer(2)); B
[ 0 -1]
[ 1  3]

Sage Live

from sage.modular.pollack_stevens.fund_domain import ManinRelations
MR = ManinRelations(11)
MR.indices_with_three_torsion()
MR = ManinRelations(13)
MR.indices_with_three_torsion()
B = MR.reps(2); B

The corresponding matrix of order three:

Sage

sage: A = MR.three_torsion_matrix(B); A
[-4 -1]
[13  3]
sage: A^3
[1 0]
[0 1]

Python

>>> from sage.all import *
>>> A = MR.three_torsion_matrix(B); A
[-4 -1]
[13  3]
>>> A**Integer(3)
[1 0]
[0 1]

Sage Live

A = MR.three_torsion_matrix(B); A
A^3

The columns of B and the columns of A*B and A^2*B give the same rational cusps:

Sage

sage: B
[ 0 -1]
[ 1  3]
sage: A*B, A^2*B
(
[-1  1]  [ 1  0]
[ 3 -4], [-4  1]
)

Python

>>> from sage.all import *
>>> B
[ 0 -1]
[ 1  3]
>>> A*B, A**Integer(2)*B
(
[-1  1]  [ 1  0]
[ 3 -4], [-4  1]
)

Sage Live

B
A*B, A^2*B

indices_with_two_torsion()

Return the indices of coset representatives whose associated unimodular path contains a point fixed by a ${\mathrm{\Gamma }}_0(N)$ element of order 2 (where the order is computed in $PS{L}_2(\mathbf{Z})$).

OUTPUT: list of integers

EXAMPLES:

Sage

sage: from sage.modular.pollack_stevens.fund_domain import ManinRelations
sage: MR = ManinRelations(11)
sage: MR.indices_with_two_torsion()
[]
sage: MR = ManinRelations(13)
sage: MR.indices_with_two_torsion()
[3, 4]
sage: MR.reps(3), MR.reps(4)
(
[-1 -1]  [-1 -2]
[ 3  2], [ 2  3]
)

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.fund_domain import ManinRelations
>>> MR = ManinRelations(Integer(11))
>>> MR.indices_with_two_torsion()
[]
>>> MR = ManinRelations(Integer(13))
>>> MR.indices_with_two_torsion()
[3, 4]
>>> MR.reps(Integer(3)), MR.reps(Integer(4))
(
[-1 -1]  [-1 -2]
[ 3  2], [ 2  3]
)

Sage Live

from sage.modular.pollack_stevens.fund_domain import ManinRelations
MR = ManinRelations(11)
MR.indices_with_two_torsion()
MR = ManinRelations(13)
MR.indices_with_two_torsion()
MR.reps(3), MR.reps(4)

The corresponding matrix of order 2:

Sage

sage: A = MR.two_torsion_matrix(MR.reps(3)); A
[  5   2]
[-13  -5]
sage: A^2
[-1  0]
[ 0 -1]

Python

>>> from sage.all import *
>>> A = MR.two_torsion_matrix(MR.reps(Integer(3))); A
[  5   2]
[-13  -5]
>>> A**Integer(2)
[-1  0]
[ 0 -1]

Sage Live

A = MR.two_torsion_matrix(MR.reps(3)); A
A^2

You can see that multiplication by A just interchanges the rational cusps determined by the columns of the matrix MR.reps(3):

Sage

sage: MR.reps(3), A*MR.reps(3)
(
[-1 -1]  [ 1 -1]
[ 3  2], [-2  3]
)

Python

>>> from sage.all import *
>>> MR.reps(Integer(3)), A*MR.reps(Integer(3))
(
[-1 -1]  [ 1 -1]
[ 3  2], [-2  3]
)

Sage Live

MR.reps(3), A*MR.reps(3)

is_unimodular_path(r1, r2)

Determine whether two (non-infinite) cusps are connected by a unimodular path.

INPUT:

• r1, r2 – rational numbers

OUTPUT:

A boolean expressing whether or not a unimodular path connects r1 to r2.

