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 fromself.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 ofA*B
andA^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 matrixMR.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
tor2
.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 inself.reps()
where each value is a list of matrices, and the entries ofD
satisfy:\[D[h][j] = \gamma_{hj} * \gamma_a\]INPUT:
l
– a primegen
– 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 primegen
– 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 ofA*B
andA^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 matrixMR.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
inself.reps_with_two_torsion()
.INPUT:
A
– a matrix inself.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
inself.reps_with_two_torsion()
.INPUT:
A
– a matrix inself.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
andr2
andr2
andr1
, 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 inreps
which are generatorsrels
– a list of list of triples(d, A, i)
, one for each coset representative ofreps
which describes how to express the elements ofreps
in terms of generators specified byindices
. Seerelations()
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 ofreps[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 ofself.reps(j)
is equivalent to the bottom row ofA
.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 inself.reps()
or all indices ifn
isNone
.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 ofd
times the divisor attached toB^{-1} * self.reps(i)
.If
A
is an integer, then return this data for theA
-th coset representative.If
A
isNone
, 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 ifn
isNone
.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]
ifa<l
and[l, 0, 0, 1]
ifa>=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]