# Pollack-Stevens’ Modular Symbols Spaces¶

This module contains a class for spaces of modular symbols that use Glenn Stevens’ conventions, as explained in [PS2011].

There are two main differences between the modular symbols in this directory and the ones in sage.modular.modsym:

• There is a shift in the weight: weight $$k=0$$ here corresponds to weight $$k=2$$ there.

• There is a duality: these modular symbols are functions from $$\textrm{Div}^0(P^1(\QQ))$$ (cohomological objects), the others are formal linear combinations of $$\textrm{Div}^0(P^1(\QQ))$$ (homological objects).

EXAMPLES:

First we create the space of modular symbols of weight 0 ($$k=2$$) and level 11:

sage: M = PollackStevensModularSymbols(Gamma0(11), 0); M
Space of modular symbols for Congruence Subgroup Gamma0(11) with sign 0 and values in Sym^0 Q^2


One can also create a space of overconvergent modular symbols, by specifying a prime and a precision:

sage: M = PollackStevensModularSymbols(Gamma0(11), p = 5, prec_cap = 10, weight = 0); M
Space of overconvergent modular symbols for Congruence Subgroup Gamma0(11) with sign 0 and values in Space of 5-adic distributions with k=0 action and precision cap 10


Currently not much functionality is available on the whole space, and these spaces are mainly used as parents for the modular symbols. These can be constructed from the corresponding classical modular symbols (or even elliptic curves) as follows:

sage: A = ModularSymbols(13, sign=1, weight=4).decomposition()[0]
sage: A.is_cuspidal()
True
sage: from sage.modular.pollack_stevens.space import ps_modsym_from_simple_modsym_space
sage: f = ps_modsym_from_simple_modsym_space(A); f
Modular symbol of level 13 with values in Sym^2 Q^2
sage: f.values()
[(-13, 0, -1),
(247/2, 13/2, -6),
(39/2, 117/2, 42),
(-39/2, 39, 111/2),
(-247/2, -117, -209/2)]
sage: f.parent()
Space of modular symbols for Congruence Subgroup Gamma0(13) with sign 1 and values in Sym^2 Q^2

sage: E = EllipticCurve('37a1')
sage: phi = E.pollack_stevens_modular_symbol(); phi
Modular symbol of level 37 with values in Sym^0 Q^2
sage: phi.values()
[0, 1, 0, 0, 0, -1, 1, 0, 0]
sage: phi.parent()
Space of modular symbols for Congruence Subgroup Gamma0(37) with sign 0 and values in Sym^0 Q^2

class sage.modular.pollack_stevens.space.PollackStevensModularSymbols_factory

Create a space of Pollack-Stevens modular symbols.

INPUT:

• group – integer or congruence subgroup

• weight – integer $$\ge 0$$, or None

• sign – integer; -1, 0, 1

• base_ring – ring or None

• p – prime or None

• prec_cap – positive integer or None

• coefficients – the coefficient module (a special type of module, typically distributions), or None

If an explicit coefficient module is given, then the arguments weight, base_ring, prec_cap, and p are redundant and must be None. They are only relevant if coefficients is None, in which case the coefficient module is inferred from the other data.

Note

We emphasize that in the Pollack-Stevens notation, the weight is the usual weight minus 2, so a classical weight 2 modular form corresponds to a modular symbol of “weight 0”.

EXAMPLES:

sage: M = PollackStevensModularSymbols(Gamma0(7), weight=0, prec_cap = None); M
Space of modular symbols for Congruence Subgroup Gamma0(7) with sign 0 and values in Sym^0 Q^2


An example with an explicit coefficient module:

sage: D = OverconvergentDistributions(3, 7, prec_cap=10)
sage: M = PollackStevensModularSymbols(Gamma0(7), coefficients=D); M
Space of overconvergent modular symbols for Congruence Subgroup Gamma0(7) with sign 0 and values in Space of 7-adic distributions with k=3 action and precision cap 10

create_key(group, weight=None, sign=0, base_ring=None, p=None, prec_cap=None, coefficients=None)

Sanitize input.

EXAMPLES:

sage: D = OverconvergentDistributions(3, 7, prec_cap=10)
sage: M = PollackStevensModularSymbols(Gamma0(7), coefficients=D) # indirect doctest

create_object(version, key)

Create a space of modular symbols from key.

INPUT:

• version – the version of the object to create

• key – a tuple of parameters, as created by create_key()

EXAMPLES:

sage: D = OverconvergentDistributions(5, 7, 15)
sage: M = PollackStevensModularSymbols(Gamma0(7), coefficients=D) # indirect doctest
sage: M2 = PollackStevensModularSymbols(Gamma0(7), coefficients=D) # indirect doctest
sage: M is M2
True

class sage.modular.pollack_stevens.space.PollackStevensModularSymbolspace(group, coefficients, sign=0)

A class for spaces of modular symbols that use Glenn Stevens’ conventions. This class should not be instantiated directly by the user: this is handled by the factory object PollackStevensModularSymbols_factory.

