Manin map

Represents maps from a set of right coset representatives to a coefficient module.

This is a basic building block for implementing modular symbols, and provides basic arithmetic and right action of matrices.

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: phi = E.pollack_stevens_modular_symbol()
sage: phi
Modular symbol of level 11 with values in Sym^0 Q^2
sage: phi.values()
[-1/5, 1, 0]

sage: from sage.modular.pollack_stevens.manin_map import ManinMap, M2Z
sage: from sage.modular.pollack_stevens.fund_domain import ManinRelations
sage: D = OverconvergentDistributions(0, 11, 10)
sage: MR = ManinRelations(11)
sage: data  = {M2Z([1,0,0,1]):D([1,2]), M2Z([0,-1,1,3]):D([3,5]), M2Z([-1,-1,3,2]):D([1,1])}
sage: f = ManinMap(D, MR, data)
sage: f(M2Z([1,0,0,1]))
(1 + O(11^2), 2 + O(11))

sage: S = Symk(0,QQ)
sage: MR = ManinRelations(37)
sage: data  = {M2Z([-2,-3,5,7]): S(0), M2Z([1,0,0,1]): S(0), M2Z([-1,-2,3,5]): S(0), M2Z([-1,-4,2,7]): S(1), M2Z([0,-1,1,4]): S(1), M2Z([-3,-1,7,2]): S(-1), M2Z([-2,-3,3,4]): S(0), M2Z([-4,-3,7,5]): S(0), M2Z([-1,-1,4,3]): S(0)}
sage: f = ManinMap(S,MR,data)
sage: f(M2Z([2,3,4,5]))
1

class sage.modular.pollack_stevens.manin_map.ManinMap(codomain, manin_relations, defining_data, check=True)

Bases: object

Map from a set of right coset representatives of \(\Gamma_0(N)\) in \(SL_2(\ZZ)\) to a coefficient module that satisfies the Manin relations.

INPUT:

• codomain – coefficient module

• manin_relations – a sage.modular.pollack_stevens.fund_domain.ManinRelations object

• defining_data – a dictionary whose keys are a superset of manin_relations.gens() and a subset of manin_relations.reps(), and whose values are in the codomain.

• check – do numerous (slow) checks and transformations to ensure that the input data is perfect.

EXAMPLES:

sage: from sage.modular.pollack_stevens.manin_map import M2Z, ManinMap
sage: D = OverconvergentDistributions(0, 11, 10)
sage: manin = sage.modular.pollack_stevens.fund_domain.ManinRelations(11)
sage: data  = {M2Z([1,0,0,1]):D([1,2]), M2Z([0,-1,1,3]):D([3,5]), M2Z([-1,-1,3,2]):D([1,1])}
sage: f = ManinMap(D, manin, data); f # indirect doctest
Map from the set of right cosets of Gamma0(11) in SL_2(Z) to Space of 11-adic distributions with k=0 action and precision cap 10
sage: f(M2Z([1,0,0,1]))
(1 + O(11^2), 2 + O(11))

apply(f, codomain=None, to_moments=False)

Return Manin map given by \(x \mapsto f(self(x))\), where \(f\) is anything that can be called with elements of the coefficient module.

This might be used to normalize, reduce modulo a prime, change base ring, etc.

INPUT:

• f – anything that can be called with elements of the coefficient module

• codomain – (default: None) the codomain of the return map

• to_moments – (default: False) if True, will apply f to each of the moments instead

EXAMPLES:

sage: from sage.modular.pollack_stevens.manin_map import M2Z, ManinMap
sage: from sage.modular.pollack_stevens.fund_domain import ManinRelations
sage: S = Symk(0,QQ)
sage: MR = ManinRelations(37)
sage: data  = {M2Z([-2,-3,5,7]): S(0), M2Z([1,0,0,1]): S(0), M2Z([-1,-2,3,5]): S(0), M2Z([-1,-4,2,7]): S(1), M2Z([0,-1,1,4]): S(1), M2Z([-3,-1,7,2]): S(-1), M2Z([-2,-3,3,4]): S(0), M2Z([-4,-3,7,5]): S(0), M2Z([-1,-1,4,3]): S(0)}
sage: f = ManinMap(S,MR,data)
sage: list(f.apply(lambda t:2*t))
[0, 2, 0, 0, 0, -2, 2, 0, 0]

compute_full_data()

Compute the values of self on all coset reps from its values on our generating set.