EXAMPLES:

Sage

sage: from sage.modular.pollack_stevens.fund_domain import ManinRelations
sage: A = ManinRelations(11)
sage: A.is_unimodular_path(0,1/3)
True
sage: A.is_unimodular_path(1/3,0)
True
sage: A.is_unimodular_path(0,2/3)
False
sage: A.is_unimodular_path(2/3,0)
False

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.fund_domain import ManinRelations
>>> A = ManinRelations(Integer(11))
>>> A.is_unimodular_path(Integer(0),Integer(1)/Integer(3))
True
>>> A.is_unimodular_path(Integer(1)/Integer(3),Integer(0))
True
>>> A.is_unimodular_path(Integer(0),Integer(2)/Integer(3))
False
>>> A.is_unimodular_path(Integer(2)/Integer(3),Integer(0))
False

Sage Live

from sage.modular.pollack_stevens.fund_domain import ManinRelations
A = ManinRelations(11)
A.is_unimodular_path(0,1/3)
A.is_unimodular_path(1/3,0)
A.is_unimodular_path(0,2/3)
A.is_unimodular_path(2/3,0)

prep_hecke_on_gen(l, gen, modulus=None)

This function does some precomputations needed to compute $T_l$.

In particular, if $\varphi$ is a modular symbol and $D_m$ is the divisor associated to the generator gen, to compute $(\varphi |T_l)(D_m)$ one needs to compute $\varphi ({\gamma }_a{D}_m)|{\gamma }_a$ where ${\gamma }_a$ runs through the $l+1$ matrices defining $T_l$. One then takes ${\gamma }_a{D}_m$ and writes it as a sum of unimodular divisors. For each such unimodular divisor, say $[M]$ where $M$ is a $S{L}_2$ matrix, we then write $M=\gamma h$ where $\gamma$ is in ${\mathrm{\Gamma }}_0(N)$ and $h$ is one of our chosen coset representatives. Then $\varphi ([M])=\varphi ([h])|{\gamma }^{-1}$. Thus, one has

$$(\varphi |{\gamma }_a)(D_m)=\sum _h\sum _j\varphi ([h])|{\gamma }_{hj}^{-1}\cdot {\gamma }_a$$

as $h$ runs over all coset representatives and $j$ simply runs over however many times $M_h$ appears in the above computation.

Finally, the output of this function is a dictionary D whose keys are the coset representatives in self.reps() where each value is a list of matrices, and the entries of D satisfy:

$$D[h][j]={\gamma }_{hj}\ast {\gamma }_a$$

INPUT:

• l – a prime

• gen – a generator

OUTPUT:

A list of lists (see above).

EXAMPLES:

Sage

sage: E = EllipticCurve('11a')
sage: phi = E.pollack_stevens_modular_symbol()
sage: phi.values()
[-1/5, 1, 0]
sage: M = phi.parent().source()
sage: w = M.prep_hecke_on_gen(2, M.gens()[0])
sage: one = Matrix(ZZ,2,2,1)
sage: one.set_immutable()
sage: w[one]
[[1 0]
[0 2], [1 1]
[0 2], [2 0]
[0 1]]

Python

>>> from sage.all import *
>>> E = EllipticCurve('11a')
>>> phi = E.pollack_stevens_modular_symbol()
>>> phi.values()
[-1/5, 1, 0]
>>> M = phi.parent().source()
>>> w = M.prep_hecke_on_gen(Integer(2), M.gens()[Integer(0)])
>>> one = Matrix(ZZ,Integer(2),Integer(2),Integer(1))
>>> one.set_immutable()
>>> w[one]
[[1 0]
[0 2], [1 1]
[0 2], [2 0]
[0 1]]

Sage Live

E = EllipticCurve('11a')
phi = E.pollack_stevens_modular_symbol()
phi.values()
M = phi.parent().source()
w = M.prep_hecke_on_gen(2, M.gens()[0])
one = Matrix(ZZ,2,2,1)
one.set_immutable()
w[one]

prep_hecke_on_gen_list(l, gen, modulus=None)

Return the precomputation to compute $T_l$ in a way that speeds up the Hecke calculation.

Namely, returns a list of the form [h,A].

INPUT:

• l – a prime

• gen – a generator

OUTPUT:

A list of lists (see above).

