Manin Relations for overconvergent modular symbols

Code to create the Manin Relations class, which solves the “Manin relations”. That is, a description of \(Div^0(P^1(\QQ))\) as a \(\ZZ[\Gamma_0(N)]\)-module in terms of generators and relations is found. The method used is geometric, constructing a nice fundamental domain for \(\Gamma_0(N)\) 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: anything that can be converted into a \(2 \times 2\) matrix.

EXAMPLES:

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]

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

Bases: sage.modular.pollack_stevens.fund_domain.PollackStevensModularDomain

This class gives a description of \(Div^0(P^1(\QQ))\) as a \(\ZZ[\Gamma_0(N)]\)-module.

INPUT:

• N – a positive integer, the level of \(\Gamma_0(N)\) to work with

EXAMPLES:

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

Large values of N are not supported:

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

fd_boundary(C)

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

Here the fundamental domain is for \(\Gamma_0(N)\). (In the case when \(\Gamma_0(N)\) 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 – a 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 \(\Gamma_0(N)\) (or nearly so in the case of 3-torsion).

EXAMPLES:

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]
]

form_list_of_cusps()

Return the intersection of a fundamental domain for \(\Gamma_0(N)\) 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 \(\Gamma_0(N)\) 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 \(\Gamma_0(N)\) with the real axis.

EXAMPLES:

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]

indices_with_three_torsion()

A list of indices of coset representatives whose associated unimodular path contains a point fixed by a \(\Gamma_0(N)\) element of order 3 in the ideal triangle directly below that path (the order is computed in \(PSL_2(\ZZ)\)).

EXAMPLES:

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]

The corresponding matrix of order three:

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

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

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

indices_with_two_torsion()

Return the indices of coset representatives whose associated unimodular path contains a point fixed by a \(\Gamma_0(N)\) element of order 2 (where the order is computed in \(PSL_2(\ZZ)\)).

OUTPUT:

A list of integers.

EXAMPLES:

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]
)

The corresponding matrix of order 2:

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

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

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

prep_hecke_on_gen(l, gen, modulus=None)

This function does some precomputations needed to compute \(T_l\).

In particular, if \(\phi\) is a modular symbol and \(D_m\) is the divisor associated to the generator gen, to compute \((\phi|T_{l})(D_m)\) one needs to compute \(\phi(\gamma_a D_m)|\gamma_a\) 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 \(SL_2\) matrix, we then write \(M=\gamma h\) where \(\gamma\) is in \(\Gamma_0(N)\) and \(h\) is one of our chosen coset representatives. Then \(\phi([M]) = \phi([h]) | \gamma^{-1}\). Thus, one has

\[(\phi | \gamma_a)(D_m) = \sum_h \sum_j \phi([h]) | \gamma_{hj}^{-1} \cdot \gamma_a\]

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:

\[D[h][j] = \gamma_{hj} * \gamma_a\]

INPUT:

• l – a prime
• gen – a generator

OUTPUT:

A list of lists (see above).

EXAMPLES:

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]]

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: 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

reps_with_three_torsion()

A list of coset representatives whose associated unimodular path contains a point fixed by a \(\Gamma_0(N)\) element of order 3 in the ideal triangle directly below that path (the order is computed in \(PSL_2(\ZZ)\)).

EXAMPLES:

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]

The corresponding matrix of order three:

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

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

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

reps_with_two_torsion()

The coset representatives whose associated unimodular path contains a point fixed by a \(\Gamma_0(N)\) element of order 2 (where the order is computed in \(PSL_2(\ZZ)\)).

OUTPUT:

A list of matrices.

EXAMPLES:

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]

The corresponding matrix of order 2:

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

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

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

three_torsion_matrix(A)

Return the matrix of order two in \(\Gamma_0(N)\) which corresponds to an A in self.reps_with_two_torsion().

INPUT:

• A – a matrix in self.reps_with_two_torsion()

EXAMPLES:

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

The corresponding matrix of order 3:

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

two_torsion_matrix(A)

Return the matrix of order two in \(\Gamma_0(N)\) which corresponds to an A in self.reps_with_two_torsion().

INPUT:

• A – a matrix in self.reps_with_two_torsion()

EXAMPLES:

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

The corresponding matrix of order 2:

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

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: 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]
)

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

Bases: sage.structure.sage_object.SageObject

The domain of a modular symbol.

INPUT:

• N – a positive integer, the level of the congruence subgroup \(\Gamma_0(N)\)
• reps – a list of \(2 \times 2\) matrices, the coset representatives of \(Div^0(P^1(\QQ))\)
• indices – a list of integers; indices of elements in reps which are generators
• rels – a 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 – a dictionary which maps normalized coordinates on \(P^1(\ZZ/N\ZZ)\) to an integer such that a matrix whose bottom row is equivalent to \([a:b]\) in \(P^1(\ZZ/N\ZZ)\) is in the coset of reps[equiv_ind[(a,b)]]

EXAMPLES:

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

P1()

Return the Sage representation of \(P^1(\ZZ/N\ZZ)\).

EXAMPLES:

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

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 \(P^1(\ZZ/N\ZZ)\).

INPUT:

• A – an element of \(SL_2(\ZZ)\)

OUTPUT:

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

EXAMPLES:

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)

equivalent_rep(A)

Return a coset representative that is equivalent to A modulo \(\Gamma_0(N)\).

INPUT:

• A – a matrix in \(SL_2(\ZZ)\)

OUTPUT:

The unique generator congruent to A modulo \(\Gamma_0(N)\).

EXAMPLES:

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]

gen(n=0)

Return the n-th generator.

INPUT:

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

EXAMPLES:

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

gens()

Return the list of coset representatives chosen as generators.

EXAMPLES:

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]
]

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 \(\ZZ[\Gamma_0(N)]\), 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: 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]

level()

Return the level \(N\) of \(\Gamma_0(N)\) that we work with.

OUTPUT:

The integer \(N\) of the group \(\Gamma_0(N)\) for which the Manin Relations are being computed.

EXAMPLES:

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

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

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 \(\ZZ[\Gamma_0(N)]\)-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: 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)]

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

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

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

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]
)

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: 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]
]

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:

• \(a\) – an integer or Infinity
• l – a prime

OUTPUT:

A \(2 \times 2\) matrix of determinant l

EXAMPLES:

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]