INPUT:

• group – congruence subgroup

• coefficients – a coefficient module

• sign – (default: 0); 0, -1, or 1

EXAMPLES:

sage: D = OverconvergentDistributions(2, 11)
sage: M = PollackStevensModularSymbols(Gamma0(2), coefficients=D); M.sign()
0
sage: M = PollackStevensModularSymbols(Gamma0(2), coefficients=D, sign=-1); M.sign()
-1
sage: M = PollackStevensModularSymbols(Gamma0(2), coefficients=D, sign=1); M.sign()
1

change_ring(new_base_ring)

Change the base ring of this space to new_base_ring.

INPUT:

• new_base_ring – a ring

OUTPUT:

A space of modular symbols over the specified base.

EXAMPLES:

sage: from sage.modular.pollack_stevens.distributions import Symk
sage: D = Symk(4)
sage: M = PollackStevensModularSymbols(Gamma(6), coefficients=D); M
Space of modular symbols for Congruence Subgroup Gamma(6) with sign 0 and values in Sym^4 Q^2
sage: M.change_ring(Qp(5,8))
Space of modular symbols for Congruence Subgroup Gamma(6) with sign 0 and values in Sym^4 Q_5^2

coefficient_module()

Return the coefficient module of this space.

EXAMPLES:

sage: D = OverconvergentDistributions(2, 11)
sage: M = PollackStevensModularSymbols(Gamma0(2), coefficients=D)
sage: M.coefficient_module()
Space of 11-adic distributions with k=2 action and precision cap 20
sage: M.coefficient_module() is D
True

group()

Return the congruence subgroup of this space.

EXAMPLES:

sage: D = OverconvergentDistributions(2, 5)
sage: G = Gamma0(23)
sage: M = PollackStevensModularSymbols(G, coefficients=D)
sage: M.group()
Congruence Subgroup Gamma0(23)
sage: D = Symk(4)
sage: G = Gamma1(11)
sage: M = PollackStevensModularSymbols(G, coefficients=D)
sage: M.group()
Congruence Subgroup Gamma1(11)

level()

Return the level $$N$$, where this space is of level $$\Gamma_0(N)$$.

EXAMPLES:

sage: D = OverconvergentDistributions(7, 11)
sage: M = PollackStevensModularSymbols(Gamma1(14), coefficients=D)
sage: M.level()
14

ncoset_reps()

Return the number of coset representatives defining the domain of the modular symbols in this space.

OUTPUT:

The number of coset representatives stored in the manin relations. (Just the size of $$P^1(\ZZ/N\ZZ)$$)

EXAMPLES:

sage: D = Symk(2)
sage: M = PollackStevensModularSymbols(Gamma0(2), coefficients=D)
sage: M.ncoset_reps()
3

ngens()

Returns the number of generators defining this space.

EXAMPLES:

sage: D = OverconvergentDistributions(4, 29)
sage: M = PollackStevensModularSymbols(Gamma1(12), coefficients=D)
sage: M.ngens()
5
sage: D = Symk(2)
sage: M = PollackStevensModularSymbols(Gamma0(2), coefficients=D)
sage: M.ngens()
2

precision_cap()

Return the number of moments of each element of this space.

EXAMPLES:

sage: D = OverconvergentDistributions(2, 5)
sage: M = PollackStevensModularSymbols(Gamma1(13), coefficients=D)
sage: M.precision_cap()
20
sage: D = OverconvergentDistributions(3, 7, prec_cap=10)
sage: M = PollackStevensModularSymbols(Gamma0(7), coefficients=D)
sage: M.precision_cap()
10

prime()

Return the prime of this space.

EXAMPLES:

sage: D = OverconvergentDistributions(2, 11)
sage: M = PollackStevensModularSymbols(Gamma(2), coefficients=D)
sage: M.prime()
11

random_element(M=None)

Return a random overconvergent modular symbol in this space with $$M$$ moments

INPUT:

• M – positive integer

OUTPUT:

An element of the modular symbol space with $$M$$ moments

Returns a random element in this space by randomly choosing values of distributions on all but one divisor, and solves the difference equation to determine the value on the last divisor.

sage: D = OverconvergentDistributions(2, 11)
sage: M = PollackStevensModularSymbols(Gamma0(11), coefficients=D)
sage: M.random_element(10)
Traceback (most recent call last):
...
NotImplementedError

sign()

Return the sign of this space.