EXAMPLES:

Sage

sage: E = EllipticCurve('11a')
sage: phi = E.pollack_stevens_modular_symbol()
sage: phi.values()
[-1/5, 1, 0]
sage: M = phi.parent().source()
sage: len(M.prep_hecke_on_gen_list(2, M.gens()[0]))
4

Python

>>> from sage.all import *
>>> E = EllipticCurve('11a')
>>> phi = E.pollack_stevens_modular_symbol()
>>> phi.values()
[-1/5, 1, 0]
>>> M = phi.parent().source()
>>> len(M.prep_hecke_on_gen_list(Integer(2), M.gens()[Integer(0)]))
4

Sage Live

E = EllipticCurve('11a')
phi = E.pollack_stevens_modular_symbol()
phi.values()
M = phi.parent().source()
len(M.prep_hecke_on_gen_list(2, M.gens()[0]))

reps_with_three_torsion()

A list of coset representatives whose associated unimodular path contains a point fixed by a ${\mathrm{\Gamma }}_0(N)$ element of order 3 in the ideal triangle directly below that path (the order is computed in $PS{L}_2(\mathbf{Z})$).

EXAMPLES:

Sage

sage: from sage.modular.pollack_stevens.fund_domain import ManinRelations
sage: MR = ManinRelations(13)
sage: B = MR.reps_with_three_torsion()[0]; B
[ 0 -1]
[ 1  3]

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.fund_domain import ManinRelations
>>> MR = ManinRelations(Integer(13))
>>> B = MR.reps_with_three_torsion()[Integer(0)]; B
[ 0 -1]
[ 1  3]

Sage Live

from sage.modular.pollack_stevens.fund_domain import ManinRelations
MR = ManinRelations(13)
B = MR.reps_with_three_torsion()[0]; B

The corresponding matrix of order three:

Sage

sage: A = MR.three_torsion_matrix(B); A
[-4 -1]
[13  3]
sage: A^3
[1 0]
[0 1]

Python

>>> from sage.all import *
>>> A = MR.three_torsion_matrix(B); A
[-4 -1]
[13  3]
>>> A**Integer(3)
[1 0]
[0 1]

Sage Live

A = MR.three_torsion_matrix(B); A
A^3

The columns of B and the columns of A*B and A^2*B give the same rational cusps:

Sage

sage: B
[ 0 -1]
[ 1  3]
sage: A*B, A^2*B
(
[-1  1]  [ 1  0]
[ 3 -4], [-4  1]
)

Python

>>> from sage.all import *
>>> B
[ 0 -1]
[ 1  3]
>>> A*B, A**Integer(2)*B
(
[-1  1]  [ 1  0]
[ 3 -4], [-4  1]
)

Sage Live

B
A*B, A^2*B

reps_with_two_torsion()

The coset representatives whose associated unimodular path contains a point fixed by a ${\mathrm{\Gamma }}_0(N)$ element of order 2 (where the order is computed in $PS{L}_2(\mathbf{Z})$).

OUTPUT: list of matrices

EXAMPLES:

Sage

sage: from sage.modular.pollack_stevens.fund_domain import ManinRelations
sage: MR = ManinRelations(11)
sage: MR.reps_with_two_torsion()
[]
sage: MR = ManinRelations(13)
sage: MR.reps_with_two_torsion()
[
[-1 -1]  [-1 -2]
[ 3  2], [ 2  3]
]
sage: B = MR.reps_with_two_torsion()[0]

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.fund_domain import ManinRelations
>>> MR = ManinRelations(Integer(11))
>>> MR.reps_with_two_torsion()
[]
>>> MR = ManinRelations(Integer(13))
>>> MR.reps_with_two_torsion()
[
[-1 -1]  [-1 -2]
[ 3  2], [ 2  3]
]
>>> B = MR.reps_with_two_torsion()[Integer(0)]

Sage Live

from sage.modular.pollack_stevens.fund_domain import ManinRelations
MR = ManinRelations(11)
MR.reps_with_two_torsion()
MR = ManinRelations(13)
MR.reps_with_two_torsion()
B = MR.reps_with_two_torsion()[0]

The corresponding matrix of order 2:

Sage

sage: A = MR.two_torsion_matrix(B); A
[  5   2]
[-13  -5]
sage: A^2
[-1  0]
[ 0 -1]

Python

>>> from sage.all import *
>>> A = MR.two_torsion_matrix(B); A
[  5   2]
[-13  -5]
>>> A**Integer(2)
[-1  0]
[ 0 -1]

Sage Live

A = MR.two_torsion_matrix(B); A
A^2

You can see that multiplication by A just interchanges the rational cusps determined by the columns of the matrix MR.reps(3):

Sage

sage: B, A*B
(
[-1 -1]  [ 1 -1]
[ 3  2], [-2  3]
)

Python

>>> from sage.all import *
>>> B, A*B
(
[-1 -1]  [ 1 -1]
[ 3  2], [-2  3]
)

Sage Live

B, A*B

three_torsion_matrix(A)

Return the matrix of order two in ${\mathrm{\Gamma }}_0(N)$ which corresponds to an A in self.reps_with_two_torsion().

