41.10 Relation matrices for ambient modular symbols spaces

Module: sage.modular.modsym.relation_matrix

Relation matrices for ambient modular symbols spaces.

Module-level Functions

T_relation_matrix_wtk_g0( syms, mod, field, weight, sparse)

Compute a matrix whose echelon form gives the quotient by 3-term T relations.

Input:

syms
- ManinSymbols
mod
- list that gives quotient modulo some two-term relations, i.e., the S relations, and if sign is nonzero, the I relations.
field
- base_ring
weight
- int

Output: A sparse matrix whose rows correspond to the reduction of the T relations modulo the S and I relations.

compute_presentation( syms, sign, field, weight, [sparse=None])

Compute the presentation for self, as a quotient of Manin symbols modulo relations.

Input:

syms
- manin_symbols.ManinSymbols
sign
- integer (-1, 0, 1)
field
- a field
weight
- integer weight

Output:
- sparse matrix whose rows give each generator in terms of a basis for the quotient
- list of integers that give the basis for the quotient
- mod: list where mod[i]=(j,s) means that x_i = s*x_j modulo the 2-term S (and possibly I) relations.

ALGORITHM:

  1. Let $ S = [0,-1; 1,0], T = [0,-1; 1,-1]$ , and $ I = [-1,0; 0,1]$ .

  2. Let $ x_0,\ldots, x_{n-1}$ by a list of all non-equivalent Manin symbols.

  3. Form quotient by 2-term S and (possibly) I relations.

  4. Create a sparse matrix $ A$ with $ m$ columns, whose rows encode the relations

    $\displaystyle [x_i] + [x_i T] + [x_i T^2] = 0.
$

    There are about n such rows. The number of nonzero entries per row is at most 3*(k-1). Note that we must include rows for *all* i, since even if $ [x_i] = [x_j]$ , it need not be the case that $ [x_i T] = [x_j T]$ , since $ S$ and $ T$ do not commute. However, in many cases we have an a priori formula for the dimension of the quotient by all these relations, so we can omit many relations and just check that there are enough at the end--if there aren't, we add in more.

  5. Compute the reduced row echelon form of $ A$ using sparse Gaussian elimination.

  6. Use what we've done above to read off a sparse matrix R that uniquely expresses each of the n Manin symbols in terms of a subset of Manin symbols, modulo the relations. This subset of Manin symbols is a basis for the quotient by the relations.

gens_to_basis_matrix( syms, relation_matrix, mod, field, sparse)

Compute echelon form of 3-term relation matrix, and read off each generator in terms of basis.

Input:

syms
- a ManinSymbols object
relation_matrix
- as output by __compute_T_relation_matrix(self, mod)
mod
- quotient of modular symbols modulo the 2-term S (and possibly I) relations
field
- base field
sparse
- (bool): whether or not matrix should be sparse

Output:
matrix
- a matrix whose ith row expresses the Manin symbol generators in terms of a basis of Manin symbols (modulo the S, (possibly I,) and T rels) Note that the entries of the matrix need not be integers.

list - integers i, such that the Manin symbols x_i are a basis.

modI_relations( syms, sign)

Compute quotient of Manin symbols by the I relations.

Input:

syms
- ManinSymbols
sign
- int (either -1, 0, or 1)
Output:
rels
- set of pairs of pairs (j, s), where if mod[i] = (j,s), then x_i = s*x_j (mod S relations)

WARNING: We quotient by the involution eta((u,v)) = (-u,v), which has the opposite sign as the involution in Merel's Springer LNM 1585 paper! Thus our +1 eigenspace is his -1 eigenspace, etc. We do this for consistency with MAGMA.

modS_relations( syms)

Compute quotient of Manin symbols by the S relations.

Here S is the 2x2 matrix [0, -1; 1, 0].

Input:

syms
- manin_symbols.ManinSymbols

Output:
rels
- set of pairs of pairs (j, s), where if mod[i] = (j,s), then x_i = s*x_j (mod S relations)

sage: from sage.modular.modsym.manin_symbols import ManinSymbolList_gamma0
sage: from sage.modular.modsym.relation_matrix import modS_relations

sage: syms = ManinSymbolList_gamma0(2, 4); syms
Manin Symbol List of weight 4 for Gamma0(2)
sage: modS_relations(syms)
set([((3, -1), (4, 1)), ((5, -1), (5, 1)), ((1, 1), (6, 1)), ((0, 1), (7,
1)), ((3, 1), (4, -1)), ((2, 1), (8, 1))])

sage: syms = ManinSymbolList_gamma0(7, 2); syms
Manin Symbol List of weight 2 for Gamma0(7)
sage: modS_relations(syms)
set([((3, 1), (4, 1)), ((2, 1), (7, 1)), ((5, 1), (6, 1)), ((0, 1), (1,
1))])

Next we do an example with Gamma1:

sage: from sage.modular.modsym.manin_symbols import ManinSymbolList_gamma1
sage: syms = ManinSymbolList_gamma1(3,2); syms
Manin Symbol List of weight 2 for Gamma1(3)
sage: modS_relations(syms)
set([((3, 1), (6, 1)), ((0, 1), (5, 1)), ((0, 1), (2, 1)), ((3, 1), (4,
1)), ((6, 1), (7, 1)), ((1, 1), (2, 1)), ((1, 1), (5, 1)), ((4, 1), (7,
1))])

relation_matrix_wtk_g0( syms, sign, field, weight, sparse)

sparse_2term_quotient( rels, n, F)

Performs Sparse Gauss elimination on a matrix all of whose columns have at most 2 nonzero entries. We use an obvious algorithm, whichs runs fast enough. (Typically making the list of relations takes more time than computing this quotient.) This algorithm is more subtle than just ``identify symbols in pairs'', since complicated relations can cause generators to surprisingly equal 0.

Input:

rels
- set of pairs ((i,s), (j,t)). The pair represents the relation s*x_i + t*x_j = 0, where the i, j must be Python int's.
n
- int, the x_i are x_0, ..., x_n-1.
F
- base field

Output:
mod
- list such that mod[i] = (j,s), which means that x_i is equivalent to s*x_j, where the x_j are a basis for the quotient.

We quotient out by the relations

$\displaystyle 3*x0 - x1 = 0,\qquad x1 + x3 = 0,\qquad x2 + x3 = 0,\qquad x4 - x5 = 0
$

to get
sage: v = [((int(0),3), (int(1),-1)), ((int(1),1), (int(3),1)), ((int(2),1),(int(3),1)), ((int(4),1),(int(5),-1))]
sage: rels = set(v)
sage: n = 6
sage: from sage.modular.modsym.relation_matrix import sparse_2term_quotient
sage: sparse_2term_quotient(rels, n, QQ)
[(3, -1/3), (3, -1), (3, -1), (3, 1), (5, 1), (5, 1)]

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