EXAMPLES:

sage: D = OverconvergentDistributions(3, 17)
sage: M = PollackStevensModularSymbols(Gamma(5), coefficients=D)
sage: M.sign()
0
sage: D = Symk(4)
sage: M = PollackStevensModularSymbols(Gamma1(8), coefficients=D, sign=-1)
sage: M.sign()
-1

source()

Return the domain of the modular symbols in this space.

OUTPUT:

EXAMPLES:

sage: D = OverconvergentDistributions(2, 11)
sage: M = PollackStevensModularSymbols(Gamma0(2), coefficients=D)
sage: M.source()
Manin Relations of level 2

weight()

Return the weight of this space.

Warning

We emphasize that in the Pollack-Stevens notation, this is the usual weight minus 2, so a classical weight 2 modular form corresponds to a modular symbol of “weight 0”.

EXAMPLES:

sage: D = Symk(5)
sage: M = PollackStevensModularSymbols(Gamma1(7), coefficients=D)
sage: M.weight()
5

sage.modular.pollack_stevens.space.cusps_from_mat(g)

Return the cusps associated to an element of a congruence subgroup.

INPUT:

• g – an element of a congruence subgroup or a matrix

OUTPUT:

A tuple of cusps associated to g.

EXAMPLES:

sage: from sage.modular.pollack_stevens.space import cusps_from_mat
sage: g = SL2Z.one()
sage: cusps_from_mat(g)
(+Infinity, 0)


You can also just give the matrix of g:

sage: type(g)
<class 'sage.modular.arithgroup.arithgroup_element.ArithmeticSubgroupElement'>
sage: cusps_from_mat(g.matrix())
(+Infinity, 0)


Another example:

sage: from sage.modular.pollack_stevens.space import cusps_from_mat
sage: g = GammaH(3, [2]).generators()[1].matrix(); g
[-1  1]
[-3  2]
sage: cusps_from_mat(g)
(1/3, 1/2)

sage.modular.pollack_stevens.space.ps_modsym_from_elliptic_curve(E, sign=0, implementation='eclib')

Return the overconvergent modular symbol associated to an elliptic curve defined over the rationals.

INPUT:

• E – an elliptic curve defined over the rationals

• sign – the sign (default: 0). If nonzero, returns either the plus (if sign == 1) or the minus (if sign == -1) modular symbol. The default of 0 returns the sum of the plus and minus symbols.

• implementation – either ‘eclib’ (default) or ‘sage’. This determines which implementation of the underlying classical modular symbols is used.

OUTPUT:

The overconvergent modular symbol associated to E

EXAMPLES:

sage: E = EllipticCurve('113a1')
sage: symb = E.pollack_stevens_modular_symbol() # indirect doctest
sage: symb
Modular symbol of level 113 with values in Sym^0 Q^2
sage: symb.values()
[-1/2, 1, -1, 0, 0, 1, 1, -1, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, 0, 0]

sage: E = EllipticCurve([0,1])
sage: symb = E.pollack_stevens_modular_symbol()
sage: symb.values()
[-1/6, 1/3, 1/2, 1/6, -1/6, 1/3, -1/3, -1/2, -1/6, 1/6, 0, -1/6, -1/6]

sage.modular.pollack_stevens.space.ps_modsym_from_simple_modsym_space(A, name='alpha')

Returns some choice – only well defined up a nonzero scalar (!) – of an overconvergent modular symbol that corresponds to A.

INPUT:

• A – nonzero simple Hecke equivariant new space of modular symbols, which need not be cuspidal.

OUTPUT:

A choice of corresponding overconvergent modular symbols; when dim(A)>1, we make an arbitrary choice of defining polynomial for the codomain field.

EXAMPLES:

The level 11 example:

sage: from sage.modular.pollack_stevens.space import ps_modsym_from_simple_modsym_space
sage: A = ModularSymbols(11, sign=1, weight=2).decomposition()[0]
sage: A.is_cuspidal()
True
sage: f = ps_modsym_from_simple_modsym_space(A); f
Modular symbol of level 11 with values in Sym^0 Q^2
sage: f.values()
[1, -5/2, -5/2]
sage: f.weight()         # this is A.weight()-2  !!!!!!
0


And the -1 sign for the level 11 example:

sage: A = ModularSymbols(11, sign=-1, weight=2).decomposition()[0]
sage: f = ps_modsym_from_simple_modsym_space(A); f.values()
[0, 1, -1]


A does not have to be cuspidal; it can be Eisenstein:

sage: A = ModularSymbols(11, sign=1, weight=2).decomposition()[1]
sage: A.is_cuspidal()
False
sage: f = ps_modsym_from_simple_modsym_space(A); f
Modular symbol of level 11 with values in Sym^0 Q^2
sage: f.values()
[1, 0, 0]