INPUT:

• A – a matrix in self.reps_with_two_torsion()

EXAMPLES:

Sage

sage: from sage.modular.pollack_stevens.fund_domain import ManinRelations
sage: MR = ManinRelations(37)
sage: B = MR.reps_with_three_torsion()[0]

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.fund_domain import ManinRelations
>>> MR = ManinRelations(Integer(37))
>>> B = MR.reps_with_three_torsion()[Integer(0)]

Sage Live

from sage.modular.pollack_stevens.fund_domain import ManinRelations
MR = ManinRelations(37)
B = MR.reps_with_three_torsion()[0]

The corresponding matrix of order 3:

Sage

sage: A = MR.three_torsion_matrix(B); A
[-11  -3]
[ 37  10]
sage: A^3
[1 0]
[0 1]

Python

>>> from sage.all import *
>>> A = MR.three_torsion_matrix(B); A
[-11  -3]
[ 37  10]
>>> A**Integer(3)
[1 0]
[0 1]

Sage Live

A = MR.three_torsion_matrix(B); A
A^3

two_torsion_matrix(A)

Return the matrix of order two in ${\mathrm{\Gamma }}_0(N)$ which corresponds to an A in self.reps_with_two_torsion().

INPUT:

• A – a matrix in self.reps_with_two_torsion()

EXAMPLES:

Sage

sage: from sage.modular.pollack_stevens.fund_domain import ManinRelations
sage: MR = ManinRelations(25)
sage: B = MR.reps_with_two_torsion()[0]

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.fund_domain import ManinRelations
>>> MR = ManinRelations(Integer(25))
>>> B = MR.reps_with_two_torsion()[Integer(0)]

Sage Live

from sage.modular.pollack_stevens.fund_domain import ManinRelations
MR = ManinRelations(25)
B = MR.reps_with_two_torsion()[0]

The corresponding matrix of order 2:

Sage

sage: A = MR.two_torsion_matrix(B); A
[  7   2]
[-25  -7]
sage: A^2
[-1  0]
[ 0 -1]

Python

>>> from sage.all import *
>>> A = MR.two_torsion_matrix(B); A
[  7   2]
[-25  -7]
>>> A**Integer(2)
[-1  0]
[ 0 -1]

Sage Live

A = MR.two_torsion_matrix(B); A
A^2

unimod_to_matrices(r1, r2)

Return the two matrices whose associated unimodular paths connect r1 and r2 and r2 and r1, respectively.

INPUT:

• r1, r2 – rational numbers (that are assumed to be connected by a unimodular path)

OUTPUT: a pair of $2\times 2$ matrices of determinant 1

EXAMPLES:

Sage

sage: from sage.modular.pollack_stevens.fund_domain import ManinRelations
sage: A = ManinRelations(11)
sage: A.unimod_to_matrices(0,1/3)
(
[ 0  1]  [1 0]
[-1  3], [3 1]
)

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.fund_domain import ManinRelations
>>> A = ManinRelations(Integer(11))
>>> A.unimod_to_matrices(Integer(0),Integer(1)/Integer(3))
(
[ 0  1]  [1 0]
[-1  3], [3 1]
)

Sage Live

from sage.modular.pollack_stevens.fund_domain import ManinRelations
A = ManinRelations(11)
A.unimod_to_matrices(0,1/3)

class sage.modular.pollack_stevens.fund_domain.PollackStevensModularDomain(N, reps, indices, rels, equiv_ind)

Bases: SageObject

The domain of a modular symbol.

INPUT:

• N – positive integer, the level of the congruence subgroup ${\mathrm{\Gamma }}_0(N)$

• reps – list of $2\times 2$ matrices, the coset representatives of $Di{v}^0(P^1(\mathbf{Q}))$

• indices – list of integers; indices of elements in reps which are generators

• rels – list of list of triples (d, A, i), one for each coset representative of reps which describes how to express the elements of reps in terms of generators specified by indices. See relations() for a detailed explanations of these triples.