EXAMPLES:

sage: from sage.modular.pollack_stevens.manin_map import M2Z, ManinMap
sage: from sage.modular.pollack_stevens.fund_domain import ManinRelations
sage: S = Symk(0,QQ)
sage: MR = ManinRelations(37); MR.gens()
[
[1 0]  [ 0 -1]  [-1 -1]  [-1 -2]  [-2 -3]  [-3 -1]  [-1 -4]  [-4 -3]
[0 1], [ 1  4], [ 4  3], [ 3  5], [ 5  7], [ 7  2], [ 2  7], [ 7  5],

[-2 -3]
[ 3  4]
]

sage: data  = {M2Z([-2,-3,5,7]): S(0), M2Z([1,0,0,1]): S(0), M2Z([-1,-2,3,5]): S(0), M2Z([-1,-4,2,7]): S(1), M2Z([0,-1,1,4]): S(1), M2Z([-3,-1,7,2]): S(-1), M2Z([-2,-3,3,4]): S(0), M2Z([-4,-3,7,5]): S(0), M2Z([-1,-1,4,3]): S(0)}
sage: f = ManinMap(S,MR,data)
sage: len(f._dict)
9
sage: f.compute_full_data()
sage: len(f._dict)
38

extend_codomain(new_codomain, check=True)

Extend the codomain of self to new_codomain. There must be a valid conversion operation from the old to the new codomain. This is most often used for extension of scalars from \(\QQ\) to \(\QQ_p\).

EXAMPLES:

sage: from sage.modular.pollack_stevens.manin_map import ManinMap, M2Z
sage: from sage.modular.pollack_stevens.fund_domain import ManinRelations
sage: S = Symk(0,QQ)
sage: MR = ManinRelations(37)
sage: data  = {M2Z([-2,-3,5,7]): S(0), M2Z([1,0,0,1]): S(0), M2Z([-1,-2,3,5]): S(0), M2Z([-1,-4,2,7]): S(1), M2Z([0,-1,1,4]): S(1), M2Z([-3,-1,7,2]): S(-1), M2Z([-2,-3,3,4]): S(0), M2Z([-4,-3,7,5]): S(0), M2Z([-1,-1,4,3]): S(0)}
sage: m = ManinMap(S, MR, data); m
Map from the set of right cosets of Gamma0(37) in SL_2(Z) to Sym^0 Q^2
sage: m.extend_codomain(Symk(0, Qp(11)))
Map from the set of right cosets of Gamma0(37) in SL_2(Z) to Sym^0 Q_11^2

hecke(ell, algorithm='prep')

Return the image of this Manin map under the Hecke operator \(T_{\ell}\).

INPUT:

• ell – a prime

• algorithm – a string, either ‘prep’ (default) or ‘naive’

OUTPUT:

• The image of this ManinMap under the Hecke operator \(T_{\ell}\)

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: phi = E.pollack_stevens_modular_symbol()
sage: phi.values()
[-1/5, 1, 0]
sage: phi.is_Tq_eigensymbol(7,7,10)
True
sage: phi.hecke(7).values()
[2/5, -2, 0]
sage: phi.Tq_eigenvalue(7,7,10)
-2

normalize()

Normalize every value of self – e.g., reduces each value’s \(j\)-th moment modulo \(p^{N-j}\)

EXAMPLES:

sage: from sage.modular.pollack_stevens.manin_map import M2Z, ManinMap
sage: D = OverconvergentDistributions(0, 11, 10)
sage: manin = sage.modular.pollack_stevens.fund_domain.ManinRelations(11)
sage: data  = {M2Z([1,0,0,1]):D([1,2]), M2Z([0,-1,1,3]):D([3,5]), M2Z([-1,-1,3,2]):D([1,1])}
sage: f = ManinMap(D, manin, data)
sage: f._dict[M2Z([1,0,0,1])]
(1 + O(11^2), 2 + O(11))
sage: g = f.normalize()
sage: g._dict[M2Z([1,0,0,1])]
(1 + O(11^2), 2 + O(11))

p_stabilize(p, alpha, V)

Return the \(p\)-stabilization of self to level \(N*p\) on which \(U_p\) acts by \(\alpha\).

INPUT:

• p – a prime.

• alpha – a \(U_p\)-eigenvalue.

• V – a space of modular symbols.

