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# Relation matrices for ambient modular symbols spaces

This file contains functions that are used by the various ambient modular symbols classes to compute presentations of spaces in terms of generators and relations, using the standard methods based on Manin symbols.

`sage.modular.modsym.relation_matrix.``T_relation_matrix_wtk_g0`(syms, mod, field, sparse)

  Compute a matrix whose echelon form gives the quotient by 3-term T relations. Despite the name, this is used for all modular symbols spaces (including those with character and those for \(\Gamma_1\) and \(\Gamma_H\) groups), not just \(\Gamma_0\).

  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
  - `sparse` - (True or False) whether to use sparse rather than dense linear algebra

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

  EXAMPLE:

      sage: from sage.modular.modsym.relation_matrix import *
      sage: L = sage.modular.modsym.manin_symbols.ManinSymbolList_gamma_h(GammaH(36, [17,19]), 2)
      sage: modS = sparse_2term_quotient(modS_relations(L), 216, QQ)
      sage: T_relation_matrix_wtk_g0(L, modS, QQ, False)
      72 x 216 dense matrix over Rational Field (use the '.str()' method to see the entries)
      sage: T_relation_matrix_wtk_g0(L, modS, GF(17), True)
      72 x 216 sparse matrix over Finite Field of size 17 (use the '.str()' method to see the entries)

`sage.modular.modsym.relation_matrix.``compute_presentation`(syms, sign, field, 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

  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

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

     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.

  EXAMPLE:

      sage: L = sage.modular.modsym.manin_symbols.ManinSymbolList_gamma0(8,2)
      sage: sage.modular.modsym.relation_matrix.compute_presentation(L, 1, GF(9,'a'), True)
      (
      [2 0 0]
      [1 0 0]
      [0 0 0]
      [0 2 0]
      [0 0 0]
      [0 0 2]
      [0 0 0]
      [0 2 0]
      [0 0 0]
      [0 1 0]
      [0 1 0]
      [0 0 1], [1, 9, 11], [(1, 2), (1, 1), (0, 0), (9, 2), (0, 0), (11, 2), (0, 0), (9, 2), (0, 0), (9, 1), (9, 1), (11, 1)]
      )

`sage.modular.modsym.relation_matrix.``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.

  EXAMPLE:

      sage: from sage.modular.modsym.relation_matrix import *
      sage: L = sage.modular.modsym.manin_symbols.ManinSymbolList_gamma1(4, 3)
      sage: modS = sparse_2term_quotient(modS_relations(L), 24, GF(3))
      sage: gens_to_basis_matrix(L, T_relation_matrix_wtk_g0(L, modS, GF(3), 24), modS, GF(3), True)
      (24 x 2 sparse matrix over Finite Field of size 3, [13, 23])

`sage.modular.modsym.relation_matrix.``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)

  EXAMPLE:

      sage: L = sage.modular.modsym.manin_symbols.ManinSymbolList_gamma1(4, 3)
      sage: sage.modular.modsym.relation_matrix.modI_relations(L, 1)
      {((0, 1), (0, -1)),
       ((1, 1), (1, -1)),
       ((2, 1), (8, -1)),
       ((3, 1), (9, -1)),
       ((4, 1), (10, -1)),
       ((5, 1), (11, -1)),
       ((6, 1), (6, -1)),
       ((7, 1), (7, -1)),
       ((8, 1), (2, -1)),
       ((9, 1), (3, -1)),
       ((10, 1), (4, -1)),
       ((11, 1), (5, -1)),
       ((12, 1), (12, 1)),
       ((13, 1), (13, 1)),
       ((14, 1), (20, 1)),
       ((15, 1), (21, 1)),
       ((16, 1), (22, 1)),
       ((17, 1), (23, 1)),
       ((18, 1), (18, 1)),
       ((19, 1), (19, 1)),
       ((20, 1), (14, 1)),
       ((21, 1), (15, 1)),
       ((22, 1), (16, 1)),
       ((23, 1), (17, 1))}

  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.

`sage.modular.modsym.relation_matrix.``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)

  EXAMPLES:

      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)
      {((0, 1), (7, 1)),
       ((1, 1), (6, 1)),
       ((2, 1), (8, 1)),
       ((3, -1), (4, 1)),
       ((3, 1), (4, -1)),
       ((5, -1), (5, 1))}

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

`sage.modular.modsym.relation_matrix.``relation_matrix_wtk_g0`(syms, sign, field, sparse)

  Compute the matrix of relations. Despite the name, this is used for all spaces (not just for Gamma0). For a description of the algorithm, see the docstring for `compute_presentation`.

  INPUT:

  - `syms`: sage.modular.modsym.manin_symbols.ManinSymbolList object
  - `sign`: integer (0, 1 or -1)
  - `field`: the base field (non-field base rings not supported at present)
  - `sparse`: (True or False) whether to use sparse arithmetic.

  Note that ManinSymbolList objects already have a specific weight, so there is no need for an extra `weight` parameter.

  OUTPUT: a pair (R, mod) where

  - R is a matrix as output by `T_relation_matrix_wtk_g0`
  - mod is a set of 2-term relations as output by `sparse_2term_quotient`

  EXAMPLE:

      sage: L =  sage.modular.modsym.manin_symbols.ManinSymbolList_gamma0(8,2)
      sage: A = sage.modular.modsym.relation_matrix.relation_matrix_wtk_g0(L, 0, GF(2), True); A
      (
      [0 0 0 0 0 0 0 0 1 0 0 0]
      [0 0 0 0 0 0 0 0 1 1 1 0]
      [0 0 0 0 0 0 1 0 0 1 1 0]
      [0 0 0 0 0 0 1 0 0 0 0 0], [(1, 1), (1, 1), (8, 1), (10, 1), (6, 1), (11, 1), (6, 1), (9, 1), (8, 1), (9, 1), (10, 1), (11, 1)]
      )
      sage: A[0].is_sparse()
      True

`sage.modular.modsym.relation_matrix.``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, which 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.

  EXAMPLE: We quotient out by the relations

  \[3*x0 - x1 = 0,\qquad x1 + x3 = 0,\qquad x2 + x3 = 0,\qquad x4 - x5 = 0\]

  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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