• equiv_ind – dictionary which maps normalized coordinates on $P^1(\mathbf{Z}/N\mathbf{Z})$ to an integer such that a matrix whose bottom row is equivalent to $[a:b]$ in $P^1(\mathbf{Z}/N\mathbf{Z})$ is in the coset of reps[equiv_ind[(a,b)]]

EXAMPLES:

Sage

sage: from sage.modular.pollack_stevens.fund_domain import PollackStevensModularDomain, M2Z
sage: PollackStevensModularDomain(2 , [M2Z([1,0,0,1]), M2Z([1,1,-1,0]), M2Z([0,-1,1,1])], [0,2], [[(1, M2Z([1,0,0,1]), 0)], [(-1,M2Z([-1,-1,0,-1]),0)], [(1, M2Z([1,0,0,1]), 2)]], {(0,1): 0, (1,0): 1, (1,1): 2})
Modular Symbol domain of level 2

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.fund_domain import PollackStevensModularDomain, M2Z
>>> PollackStevensModularDomain(Integer(2) , [M2Z([Integer(1),Integer(0),Integer(0),Integer(1)]), M2Z([Integer(1),Integer(1),-Integer(1),Integer(0)]), M2Z([Integer(0),-Integer(1),Integer(1),Integer(1)])], [Integer(0),Integer(2)], [[(Integer(1), M2Z([Integer(1),Integer(0),Integer(0),Integer(1)]), Integer(0))], [(-Integer(1),M2Z([-Integer(1),-Integer(1),Integer(0),-Integer(1)]),Integer(0))], [(Integer(1), M2Z([Integer(1),Integer(0),Integer(0),Integer(1)]), Integer(2))]], {(Integer(0),Integer(1)): Integer(0), (Integer(1),Integer(0)): Integer(1), (Integer(1),Integer(1)): Integer(2)})
Modular Symbol domain of level 2

Sage Live

from sage.modular.pollack_stevens.fund_domain import PollackStevensModularDomain, M2Z
PollackStevensModularDomain(2 , [M2Z([1,0,0,1]), M2Z([1,1,-1,0]), M2Z([0,-1,1,1])], [0,2], [[(1, M2Z([1,0,0,1]), 0)], [(-1,M2Z([-1,-1,0,-1]),0)], [(1, M2Z([1,0,0,1]), 2)]], {(0,1): 0, (1,0): 1, (1,1): 2})

P1()

Return the Sage representation of $P^1(\mathbf{Z}/N\mathbf{Z})$.

EXAMPLES:

Sage

sage: from sage.modular.pollack_stevens.fund_domain import ManinRelations
sage: A = ManinRelations(11)
sage: A.P1()
The projective line over the integers modulo 11

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.fund_domain import ManinRelations
>>> A = ManinRelations(Integer(11))
>>> A.P1()
The projective line over the integers modulo 11

Sage Live

from sage.modular.pollack_stevens.fund_domain import ManinRelations
A = ManinRelations(11)
A.P1()

equivalent_index(A)

Return the index of the coset representative equivalent to A.

Here by equivalent we mean the unique coset representative whose bottom row is equivalent to the bottom row of A in $P^1(\mathbf{Z}/N\mathbf{Z})$.

INPUT:

• A – an element of $S{L}_2(\mathbf{Z})$

OUTPUT:

The unique integer j satisfying that the bottom row of self.reps(j) is equivalent to the bottom row of A.

EXAMPLES:

Sage

sage: from sage.modular.pollack_stevens.fund_domain import ManinRelations
sage: MR = ManinRelations(11)
sage: A = matrix(ZZ,2,2,[1,5,3,16])
sage: j = MR.equivalent_index(A); j
11
sage: MR.reps(11)
[ 1 -1]
[-1  2]
sage: MR.equivalent_rep(A)
[ 1 -1]
[-1  2]
sage: MR.P1().normalize(3,16)
(1, 9)

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.fund_domain import ManinRelations
>>> MR = ManinRelations(Integer(11))
>>> A = matrix(ZZ,Integer(2),Integer(2),[Integer(1),Integer(5),Integer(3),Integer(16)])
>>> j = MR.equivalent_index(A); j
11
>>> MR.reps(Integer(11))
[ 1 -1]
[-1  2]
>>> MR.equivalent_rep(A)
[ 1 -1]
[-1  2]
>>> MR.P1().normalize(Integer(3),Integer(16))
(1, 9)

Sage Live

from sage.modular.pollack_stevens.fund_domain import ManinRelations
MR = ManinRelations(11)
A = matrix(ZZ,2,2,[1,5,3,16])
j = MR.equivalent_index(A); j
MR.reps(11)
MR.equivalent_rep(A)
MR.P1().normalize(3,16)

equivalent_rep(A)

Return a coset representative that is equivalent to A modulo ${\mathrm{\Gamma }}_0(N)$.