OUTPUT:

• The image of this ManinMap under the Hecke operator \(T_{\ell}\)

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: phi = E.pollack_stevens_modular_symbol()
sage: f = phi._map
sage: V = phi.parent()
sage: f.p_stabilize(5,1,V)
Map from the set of right cosets of Gamma0(11) in SL_2(Z) to Sym^0 Q^2

reduce_precision(M)

Reduce the precision of all the values of the Manin map.

INPUT:

• M – an integer, the new precision.

EXAMPLES:

sage: from sage.modular.pollack_stevens.manin_map import M2Z, ManinMap
sage: D = OverconvergentDistributions(0, 11, 10)
sage: manin = sage.modular.pollack_stevens.fund_domain.ManinRelations(11)
sage: data  = {M2Z([1,0,0,1]):D([1,2]), M2Z([0,-1,1,3]):D([3,5]), M2Z([-1,-1,3,2]):D([1,1])}
sage: f = ManinMap(D, manin, data)
sage: f._dict[M2Z([1,0,0,1])]
(1 + O(11^2), 2 + O(11))
sage: g = f.reduce_precision(1)
sage: g._dict[M2Z([1,0,0,1])]
1 + O(11^2)

specialize(*args)

Specialize all the values of the Manin map to a new coefficient module. Assumes that the codomain has a specialize method, and passes all its arguments to that method.

EXAMPLES:

sage: from sage.modular.pollack_stevens.manin_map import M2Z, ManinMap
sage: D = OverconvergentDistributions(0, 11, 10)
sage: manin = sage.modular.pollack_stevens.fund_domain.ManinRelations(11)
sage: data  = {M2Z([1,0,0,1]):D([1,2]), M2Z([0,-1,1,3]):D([3,5]), M2Z([-1,-1,3,2]):D([1,1])}
sage: f = ManinMap(D, manin, data)
sage: g = f.specialize()
sage: g._codomain
Sym^0 Z_11^2

sage.modular.pollack_stevens.manin_map.unimod_matrices_from_infty(r, s)

Return a list of matrices whose associated unimodular paths connect \(\infty\) to r/s.

INPUT:

• r, s – rational numbers

OUTPUT:

• a list of \(SL_2(\ZZ)\) matrices

EXAMPLES:

sage: v = sage.modular.pollack_stevens.manin_map.unimod_matrices_from_infty(19,23); v
[
[ 0  1]  [-1  0]  [-4  1]  [-5 -4]  [-19   5]
[-1  0], [-1 -1], [-5  1], [-6 -5], [-23   6]
]
sage: [a.det() for a in v]
[1, 1, 1, 1, 1]

sage: sage.modular.pollack_stevens.manin_map.unimod_matrices_from_infty(11,25)
[
[ 0  1]  [-1  0]  [-3  1]  [-4 -3]  [-11   4]
[-1  0], [-2 -1], [-7  2], [-9 -7], [-25   9]
]

ALGORITHM:

This is Manin’s continued fraction trick, which gives an expression \(\{\infty,r/s\} = \{\infty,0\} + ... + \{a,b\} + ... + \{*,r/s\}\), where each \(\{a,b\}\) is the image of \(\{0,\infty\}\) under a matrix in \(SL_2(\ZZ)\).

sage.modular.pollack_stevens.manin_map.unimod_matrices_to_infty(r, s)

Return a list of matrices whose associated unimodular paths connect \(0\) to r/s.

INPUT:

• r, s – rational numbers

OUTPUT:

• a list of matrices in \(SL_2(\ZZ)\)

EXAMPLES:

sage: v = sage.modular.pollack_stevens.manin_map.unimod_matrices_to_infty(19,23); v
[
[1 0]  [ 0  1]  [1 4]  [-4  5]  [ 5 19]
[0 1], [-1  1], [1 5], [-5  6], [ 6 23]
]
sage: [a.det() for a in v]
[1, 1, 1, 1, 1]

sage: sage.modular.pollack_stevens.manin_map.unimod_matrices_to_infty(11,25)
[
[1 0]  [ 0  1]  [1 3]  [-3  4]  [ 4 11]
[0 1], [-1  2], [2 7], [-7  9], [ 9 25]
]

ALGORITHM:

This is Manin’s continued fraction trick, which gives an expression \(\{0,r/s\} = \{0,\infty\} + ... + \{a,b\} + ... + \{*,r/s\}\), where each \(\{a,b\}\) is the image of \(\{0,\infty\}\) under a matrix in \(SL_2(\ZZ)\).