We create the simplest weight 2 example in which A has dimension bigger than 1:

sage: A = ModularSymbols(23, sign=1, weight=2).decomposition()[0]
sage: f = ps_modsym_from_simple_modsym_space(A); f.values()
[1, 0, 0, 0, 0]
sage: A = ModularSymbols(23, sign=-1, weight=2).decomposition()[0]
sage: f = ps_modsym_from_simple_modsym_space(A); f.values()
[0, 1, -alpha, alpha, -1]
sage: f.base_ring()
Number Field in alpha with defining polynomial x^2 + x - 1


We create the +1 modular symbol attached to the weight 12 modular form Delta:

sage: A = ModularSymbols(1, sign=+1, weight=12).decomposition()[0]
sage: f = ps_modsym_from_simple_modsym_space(A); f
Modular symbol of level 1 with values in Sym^10 Q^2
sage: f.values()
[(-1620/691, 0, 1, 0, -9/14, 0, 9/14, 0, -1, 0, 1620/691), (1620/691, 1620/691, 929/691, -453/691, -29145/9674, -42965/9674, -2526/691, -453/691, 1620/691, 1620/691, 0), (0, -1620/691, -1620/691, 453/691, 2526/691, 42965/9674, 29145/9674, 453/691, -929/691, -1620/691, -1620/691)]


And, the -1 modular symbol attached to Delta:

sage: A = ModularSymbols(1, sign=-1, weight=12).decomposition()[0]
sage: f = ps_modsym_from_simple_modsym_space(A); f
Modular symbol of level 1 with values in Sym^10 Q^2
sage: f.values()
[(0, 1, 0, -25/48, 0, 5/12, 0, -25/48, 0, 1, 0), (0, -1, -2, -119/48, -23/12, -5/24, 23/12, 3, 2, 0, 0), (0, 0, 2, 3, 23/12, -5/24, -23/12, -119/48, -2, -1, 0)]


A consistency check with sage.modular.pollack_stevens.space.ps_modsym_from_simple_modsym_space():

sage: from sage.modular.pollack_stevens.space import ps_modsym_from_simple_modsym_space
sage: E = EllipticCurve('11a')
sage: f_E = E.pollack_stevens_modular_symbol(); f_E.values()
[-1/5, 1, 0]
sage: A = ModularSymbols(11, sign=1, weight=2).decomposition()[0]
sage: f_plus = ps_modsym_from_simple_modsym_space(A); f_plus.values()
[1, -5/2, -5/2]
sage: A = ModularSymbols(11, sign=-1, weight=2).decomposition()[0]
sage: f_minus = ps_modsym_from_simple_modsym_space(A); f_minus.values()
[0, 1, -1]


We find that a linear combination of the plus and minus parts equals the Pollack-Stevens symbol attached to E. This illustrates how ps_modsym_from_simple_modsym_space is only well-defined up to a nonzero scalar:

sage: (-1/5)*vector(QQ, f_plus.values()) + (1/2)*vector(QQ, f_minus.values())
(-1/5, 1, 0)
sage: vector(QQ, f_E.values())
(-1/5, 1, 0)


The next few examples all illustrate the ways in which exceptions are raised if A does not satisfy various constraints.

First, A must be new:

sage: A = ModularSymbols(33,sign=1).cuspidal_subspace().old_subspace()
sage: ps_modsym_from_simple_modsym_space(A)
Traceback (most recent call last):
...
ValueError: A must be new


A must be simple:

sage: A = ModularSymbols(43,sign=1).cuspidal_subspace()
sage: ps_modsym_from_simple_modsym_space(A)
Traceback (most recent call last):
...
ValueError: A must be simple


A must have sign -1 or +1 in order to be simple:

sage: A = ModularSymbols(11).cuspidal_subspace()
sage: ps_modsym_from_simple_modsym_space(A)
Traceback (most recent call last):
...
ValueError: A must have sign +1 or -1 (otherwise it is not simple)


The dimension must be positive:

sage: A = ModularSymbols(10).cuspidal_subspace(); A
Modular Symbols subspace of dimension 0 of Modular Symbols space of dimension 3 for Gamma_0(10) of weight 2 with sign 0 over Rational Field
sage: ps_modsym_from_simple_modsym_space(A)
Traceback (most recent call last):
...
ValueError: A must have positive dimension


We check that forms of nontrivial character are getting handled correctly:

sage: from sage.modular.pollack_stevens.space import ps_modsym_from_simple_modsym_space
sage: f = Newforms(Gamma1(13), names='a')[0]
sage: phi = ps_modsym_from_simple_modsym_space(f.modular_symbols(1))
sage: phi.hecke(7)
Modular symbol of level 13 with values in Sym^0 (Number Field in alpha with defining polynomial x^2 + 3*x + 3)^2 twisted by Dirichlet character modulo 13 of conductor 13 mapping 2 |--> -alpha - 1
sage: phi.hecke(7).values()
[0, 0, 0, 0, 0]