INPUT:

• A – a matrix in $S{L}_2(\mathbf{Z})$

OUTPUT:

The unique generator congruent to A modulo ${\mathrm{\Gamma }}_0(N)$.

EXAMPLES:

Sage

sage: from sage.modular.pollack_stevens.fund_domain import ManinRelations
sage: A = matrix([[5,3],[38,23]])
sage: ManinRelations(60).equivalent_rep(A)
[-7 -3]
[26 11]

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.fund_domain import ManinRelations
>>> A = matrix([[Integer(5),Integer(3)],[Integer(38),Integer(23)]])
>>> ManinRelations(Integer(60)).equivalent_rep(A)
[-7 -3]
[26 11]

Sage Live

from sage.modular.pollack_stevens.fund_domain import ManinRelations
A = matrix([[5,3],[38,23]])
ManinRelations(60).equivalent_rep(A)

gen(n=0)

Return the $n$-th generator.

INPUT:

• n – integer (default: 0); which generator is desired

EXAMPLES:

Sage

sage: from sage.modular.pollack_stevens.fund_domain import ManinRelations
sage: A = ManinRelations(137)
sage: A.gen(17)
[-4 -1]
[ 9  2]

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.fund_domain import ManinRelations
>>> A = ManinRelations(Integer(137))
>>> A.gen(Integer(17))
[-4 -1]
[ 9  2]

Sage Live

from sage.modular.pollack_stevens.fund_domain import ManinRelations
A = ManinRelations(137)
A.gen(17)

gens()

Return the tuple of coset representatives chosen as generators.

EXAMPLES:

Sage

sage: from sage.modular.pollack_stevens.fund_domain import ManinRelations
sage: A = ManinRelations(11)
sage: A.gens()
(
[1 0]  [ 0 -1]  [-1 -1]
[0 1], [ 1  3], [ 3  2]
)

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.fund_domain import ManinRelations
>>> A = ManinRelations(Integer(11))
>>> A.gens()
(
[1 0]  [ 0 -1]  [-1 -1]
[0 1], [ 1  3], [ 3  2]
)

Sage Live

from sage.modular.pollack_stevens.fund_domain import ManinRelations
A = ManinRelations(11)
A.gens()

indices(n=None)

Return the $n$-th index of the coset representatives which were chosen as our generators.

In particular, the divisors associated to these coset representatives generate all divisors over $\mathbf{Z}[{\mathrm{\Gamma }}_0(N)]$, and thus a modular symbol is uniquely determined by its values on these divisors.

INPUT:

• n – integer (default: None)

OUTPUT:

The n-th index of the generating set in self.reps() or all indices if n is None.

EXAMPLES:

Sage

sage: from sage.modular.pollack_stevens.fund_domain import ManinRelations
sage: A = ManinRelations(11)
sage: A.indices()
[0, 2, 3]

sage: A.indices(2)
3

sage: A = ManinRelations(13)
sage: A.indices()
[0, 2, 3, 4, 5]

sage: A = ManinRelations(101)
sage: A.indices()
[0, 2, 3, 4, 5, 6, 8, 9, 11, 13, 14, 16, 17, 19, 20, 23, 24, 26, 28]

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.fund_domain import ManinRelations
>>> A = ManinRelations(Integer(11))
>>> A.indices()
[0, 2, 3]

>>> A.indices(Integer(2))
3

>>> A = ManinRelations(Integer(13))
>>> A.indices()
[0, 2, 3, 4, 5]

>>> A = ManinRelations(Integer(101))
>>> A.indices()
[0, 2, 3, 4, 5, 6, 8, 9, 11, 13, 14, 16, 17, 19, 20, 23, 24, 26, 28]

Sage Live

from sage.modular.pollack_stevens.fund_domain import ManinRelations
A = ManinRelations(11)
A.indices()
A.indices(2)
A = ManinRelations(13)
A.indices()
A = ManinRelations(101)
A.indices()

level()

Return the level $N$ of ${\mathrm{\Gamma }}_0(N)$ that we work with.

OUTPUT:

The integer $N$ of the group ${\mathrm{\Gamma }}_0(N)$ for which the Manin Relations are being computed.

EXAMPLES:

Sage

sage: from sage.modular.pollack_stevens.fund_domain import ManinRelations
sage: A = ManinRelations(11)
sage: A.level()
11

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.fund_domain import ManinRelations
>>> A = ManinRelations(Integer(11))
>>> A.level()
11

Sage Live

from sage.modular.pollack_stevens.fund_domain import ManinRelations
A = ManinRelations(11)
A.level()

ngens()

Return the number of generators.

OUTPUT:

The number of coset representatives from which a modular symbol’s value on any coset can be derived.

EXAMPLES:

Sage

sage: from sage.modular.pollack_stevens.fund_domain import ManinRelations
sage: A = ManinRelations(1137)
sage: A.ngens()
255

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.fund_domain import ManinRelations
>>> A = ManinRelations(Integer(1137))
>>> A.ngens()
255

Sage Live

from sage.modular.pollack_stevens.fund_domain import ManinRelations
A = ManinRelations(1137)
A.ngens()

relations(A=None)

Express the divisor attached to the coset representative of A in terms of our chosen generators.

INPUT:

• A – None, an integer, or a coset representative (default: None)

OUTPUT:

A $\mathbf{Z}[{\mathrm{\Gamma }}_0(N)]$-relation expressing the divisor attached to A in terms of the generating set. The relation is given as a list of triples (d, B, i) such that the divisor attached to A is the sum of d times the divisor attached to B^{-1} * self.reps(i).

If A is an integer, then return this data for the A-th coset representative.

If A is None, then return this data in a list for all coset representatives.

Note

These relations allow us to recover the value of a modular symbol on any coset representative in terms of its values on our generating set.

EXAMPLES:

Sage

sage: from sage.modular.pollack_stevens.fund_domain import ManinRelations
sage: MR = ManinRelations(11)
sage: MR.indices()
[0, 2, 3]
sage: MR.relations(0)
[(1, [1 0]
[0 1], 0)]
sage: MR.relations(2)
[(1, [1 0]
[0 1], 2)]
sage: MR.relations(3)
[(1, [1 0]
[0 1], 3)]

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.fund_domain import ManinRelations
>>> MR = ManinRelations(Integer(11))
>>> MR.indices()
[0, 2, 3]
>>> MR.relations(Integer(0))
[(1, [1 0]
[0 1], 0)]
>>> MR.relations(Integer(2))
[(1, [1 0]
[0 1], 2)]
>>> MR.relations(Integer(3))
[(1, [1 0]
[0 1], 3)]

Sage Live

from sage.modular.pollack_stevens.fund_domain import ManinRelations
MR = ManinRelations(11)
MR.indices()
MR.relations(0)
MR.relations(2)
MR.relations(3)

The fourth coset representative can be expressed through the second coset representative:

Sage

sage: MR.reps(4)
[-1 -2]
[ 2  3]
sage: d, B, i = MR.relations(4)[0]
sage: P = B.inverse()*MR.reps(i); P
[ 2 -1]
[-3  2]
sage: d # the above corresponds to minus the divisor of A.reps(4) since d is -1
-1

Python

>>> from sage.all import *
>>> MR.reps(Integer(4))
[-1 -2]
[ 2  3]
>>> d, B, i = MR.relations(Integer(4))[Integer(0)]
>>> P = B.inverse()*MR.reps(i); P
[ 2 -1]
[-3  2]
>>> d # the above corresponds to minus the divisor of A.reps(4) since d is -1
-1

Sage Live

MR.reps(4)
d, B, i = MR.relations(4)[0]
P = B.inverse()*MR.reps(i); P
d # the above corresponds to minus the divisor of A.reps(4) since d is -1

The sixth coset representative can be expressed as the sum of the second and the third:

Sage

sage: MR.reps(6)
[ 0 -1]
[ 1  2]
sage: MR.relations(6)
[(1, [1 0]
[0 1], 2), (1, [1 0]
[0 1], 3)]
sage: MR.reps(2), MR.reps(3) # MR.reps(6) is the sum of these divisors
(
[ 0 -1]  [-1 -1]
[ 1  3], [ 3  2]
)

Python

>>> from sage.all import *
>>> MR.reps(Integer(6))
[ 0 -1]
[ 1  2]
>>> MR.relations(Integer(6))
[(1, [1 0]
[0 1], 2), (1, [1 0]
[0 1], 3)]
>>> MR.reps(Integer(2)), MR.reps(Integer(3)) # MR.reps(6) is the sum of these divisors
(
[ 0 -1]  [-1 -1]
[ 1  3], [ 3  2]
)

Sage Live

MR.reps(6)
MR.relations(6)
MR.reps(2), MR.reps(3) # MR.reps(6) is the sum of these divisors

reps(n=None)

Return the n-th coset representative associated with our fundamental domain.

INPUT:

• n – integer (default: None)

OUTPUT:

The n-th coset representative or all coset representatives if n is None.

EXAMPLES:

Sage

sage: from sage.modular.pollack_stevens.fund_domain import ManinRelations
sage: A = ManinRelations(11)
sage: A.reps(0)
[1 0]
[0 1]
sage: A.reps(1)
[ 1  1]
[-1  0]
sage: A.reps(2)
[ 0 -1]
[ 1  3]
sage: A.reps()
[
[1 0]  [ 1  1]  [ 0 -1]  [-1 -1]  [-1 -2]  [-2 -1]  [ 0 -1]  [ 1  0]
[0 1], [-1  0], [ 1  3], [ 3  2], [ 2  3], [ 3  1], [ 1  2], [-2  1],

[ 0 -1]  [ 1  0]  [-1 -1]  [ 1 -1]
[ 1  1], [-1  1], [ 2  1], [-1  2]
]

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.fund_domain import ManinRelations
>>> A = ManinRelations(Integer(11))
>>> A.reps(Integer(0))
[1 0]
[0 1]
>>> A.reps(Integer(1))
[ 1  1]
[-1  0]
>>> A.reps(Integer(2))
[ 0 -1]
[ 1  3]
>>> A.reps()
[
[1 0]  [ 1  1]  [ 0 -1]  [-1 -1]  [-1 -2]  [-2 -1]  [ 0 -1]  [ 1  0]
[0 1], [-1  0], [ 1  3], [ 3  2], [ 2  3], [ 3  1], [ 1  2], [-2  1],
<BLANKLINE>
[ 0 -1]  [ 1  0]  [-1 -1]  [ 1 -1]
[ 1  1], [-1  1], [ 2  1], [-1  2]
]

Sage Live

from sage.modular.pollack_stevens.fund_domain import ManinRelations
A = ManinRelations(11)
A.reps(0)
A.reps(1)
A.reps(2)
A.reps()

sage.modular.pollack_stevens.fund_domain.basic_hecke_matrix(a, l)

Return the $2\times 2$ matrix with entries [1, a, 0, l] if a<l and [l, 0, 0, 1] if a>=l.

INPUT:

• a – integer or Infinity

• l – a prime

OUTPUT: a $2\times 2$ matrix of determinant l

EXAMPLES:

Sage

sage: from sage.modular.pollack_stevens.fund_domain import basic_hecke_matrix
sage: basic_hecke_matrix(0, 7)
[1 0]
[0 7]
sage: basic_hecke_matrix(5, 7)
[1 5]
[0 7]
sage: basic_hecke_matrix(7, 7)
[7 0]
[0 1]
sage: basic_hecke_matrix(19, 7)
[7 0]
[0 1]
sage: basic_hecke_matrix(infinity, 7)
[7 0]
[0 1]

Python

>>> from sage.all import *
>>> from sage.modular.pollack_stevens.fund_domain import basic_hecke_matrix
>>> basic_hecke_matrix(Integer(0), Integer(7))
[1 0]
[0 7]
>>> basic_hecke_matrix(Integer(5), Integer(7))
[1 5]
[0 7]
>>> basic_hecke_matrix(Integer(7), Integer(7))
[7 0]
[0 1]
>>> basic_hecke_matrix(Integer(19), Integer(7))
[7 0]
[0 1]
>>> basic_hecke_matrix(infinity, Integer(7))
[7 0]
[0 1]

Sage Live

from sage.modular.pollack_stevens.fund_domain import basic_hecke_matrix
basic_hecke_matrix(0, 7)
basic_hecke_matrix(5, 7)
basic_hecke_matrix(7, 7)
basic_hecke_matrix(19, 7)
basic_hecke_matrix(infinity, 7)

Make live